text:
# Please avoid using the RAI acronym, or use it carefully & be clear in context

This meta exists to explain some issues with the RAI acronym. It has a couple of issues with its usage, and often get asked to clarify it. You've probably been pointed to this meta in relation to a request to clarify some usage of the acronym in a post. It attempts to summarise a couple of issues people will regularly have to cover in requesting such a clarification — this meta exists to save them the effort of re-explanation when it's difficult to explain in the size of a comment what's going on behind their request.

If they've pointed out which issue's going on, please refer to that section.

## First issue: RAI can mean two completely different things.

It can stand for either of these:

• Rules As Intended — an assertion about what the person who wrote the rules meant by them.
• Rules As Interpreted — what you think they say.

It can also have other readings, and some readers may not be aware of the reading you intended to convey.

Rather than use the "RAI" acronym, it's often better to say “rules as intended” or “rules as interpreted” in full so that people know what you mean. If you do want to use the acronym, please make sure it's very clear in context, such as by only using the acronym as shorthand after you've already spelled out those words.

Bear in mind that while we're a site by experts, we're also trying to assist and guide beginner and intermediate audiences. They may be unsure what you mean based on their colloquial experiences.

## Second issue: Making assumptions for Rules As Intended.

There's lots of precedent for errors in RPG writing. An author picks some ambiguous wording which reads differently to what they intended; mechanics get changed or dropped during production which changes how a feature functions; an editor “helpfully” makes a revision which breaks something; or an author has a deadline of ten spells to finish by sundown and are focused on quantity without time to spare on polish (as RPG designers sometimes describe in podcasts).

If we're honest, we usually really don't know what designers intended for the words we're reading.

Despite that, there's a common trend in RPG advice: an expert will make an educated guess about what the writer intended, conclude their guess is correct because it makes sense, and then provide advice asserting what they guessed is the case. That's done with the best of intentions, but really, there's no guarantee the guess is correct, so doing this can be misleading and problematic.

We prefer (and may require) that if you make an assertion about an author's intentions, you back it up with evidence of their intentions: a reference to the game text discussing the game's philosophy, a quote from the designer, etc. Doing so improves confidence in your answer and teaches people more about the game they're playing. If you can't produce such a citation, it's probably best to drop the part where you're asserting about their intentions, and focus instead on working with what you do know for sure. It's okay if you don't know for sure what they meant.

• I don't remember seeing RAI standing for "Rules As Interpreted", even if (because of the second problem you describe) it certainly could sometimes be. Oct 20 '17 at 13:42
• @AnneAunyme This acronym list even mentions only Rules As Interpreted, and not Rules As Intended. :) It's usually the first result on a google search for rai acronym, too. Oct 20 '17 at 13:59
• +1000 if I could. This is one of my personal bugbears, and you very clearly lay out the issues with the acronym Oct 20 '17 at 14:08
• Oct 21 '17 at 12:48
• This is a low grade problem that's recurred over time as a cursory survey of Meta indicates. Oct 21 '17 at 12:49
• In my experience, rules as interpreted is used (intended?) vastly less often than rules as intended. It seems to exist solely as an attempt to redefine RAI to mean what people mean when they use it, which is nice except that it doesn’t have a whole lot of acceptance and most people still assume RAI is used to refer to intent. Oct 23 '17 at 14:30
• @KRyan It's also my experience "interpreted" gets used considerably less often. Oct 23 '17 at 14:32
• Could someone clarify the difference between RAW and "rules as interpreted"? The rules have to be interpreted to be understood, so isn't RAW the same as "rules as interpreted"? This is distinct "rules as intended" which usually comes in the form "a designer said on twitter ..." Oct 18 '19 at 0:34
• @jgn That sounds like a [terminology] question for mainsite. Oct 18 '19 at 17:25
• @SevenSidedDie Good idea: What does “Rules As Interpreted” mean and how is it different to RAW and RAI (from D&D)? Oct 18 '19 at 17:36
• @doppelgreener My circle of RPG players/masters recently had a discussion around this same subject. We settled in using RAIA (Rules as intended by author) and RAIM (Rules as Interpreted by myself) acronyms instead of the ambyguous RAI. We also use HCU (Houserule currently in use) when talking about homebrew or variant rulesets that are "valid" for a given game. Oct 30 '19 at 14:37

I agree.

While I personally have never heard "Rules As Interpreted" for RAI, and always knew it as "Intended" - ironically, people generally substitute their own opinion for "intended" anyway. Unless the question is asking for developer citations and is tagged , "RAI" is a signpost saying to me "look at this question and see if it needs to be closed as opinion-based."

If you really want to know intent ask a question that fits the tag. If you are just looking for a reasonable interpretation or you are just saying "how you read it" - say that, you don't need an ambigious abbreviation for it.

• Are you referring only to RAI as the basis of a question, or also as part of an answer? If the former, I understand this answer. My initial objection to this meta was (in part) the spray paint rather than brush paint approach to the use of the acronym generally. Oct 24 '17 at 18:46
• @Korvin The 'people substituting their own opinion for "intended"' thing mainly happens in answers. Oct 25 '17 at 15:50
• Hmm, now that designer reasons have been put off limits ... this answer seems to have been overcome by events. Oct 18 '19 at 21:48

## One does not have to ask the author to analyze their intent

If that were the case, studying Shakespeare would be pointless (or more pointless if you are of a scientific rather than humanities bent :-))

There are some ways of thinking of this that may (or may not) bring clarity:

## Literary Theory

If the rules are to be considered as a literary work then Rules as Intended is a function of how to deal with Authorial Intent.

These range from the New Criticism view which holds that the intent of the author is irrelevant "the text is the primary source of meaning, and any details of the author's desires or life are secondary". This is essentially what we mean by Rules as Written

To the Cambridge School contextualism which states that we need to consider the social environment in which the work was made - for role-playing games this would include the prior works that influenced it and the social norms of the community at the time it was written. This is possibly what we mean by Rules as Intended but it spreads the net wider than authorial intent.

Weak Internationalism which says that both the readers' response and author's intent must be combined - generously this could be what the OP refers to when saying "an expert will make an educated guess about what the writer intended".

And then there are the strange ways of looking at it - I would love to see a Marxist interpretation of D&D.

## Legal Theory

Alternatively (or as well as), RPG rules could be thought of as akin to laws and the techniques that courts use to interpret them can be applied. The primary methods are:

• the literal approach - our RAW.
• the golden rule - reject RAW if it results in an absurdity or is inconsistent with the rest of the rules.
• the mischief rule - what is the problem this rule is trying to fix? This is really only applicable to errata.
• purposive approach - what are the rules trying to achieve? Interpret them so that is achieved.

In a legal context, these rules are applied in the context of looking beyond the statute to the "intent" of the legislators to the extent that this can be determined (e.g. from speeches made or press releases explaining the new law etc.).

• This seems to be discussing the concept of RAW/RAI₁/RAI₂ rather than discussing or responding to the proposal/instruction of the meta post. Am I missing something that's leading me to misunderstand this? Oct 23 '17 at 17:35
• @SevenSidedDie I think the (implied) point is that it's reasonable to discuss the intended meaning of rules even without designer quotations via the techniques of literary analysis, which is something I agree with. I think this is a response to the potential pile-on effect of our agreeing with the idea that people are using 'RAI' badly that it becomes unfashionable to discuss RAI without establishing authorial intent solely via Word Of God rather than all the other cool well-established tools for doing that literarily. Oct 30 '17 at 6:03
• @thedark That's possible. If it's saying that, I think it would be improved by stating that thesis explicitly. Oct 30 '17 at 6:24
• I do agree that all of those ways of looking at things are helpful. However, "rules as intended" implies some sort of proof that you have the intent correct. Literature isn't rules, so it's a bit more flexible. And even the legal concept of intent tends to be debatable, with people taking two different side. There's no harm in looking at these things, but I think the phrase "rules as intended" should be reserved for when we are sure. Nov 1 '17 at 21:32

For D&D 5E, the term is defined in Sage Advice Compendium. RAI means Rules As Intended by the game developers:

RAI Some of you are especially interested in knowing the intent behind a rule. That’s where RAI comes in: “rules as intended.” This approach is all about what the designers meant when they wrote something. In a perfect world, RAW and RAI align perfectly, but sometimes the words on the page don’t succeed at communicating the designers’ intent. Or perhaps the words succeed with one group of players but not with another.

• We're not D&D 5e Stack Exchange though, and we're not expecting our D&D 5e readers to have familiarised themselves with this glossary entry. Nov 20 '17 at 21:38
• Should we then explain that HP means Health Points and XP means Experience Pointes, DEX means Dexterity? Really? Nov 20 '17 at 21:44
• No, since those are described in the manuals they're using, and used consistently colloquially. We can however just spell out "interpretation" or "intention" given the community uses this acronym colloquially in more than one way and the definition you quote is in supplementary material beginners are unlikely to have come across. Nov 20 '17 at 21:46
• @J.A.Streich If there are other common meanings relevant to a question or roleplaying in general, yes, those should be expanded. However, I'm unaware of any ambiguity regarding HP, XP, or Dex, and haven't observed any practical problems on the site caused by posts using those acronyms. This reflects our site's philosophy that 1) practical site problems are those that need addressing, and hypotheticals can be ignored until they become non-hypothetical, 2) an RPG's publisher solving a practical problem for their readers might not solve it for our readers, who are overlapping but separate. Nov 20 '17 at 22:34
• Besides "thefreedictionary.com" is there any source, particularly a rpg gaming source that officially states RAI as meaning "rules as interpreted"? Oct 18 '19 at 0:36
• @jgn It exists in the wider RPG community, and taste enough to have to accommodate it. We’re a pragmatic site, so we have to deal with the state of the wider RPG community as it is, rather than an idealised version. As a sample of descriptivist attestation, here are 8 questions and 19 answers just on this site that for one reason or another use the term “rules as interpreted”. Oct 18 '19 at 17:28
• The question remains whether or not people using "rules as interpreted" actually mean "rules as intended" and they are just misheard, misread, or learned the term wrong. I am asking if there are any sources that use it because I am questioning if it is a real thing. I came to this question after reading every one of those questions/answers and realising they are all about D&D where RAI has already been officially designated to mean Rules As Intended. Could it be this confusion is just because some people created a mistaken reverse-acronym? There are hundreds of correct usages of RAI. Oct 18 '19 at 17:42
• ", since those are described in the manuals they're using" So is RAI for D&D.... Oct 24 '19 at 15:48
• @pllpnakjlx as I review the objections to this answer, I see some of the pointless anti-D&D threads that you mentioned elsewhere. Apr 16 '20 at 13:10

Perhaps "Rules As Interpreted" needs to be discouraged or never written as an abbreviation.

1. It's a synonym for RAW. When explaining how a rule is written, what you are explaining is how you interpret the rule. There is no difference between "Rules As Written" and "Rules As Interpreted."
2. Very few people have ever heard "Rules As Interpreted", few even seem to use it. On rpg.stackexchange searching "Rules As Intended" gives 238 questions, "Rules As Interpreted" gives 27 results. "RAI" is used 784 times, 94 times in dnd-3.5e, 15 in dnd-4e, 516 in dnd-5e, 3 times in dnd - making up more than 80% of RAI usage. We can conclude that the vast majority of usage of RAI means "Rules As Intended". Even discussing "Rules As Interpeted" only serves to confuse people because of the similarity to "Rules As Intended".
3. "Rules As Intended" is official WotC terminology. There are more than 25,000 questions about dnd. Maybe the the popularity of the franchise and its dominance on this board should lend some favour to not confusing askers with terminology that only really exists here.
4. Rather than telling people not to use RAI, it would be less hostile to new users to let them continue to use common terminology on the site.
• We can't retire a term we don't have control over. Oct 18 '19 at 13:50
• @doppelgreener But you can mandate that people can't use "RAI"? I don't think you have any more control over "RAI" than you do "Rules As Interpreted". At the very least you can discourage the use of "Rules As Interpreted" or using "RAI" to mean "Rules As Interpreted". Oct 18 '19 at 13:55
• I haven't mandated anything. I wrote this meta asking people to be careful using an acronym we found ourselves often having to ask for clarification over. Oct 18 '19 at 13:56
• Unfortunately only a small portion of the userbase reads meta (many don't even know it exists yet), and only some users will have seen this thread. This means the vast majority of our users will be completely unfamiliar with such a declaration and will continue to use RAI however, perhaps or perhaps not meaning “as interpreted”. So we still have to check. I wrote this meta as describing the issues so that people could link it while checking if they needed to. Oct 18 '19 at 14:04
• This advice was written in response to actual incidents of confusion, where RAI did get used for the rarer “Interpreted” and it caused problems for the asker. It became clear that we could not always assume what the TLA meant when Jane Random Asker used it. It’s a “no glass bottles sign” situation, not a “no elephants sign” one. That’s the kind of uncertainty that results in questions being put on hold, so it had to be addressed—and a page like this helps quickly convey that need in a comment to Jane R.A. Oct 18 '19 at 15:02
• Very often, here and elsewhere, “RAI” is used for “not RAW.” “Rules as interpreted” is a backronym (kinda) for that sense of it, though most people still say “intended” even when they have absolutely no basis for the claim that their interpretation was actually what the authors intended. A lot of people seem to genuinely think that they really do know what was intended, and it just so happens to match their personal interpretation. Hence arguments. Oct 18 '19 at 15:22
• At any rate, the rampant misuse, the difficulty backing up any intent (and the off-topic nature of questions about it), and the fact that we default to looking for interpretation so there isn’t any particular need to specify “not RAW” when that’s the default all add up to the conclusion that it’s just a term best avoided. It adds little-to-no value and causes miscommunication and arguments. Oct 18 '19 at 15:23
• @jgn The former is a gross mischaracterization of the statements being made and I think you know that. I have strong objections and problems with how segments of this community treat the rules vs. their interpretations vs. their personal preferences, I have strong objections to how certain users have expressed and pushed those positions (objections strong enough to see me temporarily suspended, to be clear). But you are overstating things by a significant margin. And yes, RAI should be treated that way, but history has shown that it isn’t, and it becomes too large a moderation burden. Oct 18 '19 at 16:42
• @jgn No, users are supposed to back up why they think that approach works best—answers that just say “this is what I would do” are typically downvoted and may even be deleted. A good answer would include rules support, observe any failings in those rules, and justify any divergence from the rules via backed-up argumentation: demonstration of balance or narrative concerns with the official rules, description of personal experience trying the official rules and how things improve by deviating, etc. and so on. Some form of backing up is required. Poor backing up leads to poor votes. Oct 18 '19 at 17:08
• @jgn We have a large, detailed architecture for how the site works, and you’re poking at and accidentally asking permission to disassemble load-bearing parts of it, sometimes for reasons it already solves for in another way, due to honest unfamiliarity. You might want to observe the system a bit more before vigorous attempts to change things, to avoid reinventing the wheel. The FAQ Index for RPG.se is a start. Browsing various meta tags related to your concerns is also helpful to get oriented. Oct 18 '19 at 17:37
• @jgn Yes, I described the very-best answers, the ideal that is strived for. Most answers fail to live up to the very best of our ideals—but the way the system works is that the closer to that ideal that an answer gets, the more votes it receives; the further from that ideal that an answer gets, the fewer votes it receives. There are other factors in voting—whether you think it’s a good idea or not, for example, regardless of how well backed-up it is—but for the most part the mechanism for encouraging better evidence for things is voting, and for the most part, voting accomplishes that. Oct 18 '19 at 17:51
• @jgn Yes, this is hand-wave-y. Yes, it’s not perfect. Yes, there are cases where things get badly misaligned, where votes for something undeserving happen. But it’s the best system that anyone’s come up with in my experience. What other mechanism would you have? Having moderators delete things that aren’t sufficiently backed up in their opinion? No. The system is intentionally decentralized, both to reduce the burden on the volunteer moderators and to limit how much influence they have. Only the most extreme cases should have moderator intervention. Oct 18 '19 at 17:53
• @jgn second observation on that hyperbolic mischaracterization. Is this the kind of first impression you want to make: "Hi. I've been interacting here three days, everyone else is wrong, I am right." (Not saying you have done that, but it's a risk). Seven Sided is giving you some great guidance/advice. I appreciate the passion you bring to your participation, regardless of what we will agree on, or disagree on. Oct 18 '19 at 21:56
• @jgn Perhaps as experienced users it is difficult for you to see the barriers new users face Nope. That's not it. Certainly not from my perspective. You may find the comment stream under that answer useful, or useless. Not sure. Oct 19 '19 at 1:54
• @jgn Nobody is telling you not to complain; I made a post (which amounted to a complaint) here recently that was not well received by a number of people who pushed back. Oct 19 '19 at 12:54


text:
# Algebraic Expressions and Identities

9.1. INTRODUCTION

a) What are expressions
In earlier classes, we have already become familiar with what algebraic expressions  (or simply expressions) are.
Examples of expressions are: etc.

We can form many more expressions. As we know expressions are formed from variables and constants.
The expression 2y – 5 is formed from the variable y and constants 2 and 5. The expression 4xy + 7 is formed from variables x and y and constants 4 and 7.
We know that, the value of y in the expression 2y – 5, may be anything. It can be etc.; actually countless different values. The value of an expression changes with the value chosen for the variables it contains. Thus as y takes on different values, the value of 2y – 5 goes on changing. When y = 2, 2y – 5 = 2(2) – 5 = –1; when y = 0, 2y – 5 = 2 × 0 –5 = –5, etc.

Number line and an expression
Consider the expression x + 5. Let us say the variable x has a position X on the number line;

X may be anywhere on the number line, but it is definite that the value of x + 5 is given by a point P, 5 units to the right of X. Similarly, the value of x – 4 will be 4 units to the left of X and so on.

Let’s see the position of 4x and 4x + 5

The position of 4x will be point C; the distance of C from the origin will be four times the distance of X from the origin. The position D of 4x + 5 will be 5 units to the right of C.

b) Terms, factors and Coefficients
Take the expression 4x + 5. This expression is made up of two terms, 4x and 5. Terms are added to form expressions. Terms themselves can be formed as the product of factors. The term 4x is the product of its factors 4 and x. The term 5 is made up of just one factor, i.e., 5.
The expression 7xy – 5x has two terms 7xy and –5x. The term 7xy is a product of factors 7, x and y. The numerical factor of a term is called its coefficient. The coefficient in the term 7xy is 7 and the coefficient in the term –5x is –5.

c) Monomials, binomials and polynomials
Expression that contains only one term is called a monomial. Expression that contains two terms is called a binomial. An expression containing three terms is a trinomial and so on. In general, an expression containing, one or more terms with non-zero coefficient (with variables having non negative exponents) is called a polynomial. A polynomial may contain any number of terms, one or more than one.

Examples of monomials: , etc.

Examples of binomials: , etc.

Examples of trinomials:

Examples of polynomials: a + b + c + d, 3xy, 7xyz – 10, 2x + 3y + 7z, etc.

d) Like and Unlike terms
Look at the following expressions:
$7x,14x,–13x,5{x}^{2},7y,7xy,–9{y}^{2},–9{x}^{2},–5yx$
Like terms from these are:
(i) 7x, 14x, –13x are like terms.
(ii) $5{x}^{2}$ and $–9{x}^{2}$ are like terms.
(iii) 7xy and –5yx are like terms.
Unlike terms from these are:
(i) 7x and 7y are unlike terms.
(ii) 7x and 7xy are unlike terms.
(iii) 7x and $5{x}^{2}$ are unlike terms.

9.2. ADDITION AND SUBTRACTION OF ALGEBRAIC EXPRESSIONS
Let’s add $7{x}^{2}–4x+5$and 9x – 10, we do

During addition of algebraic expression, we write each expression to be added in a separate row. While doing so we write like terms one below the other, and add them, as shown above
Thus 5 + (–10) = 5 –10 = –5.
Similarly, – 4x + 9x = (– 4 + 9)x = 5x.
Let us take some more examples.

9.3. MULTIPLICATION OF MONOMIALS WITH ALGEBRAIC EXPRESSIONS
I. Multiplying two monomials
We know that
4 × x = x + x + x + x = 4x as seen earlier.
Similarly, 4 × (3x) = 3x + 3x + 3x + 3x = 12x
Let’s observe the multiplication of few monomials.
(i) x × 3y = x × 3 × y = 3 × x × y = 3xy
(ii) 5x × 3y = 5 × x × 3 × y = 5 × 3 × x × y = 15xy
(iii) 5x × (–3y) = 5 × x × (–3) × y
= 5 × (–3) × x × y = –15xy
Note: We observe that all the three products of monomials, 3xy, 15xy, –15xy, are also monomials.
(iv) 5x × 4x2 = (5 × 4) × (x × x2)
= 20 × x3 = 20x3
Note: 5 × 4 = 20
i.e., coefficient of product = coefficient of first monomial × coefficient of second monomial; and x × x2 = x3
algebraic factor of product = algebraic factor of first monomial × algebraic factor of second monomial.
(v) 5x × (– 4xyz) = (5 × – 4) × (x × xyz)
= –20 × (x × x × yz) = –20x2 yz
Let see few more examples.
Suppose we have two monomials 4x and 5y. By multiplying them, we get
4x × 5y = (4 × x) × (5 × y)
= (4 × 5) × (x × y) = 20xy
Hence, we can say that 4x × 5y = 20xy
Now, 10x × 5x2z = (10 × x) × (5 × x2 × z)
This becomes (10 × 5) × (x × x2 × z) = 50x3z
II. Multiplying three or more monomials
Observe the following examples.
(i) 2x × 5y × 7z = (2x × 5y) × 7z = 10xy × 7z = 70xyz
(ii) 4xy × 5x2 y2 × 6x3y3 = (4xy × 5x2y2 ) × 6x3y3
= 20x3y3 × 6x3y3 = 120x3y3 × x3y3
= 120 (x3×x3) × (y3×y3) = 120x6×y6 = 120x6y6
It is clear that we first multiply the first two monomials and then multiply the resulting monomial by the third monomial. This method can be extended to the product of any number of monomials.
Example: Complete the table for area of a rectangle with given length and breadth.
Solution

 Length Breadth Area 3x 9y 4ab 2l2m 5y 4y2 5bc 3lm2 3x×5y = 15xy 9y×4y2 = 36y3 4ab×5bc = 20ab2c 2l2m×3lm2 = 6l3m3

III. Multiplication of a monomial with a polynomial
a) Multiplying a monomial by a binomial
Let us multiply the monomial 3x by the binomial
5y + 2, i.e., find 3x × (5y + 2)
Recall that 3x and (5y + 2) represent numbers. Therefore, using the distributive law,
3x × (5y + 2) = (3x × 5y) + (3x × 2) = 15xy + 6x
We commonly use distributive law in our calculations.
For example
7 × 106 = 7 × (100 + 6) = 7 × 100 + 7 × 6
(Here, we used distributive law)
= 700 + 42 = 742
7 × 38 = 7 × (40 – 2)
= 7 × 40 – 7 × 2
(Here, we used distributive law)
= 280 – 14 = 266
Similarly, (–3x) × (–5y + 2)
= (–3x) × (–5y) + (–3x) × (2) = 15xy – 6x
and 5xy × (y2 + 3) = (5xy × y2 ) + (5xy × 3)
= 5xy3 + 15xy.
Multiplication of binomial × monomial?
For example, (5y + 2) × 3x = ?
We may use commutative law as : 7 × 3 = 3 × 7 (or) in
general a × b = b × a
Similarly, (5y + 2) × 3x = 3x × (5y + 2) = 15xy + 6x
as before.
b) Multiplying a monomial by a trinomial
Consider 3p × (4p2 + 5p + 7). As in the earlier case, we use distributive law;
3p × (4p2 + 5p + 7) = (3p × 4p2 ) + (3p × 5p) + (3p × 7)
= 12p3 + 15p2 + 21p
Multiply each term of the trinomial by the monomial and add products.
Observe, by using the distributive law, we are able to carry out the multiplication term by term.
Example: Simplify the expressions and evaluate them as directed:
(i) x (x – 3) + 2 for x = 1
(ii) 3y (2y – 7) – 3 (y – 4) – 63 for y = –2
Solution: (i) x (x – 3) + 2 = x2 – 3x + 2
For x = 1, x2 – 3x + 2 = (1)2 – 3 (1) + 2
= 1 – 3 + 2 = 3 – 3 = 0
(ii) 3y (2y – 7) – 3 (y – 4) – 63 = 6y2 – 21y – 3y + 12 – 63
= 6y2 – 24y – 51
For y = –2, 6y2 – 24y – 51 = 6 (–2)2 – 24(–2) – 51
= 6 × 4 + 24 × 2 – 51
= 24 + 48 – 51 = 72 – 51 = 21

9.4. MULTIPLICATION OF BINOMIALS WITH BINOMIALS AND TRINOMIALS
I. Multiplication of a binomial by a binomial
Let us multiply one binomial (2a + 3b) by another binomial, say (3a + 4b).
Observe, every term in one binomial multiplies every term in the other binomial.
(3a + 4b) × (2a + 3b) = 3a × (2a + 3b) + 4b × (2a + 3b)
= (3a × 2a) + (3a × 3b) + (4b × 2a) + (4b × 3b)
= 6a2 + 9ab + 8ba + 12b2
= 6a2 + 17ab + 12b2
(Since ba = ab)
When we carry out term by term multiplication, we expect 2 × 2 = 4 terms to be present. But two of these are like terms, which are combined, and hence we get 3 terms.
In multiplication of polynomials with polynomials, we should always look for like terms, if any, and combine them.
In multiplication of polynomials with polynomials, we sholud always look for like terms, if any, and combine them.
II. Multiplication of a binomial with a trinomial
In this multiplication, we shall have to multiply each of the three terms in the trinomial by each of the two terms in the binomial. We shall get in all 3 × 2 = 6 terms, which may reduce to 5 or less, if the term by term multiplication results in like terms. Consider

= a × (a + 3a + 5) + 7 × (a + 3a + 5)
[using the distributive law]
= a3 + 3a2 + 5a + 7a2 + 21a + 35
= a3 + (3a2 + 7a2 ) + (5a + 21a) + 35
= a3 + 10a2 + 26a + 35

9.5. IDENTITY
Consider the equality (a + 1) (a +2) = a2 + 3a + 2
We shall evaluate both sides of this equality for some value of a, say a = 10.
For a = 10,
LHS = (a + 1) (a + 2) = (10 + 1) (10 + 2)
= 11 × 12 = 132
RHS = a2 + 3a + 2 = 102 + 3 × 10 + 2
= 100 + 30 + 2 = 132
Thus, the values of the two sides of the equality are equal for a = 10.
Let us now take a = –5
LHS = (a + 1) (a + 2) = (–5 + 1) (–5 + 2)
= (– 4) × (–3) = 12
RHS = a2 + 3a + 2 = (–5)2 + 3 (–5) + 2
= 25 – 15 + 2 = 10 + 2 = 12
Thus, for a = –5, also LHS = RHS.
We shall find that for any value of a, LHS = RHS Such an equality, true for every value of the variable in it, is called an identity. Thus,
(a + 1) (a + 2) = a2 + 3a + 2 is an identity.
An equation is true for only certain values of the variable in it. It is not true for all values of the variable. For example, consider the equation
a2 + 3a + 2 = 132
It is true for a = 10, as seen above, but it is not true for a = –5 or for a = 0 etc.

9.6. STANDARD IDENTITIES
I. Use of the Identities (a + b)2 and (a – b)2
Let us try to find the square of the number 102. The square of a number, as we know, is the product of the number with itself. One way to do this is by writing the numbers one below the other, and then multiplying them as we normally do. The other way is to break the numbers and then apply distributive property. This will make our work much easier.
Let us see how.
1022 = 102 × 102
= (100 + 2) (100 + 2)
= 100 (100 + 2) + 2 (100 + 2)
= 100 × 100 + 100 × 2 + 2 × 100 + 2 × 2
= 10000 + 200 + 200 + 4
= 10404
Let us try to find the square of the expression (a + b).
(a + b)2 = (a + b) (a + b) = a (a + b) + b (a + b)
= a × a + a × b + b × a + b × b
= a2 + ab + ab + b2
= a2 + 2ab + b2

This is, in fact, an important identity. It is known as an identity because for any value of a and b, the LHS is always equal to the RHS. The difference between an identity and an equation is that for an equation, its LHS and RHS are equal only for some values of the variable.
On the other hand, as we discussed, for an identity, the LHS equals the RHS for any value of the variable.
There is another important identity for the square of the expression (a – b). Let us try and simplify it.
(a – b)2 = (a – b) (a – b)
= a (a – b) – b (a – b)
= a × a + a × (–b) – b × a – b × (–b)
= a2 – ab – ab + b2
= a2 – 2ab + b2

Many a times, these identities help in shortening our calculations.
II. Use of the Identity (x + a) (x + b)
A very important identity that we have to learn is regarding the expression (x + a) (x + b).
We know how to multiply binomials. By finding the product of these binomials, we can find the identity.
Let’s understand this identity from the following examples:
Example: Use the appropriate identity to simplify the following expressions.
(a) (3p + 5q) (3p – 7z)

(b)
Solution
(a) The given expression is (3p + 5q)(3p – 7z).
This expression is of the form (x + a) (x + b).
Thus, we can use the identity (x + a) (x + b) = x2 + (a + b) x + ab
Here, x = 3p, a = 5q, b = –7z
$\therefore$ (3p + 5q)(3p – 7z)
= (3p)2 + (5q – 7z)(3p) + (5q)(–7z)
= 9p2 + (5q)(3p) – (7z)(3p) – 35qz
= 9p2 + 15pq – 21pz – 35qz

(b)The given expression is $\left({z}^{2}–\frac{4}{3}\right)\left({z}^{2}–\frac{3}{2}\right)$

This expression is of the form (x + a) (x + b).
Thus, we can use the identity (x + a) (x + b) = x2 + (a + b) x + ab
Here, x = z2,

$\therefore \left({z}^{2}–\frac{4}{3}\right)\left({z}^{2}–\frac{3}{2}\right)$

$={\left({z}^{2}\right)}^{2}+\left\{\left(–\frac{4}{3}\right)+\left(–\frac{3}{2}\right)\right\}{z}^{2}+\left(–\frac{4}{3}\right)×\left(–\frac{3}{2}\right)$

$={z}^{4}+\left(\frac{–8–9}{6}\right){z}^{2}+2={z}^{4}–\frac{17}{6}{z}^{2}+2$

III. Use of the Identity (a + b) (a – b)
Suppose we need to find the product of the numbers 79 and 81. Instead of multiplying these two numbers, we can use the identity (a + b) (a – b). This identity is very important and is applicable in various situations.
Let us first understand this identity.
(a + b) (a – b) = a (a – b) + b (a – b)
(By distributive property)
= a2 – ab + ba – b2
= a2 – ab + ab – b2 (ab = ba)
= a2 – b2

$\overline{)\therefore \left(a+b\right)\left(a–b\right)={a}^{2}–{b}^{2}}$

Now, let us solve some examples in which the above identity can be applied.
Example: Simplify the following expressions.

(a) $\left(\frac{3}{7}l+\frac{4}{5}m\right)\left(\frac{3}{7}l–\frac{4}{5}m\right)$

(b) (x2 – y3)(x2 + y3) + (y3 – z4)(y3 + z4) + (z4 – x2)(z4 + x2)

Solution
(a) The given expression is $\left(\frac{3}{7}l+\frac{4}{5}m\right)\left(\frac{3}{7}l–\frac{4}{5}m\right)$.

This expression is of the form (a + b) (a – b).

Hence, we can use the identity (a + b) (a – b) = a2 – b2

$=\frac{9}{49}{l}^{2}–\frac{16}{25}{\mathrm{m}}^{2}$

(b) (x2 – y3)(x2 + y3) + (y3 – z4)(y3 + z4) + (z4 – x2)(z4 + x2)
On using the identity (a + b) (a – b) = a2 – b2, we get

$\left\{{\left({x}^{2}\right)}^{2}–{\left({y}^{3}\right)}^{2}\right\}+\left\{{\left({y}^{3}\right)}^{2}–{\left({z}^{4}\right)}^{2}\right\}+\left\{{\left({z}^{4}\right)}^{2}–{\left({x}^{2}\right)}^{2}\right\}$

$={x}^{4}–{y}^{6}+{y}^{6}–{z}^{8}+{z}^{8}–{x}^{4}$

$=\left({x}^{4}–{x}^{4}\right)+\left(–{y}^{6}+{y}^{6}\right)+\left(–{z}^{8}+{z}^{8}\right)$

= 0.

format: plain


text:
## lilsis76 Group Title How can I re-write the sum's suing Sigma Notation for: 2 - 2^2 + 2^3 -2^4 +2^5 I know the reoccuring number is going to be 2 which goes to the right of the sigma. on the bottom of sigma it would be 1, and top of sigma would be 5 right? one year ago one year ago

1. lilsis76 Group Title

|dw:1353786244962:dw|

2. cnknd Group Title

$\sum_{n=0}^{4}2(-2)^n$ is one way of writing it.

3. lilsis76 Group Title

oh. so that (-2), is that how it would go from negative to positive?

4. cnknd Group Title

expand what i wrote and check for yourself

5. lilsis76 Group Title

okay: -2^0 = 0 *2 = 0 -2^1=-2*2=-4 -2^2=-4*2=-8 -2^3=-8*2=-16

6. lilsis76 Group Title

@cnknd is this how it would go?

7. cnknd Group Title

ur doing the exponents wrong... this is basics...

8. lilsis76 Group Title

|dw:1353786706417:dw|okay the first one would be>....< @cnknd

9. cnknd Group Title

in what i put down, the index n goes from 0 to 4... so it's 2(-2)^0 + 2(-2)^1 + 2(-2)^2 + 2(-2)^3 + 2(-2)^4 which is 2(1) + 2(-2) + 2(4) + 2(-8) + 2(16)

10. lilsis76 Group Title

OH SHOOT! i just noticed i started at 1. i just now noticed it starts at 0

11. lilsis76 Group Title

okay okay, i got it now. it makes more sense starting at zero haha thank you. Medal for you :)


text:
Categories

## Screw axis in P21c space group

I am having some trouble with understanding glide axis. The following is a question asking to reconstruct the full unit cell: And the following is the answer to the question: I have highlighted an element which I want to focus on. I understand the rotation of 180 degrees and translation of 1/2 of a unit […]

Categories

## Screw axis in P21c space group

I am having some trouble with understanding glide axis. The following is a question asking to reconstruct the full unit cell: And the following is the answer to the question: I have highlighted an element which I want to focus on. I understand the rotation of 180 degrees and translation of 1/2 of a unit […]

Categories

## Reason behind more Reactivity of Unsaturated Hydrocarbons

My textbook says that unsaturated hydrocarbons are more reactive than saturated ones. But double and triple bonds are stronger than single bonds. What is the explanation for this?

Categories

## Which is a stronger nucleophile in aprotic solvent: water or hydrogen sulfide?

Which one is the stronger nucleophile in aprotic solvent: $\ce{H2O}$ or $\ce{H2S}?$ According to my notes, nucleophilicity order is directly proportional to basicity order in an aprotic medium, and is inversely proportional to basicity order in protic medium. And $\ce{H2O}$ is a stronger base than $\ce{H2S},$ so it should be the strong nucleophile, but the […]

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## Can glass be deionized?

Is there a chemical treatment that could remove sodium and calcium ions from the surface of soda-lime glass to turn it into quartz glass, increasing the hardness?

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## What does HClO2 aqueous solution decompose to in an open environment?

I looked up the decompositions of diluted $\ce{HClO2}$ aqueous solution (used in a disinfectant) to know if it is safe to use on some substances, such as metal, wood, leather or skin. However, the chain I found leading to $\ce{HClO4},$ which is a very strong acid.  \begin{align} \ce{5 HClO2 &→ 4 ClO2 + HCl […]

Categories

## Intermolecular Forces of Attraction

Imagine a bottle 85% filled with water and the rest is just air, and the cap is closed. Here are now my questions. Since the cap is closed completely, the air bubble is formed. Is that air bubble compressed? In other words, since air has the least force of attraction between its molecules, they are […]

Categories

## Will NH3 gas decompose into N2 and H2 at 250 degrees celcius?

If I heat ammonia gas to 250 degrees celsius, will it decompose into its basic components – N2 and H2? What temperature is required for the reaction to occur?

Categories

## How were the bond lengths of benzene determined?

How did people manage to find out the bond length of benzene? What were the methods and apparatus they used to figure it out? This question arose while reading this passage from Organic Chemistry, seventh Edition by Paula Yurkanis Bruice Controversy over the structure of benzene continued until the 1930s, when the new techniques of […]

Categories

## What is the specific heat capacity of regular corrugated cardboard?

I am testing out different materials with different specific heat capacities and their correlation with soundproofing (e.g. wood, cork, styrofoam, etc.) but I can’t seem to find the specific heat capacity of normal cardboard anywhere.


text:
Journal article

### Study of the decay Bs(0)-->Ds(*)Ds(*).

Abstract:

We report a study of the decay Bs(0)-->Ds(*)Ds(*) using a data sample corresponding to 1.3 fb(-1) of integrated luminosity collected by the D0 experiment in 2002-2006 during run II of the Fermilab Tevatron collider. One Ds(*) meson was partially reconstructed in the decay Ds-->phi mu nu, and the other Ds(*) meson was identified using the decay Ds-->phi pi where no attempt was made to distinguish Ds and Ds(*) states. For the branching fraction Br(Bs(0)-->Ds(*)Ds(*)) we obtain a 90%...

Publication status:
Published

### Authors

Journal:
Physical Review Letters
Volume:
99
Issue:
24
Pages:
241801
Publication date:
2007-12-01
DOI:
EISSN:
1079-7114
ISSN:
0031-9007
Source identifiers:
12102
Language:
English
Keywords:
Pubs id:
pubs:12102
UUID:
uuid:a4bb39f2-d4c9-4f03-bc9f-799578c4693c
Local pid:
pubs:12102
Deposit date:
2012-12-19


text:
# Which set operation is used to find equivalence of two constraint networks?

I was given a question

Given the following constraint networks $$X$$, $$Y$$ and $$Z$$ with four variables $$x_1, x_2, x_3$$ and $$x_4$$ all defined on the same domain values $$\{\text{red}, \text{blue} \}$$. The constraints in the network are as follows:

• X: R13 = R14 = R23 = R24 = {(red, blue) (blue, red)}
• Y: R13 = R14 = R23 = R24 = {(red, blue) (blue, red)}, R12={(red, red) (blue, blue)}
• Z: R13 = R14 = R23 = R24 = {(red, blue) (blue, red)} , R34={(red, blue) (blue, red)}
1. Identify which of them are equivalent:

2. In the above question, the equivalent-networks are equivalent because some operation between some of the relations of these networks makes them the same. What is the name of the operation?

• Composition
• Union
• Intersection
• Set-difference

Now, I know that $$X$$ and $$Y$$ are equivalent because $$Y$$ is a tighter form of $$X$$. But I don't know the answer to the second question. Which of the options above are correct?

I think the answer is intersection, but not sure as to how to prove it.

• Maybe you should describe what $R12$, etc., are here, and how are they related to the variable $x_1$. Moreover, for people not familiar with the topic (like me), I would suggest that you provide a reference (that you're using) that describes the "constraint network", so that we are on the same page, and it could be helpful to those attempting to answer this question (even though this seems to be a standard approach to solve CSPs, e.g. ics.uci.edu/~csp/r17-survey.pdf). – nbro Feb 6 at 16:21


text:
Question

# The half life period of a first order reaction is $20\text{ }minutes$ . The time required for the concentration of the reactant to change from $0.16\text{ }M\text{ }to\text{ }0.02\text{ }M$ is A) $\text{80 }minutes$B) $60\text{ }minutes$C) $40\text{ }minutes$D) $20\text{ }minutes$

Verified
129k+ views
Hint: The following two formulae are needed to solve this question.

\begin{align} & k=\dfrac{0.693}{{{t}_{1/2}}} \\ & t=\dfrac{2.303}{k}\log \dfrac{a}{a-x} \\ \end{align}

The relationship between the rate constant k and the half life period for a first order reaction is as follows:
$k=\dfrac{0.693}{{{t}_{1/2}}}$
The half life period of a first order reaction is 20 minutes.

${{t}_{1/2}}=20\text{ minutes}$

Calculate the rate constant for the first order reaction:

\begin{align} & k=\dfrac{0.693}{{{t}_{1/2}}} \\ & k=\dfrac{0.693}{20\text{ minutes}} \\ & k=0.03465\text{ minut}{{\text{e}}^{-1}} \\ \end{align}

The relationship between the time t required for a certain change in concentration of reactant and the rate constant k for the first order reaction is as follows:
$t=\dfrac{2.303}{k}\log \dfrac{a}{a-x}$ ... ...(1)

Here, a is the initial concentration and a-x is the concentration at time t.
The initial concentration of the reactant is 0.16 M.

$a=0.16M$

The concentration of the reactant at time t is 0.02 M.

$a-x=0.02M$

Calculate the ratio of the initial concentration of the reactant to the concentration of the reactant at time t.

$\dfrac{a}{a-x}=\dfrac{0.16M}{0.02M}=8$

Calculate the logarithm of the ratio

$\log \dfrac{a}{a-x}=\log 8=0.9031$

Substitute values in equation (1)

\begin{align} & t=\dfrac{2.303}{k}\log \dfrac{a}{a-x} \\ & t=\dfrac{2.303}{0.03465\text{ minut}{{\text{e}}^{-1}}}\times 0.9031 \\ & t=60\text{ minute} \\ \end{align}

The time required to decrease the concentration of the reactant from $0.16\text{ }M\text{ }to\text{ }0.02\text{ }M$is $60\text{ }minutes$.

So, the correct answer is “Option B”.

Note:
Another approach to solve the same problem is to determine the number of half lives needed for the concentration of the reactant to change from $0.16\text{ }M\text{ }to\text{ }0.02\text{ }M$.
\begin{align} & \dfrac{0.02}{0.16}=\dfrac{1}{8}={{\left( \dfrac{1}{2} \right)}^{3}}={{\left( \dfrac{1}{2} \right)}^{n}} \\ & n=3 \\ \end{align}

Three half life periods are needed. The half life period is $20\text{ }minutes$. For three half life periods, time will be $60\text{ }minutes$.


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# Orphaned pages¶

Usually it is desired to have each and every .rst- file appear in the menu somewhere. Sphinx will issue a warning if this isn’t the case.

Hint

If really needed, an .rst file may be tagged as :orphan: right at the beginning. This will suppress the warning.

:orphan:


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# Math Help - Model with First Order Differential Equation

1. ## Model with First Order Differential Equation

Dear all,

I'm having some trouble with the following question involving modelling with a first-order differential equation. My solution is below but it differs from the given answer. I also can't see where I might have erred.

Thank you.

----

18. Consider an insulated box (a building, perhaps) with internal temperature $u(t)$. According to Newton's law of cooling, $u$ satisfies the differential equation

$\cfrac{du}{dt} = -k[u - T(t)]$,

where $T(t)$ is the external temperature. Suppose that $T(t)$ varies sinusoidally; assume that $T(t) = T_0 + T_1cos(wt)$

a) Solve the differential equation above and express $u(t)$ in terms of $t,k,T_0,T_1, w$.

My solution:

The above ODE is a linear equation. Rearrangement gives:
$\dfrac{du}{dt} + ku = k(T_0 + T_1\cos{wt})$.

This linear ODE has integrating factor $e^{\int k dt} = e^{kt}$.

Therefore, $\dfrac{d}{dt}(e^{kt}u) = e^{kt}k(T_0 + T_1\cos{wt})$

$\Rightarrow (e^{kt}u)=k\left(\int T_0e^{kt} dt + \int T_1\cos{wt} dt \right)$

$\Rightarrow (e^{kt}u)=k\left(T_0\frac{e^{kt}}{k} + T_1\frac{\sin{wt}}{w} \right)$ + C, where C has absorbed the product of integration constants from both integrals with $k$.

$\Rightarrow u = T_0 - T_1\dfrac{k\sin{wt}}{we^{kt}} + Ce^{-kt}$.

However, the given answer is: $u(t) = ce^{-kt} + T_0 + \frac{kT_1}{k^2 + w^2}(k\cos{wt} + w\sin{wt})$

2. Originally Posted by scherz0
Dear all,

I'm having some trouble with the following question involving modelling with a first-order differential equation. My solution is below but it differs from the given answer. I also can't see where I might have erred.

Thank you.

----

18. Consider an insulated box (a building, perhaps) with internal temperature $u(t)$. According to Newton's law of cooling, $u$ satisfies the differential equation

$\cfrac{du}{dt} = -k[u - T(t)]$,

where $T(t)$ is the external temperature. Suppose that $T(t)$ varies sinusoidally; assume that $T(t) = T_0 + T_1cos(wt)$

a) Solve the differential equation above and express $u(t)$ in terms of $t,k,T_0,T_1, w$.

My solution:

The above ODE is a linear equation. Rearrangement gives:
$\cfrac{du}{dt} + ku = k(T_0 + T_1\cos{wt})$.
This linear ODE has integrating factor $e^{\int k dt} = e^{kt}$.
Therefore, $\frac{d}{dt}(e^{kt}u) = e^{kt}k(T_0 + T_1\cos{wt})$
$\Rightarrow (e^{kt}u)=k(\int T_0e^{kt} dt + \int T_1\cos{wt} dt)$
This step is incorrect. The exponential multiplies the trig function integral as well.

$\Rightarrow u = T_0e^{kt} - T_1\frac{\sin{wt}}{wk}$.

However, the given answer is: $u(t) = ce^{-kt} + T_0 + \frac{kT_1}{k^2 + w^2}(k\cos{wt} + w\sin{wt})$

3. Thanks for your reply, ackbeet. I've since corrected my solution, but it still differs from the given answer.

5. Thanks for your reply, ackbeet. I've shown my integral evaluations. But I still don't understand the difference between my solution and the given answer.

6. Originally Posted by scherz0
Thanks for your reply, ackbeet. I've shown my integral evaluations. But I still don't understand the difference between my solution and the given answer.
If you edited the OP as "showing your integral evaluations", I still don't buy your calculations. You're still not doing what I suggested in Post # 2. Please do NOT edit your OP again. Post your workings in a new post in this thread.

You earlier requested that I show how I evaluate the integrals. Here it is below.

$\Rightarrow (e^{kt}u)=k\left(\int T_0e^{kt} dt + \int T_1\cos{wt} dt \right)$

$\Rightarrow (e^{kt}u)=k\left(T_0 \int e^{kt} dt + T_1 \int \cos{wt} dt \right)$, where $T_0, T_1$ are constants.

Since $\int e^{kt} dt = \frac{e^{kt}}{k}$ and $\int \cos{wt} dt = \frac{sin {wt}}{w}$

Therefore $(e^{kt}u)=k\left(T_0\frac{e^{kt}}{k} + T_1\frac{\sin{wt}}{w} \right)$ + C, where C has absorbed the product of integration constants from both integrals with $k$.

$\Rightarrow u = T_0 - T_1\dfrac{k\sin{wt}}{we^{kt}} + Ce^{-kt}$.

However, the given answer is: $u(t) = ce^{-kt} + T_0 + \frac{kT_1}{k^2 + w^2}(k\cos{wt} + w\sin{wt})$

8. Originally Posted by scherz0
$\Rightarrow (e^{kt}u)=k\left(\int T_0e^{kt} dt + \int T_1\cos{wt} dt \right)$
$\Rightarrow (e^{kt}u)=k\left(\int T_0 e^{kt}\,dt + \int T_1 e^{kt}\cos(wt)\, dt \right).$
You failed to distribute the exponential $e^{kt}$ into your second integrand. Carry that correction through and see what happens.


text:
Question

If 99 × 21 - $\sqrt[3]{x}$ = 1968. Then find the value of x. \begin{align} & \text{a) 1367631} \\ & \text{b) 1366731} \\ & \text{c) 1367} \\ & \text{d) 111} \\ \end{align}

Hint: Now we have one equation in one variable. Hence we can easily find x by just shifting the terms. But here we have a cube root of x hence once we find the value of the cube root is x we will just take the cube on both the sides and hence find x.

Now the given equation is 99 × 21 - $\sqrt[3]{x}$ = 1968 ……………(1)
Here using distributive property we get 99 × 21 = (100-1) × 21 = 100 × 21 – 1 × 21
Hence 99 × 21 = 2100 – 21 = 2079
Substituting this in equation (1) we get 2079 - $\sqrt[3]{x}$=1968
Now taking2079 to the right hand side we get
-$\sqrt[3]{x}$=1968-2079
-$\sqrt[3]{x}$=-111
Now multiplying the above equation by -1.
$\sqrt[3]{x}$=111
Now taking cube on both the sides we get
${{\left( \sqrt[3]{x} \right)}^{3}}={{111}^{3}}$
Now we know that cube of cube root of x equals to x, using this we get.
\begin{align} & x=111\times 111\times 111 \\ & x=12321\times 111 \\ & x=1367631 \\ \end{align}

Hence we get the value of x as 1367631

Note: At the first step we might feel like directly taking cube on both the sides though the equation can also be solved taking cube in the first step it will be a little tedious to do so hence it is better to first take everything except cube root on RHS and then solve.


text: It was a relevant question...., द कॉस्ट ऑफ 20 मीटर क्लॉथ इज 1218 ₹40 फाइंड द कॉस्ट ऑफ 1 मीटर क्लॉथ, koi toh aajao7 0 2 0 4 1 7 4 6 80819 aajao jldI..❤❤❤, Simplest form of rationalising factor for the given surd $$\frac{7}{5} \sqrt{13}$$. What is the area of parallelogram ABCD? In the accompanying diagram of a parallelogram ABCD, mLA = 17 and ml-C = — 4. 080310b, P.r. In the accompanying diagram of rhombus ABCD, diagonal AC is drawn. 2. In the accompanying diagram,! In the accompanying diagram, line ‘ is parallel to line m, and lines s and t are transversals that intersect at a point on line m. Which statement must be true? Given: Quadrilateral DEBF is a parallelogram; AEFC, and AECF≅ Prove: Quadrilateral ABCD is a parallelogram. You can refuse to use cookies by setting the necessary parameters in your browser. Julian wrote the following numeric pattern on the board3,10,17,24,31,38.what numbers of julian's pattern are compound numbers. Add your answer and earn points. The length of each side of the square is 56 4 =14. 3. In a certain quadrilateral, two opposite sides are parallel, and the other two opposite sides are not congruent. can you conclude that more than half of the residents in this town are opposed to constructing a new shopping mall? In the accompanying diagram, ABCD is a parallelogram with altitude DE drawn to side AB. Express both the perimeter and the area of the rectangle shown in the accompanying diagram as polynomials in simplest form. For this case we have the following diagonals: In a survey of 500 residents in a certain town, 275 said they were opposed to constructing a new shopping mall. Add your answer and earn points. b. Regents Exam Questions G.SRT.B.5: Triangle Proofs 1 Name: _____ www.jmap.org 3 9 In the accompanying diagram of triangles BAT and FLU, ∠B ≅∠F and BA ≅FL. m


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# Talking primes using Dirk Brockmann’s “Prime Time” explorable

I’ve been a huge fan of Dirk Brockmann’s explorable math activities since I first learned about them. The full list is here:

Dirk Brockmann’s Explorables

Today’s project was inspired by the “Prime Time” program – direct link here:

Dirk Brockmann’s Prime Time Explorable

I started the project today by asking my son to tell me some things he knew about primes. He gave the definition of a prime numbers, explained how we know that there are infinitely many primes, and talked about twin primes, though he apologized for not knowing how to prove that there were infinitely many twin primes:

Next I showed him the polynomial $n^2 + n + 41$ and we talked about this equation producing a lot of primes.

Now we went to the “prime time” explorable and my son talked about what he saw in the first two examples -> the Ulam spiral and the Sack spiral.

Finally we looked at the last two patterns -> the Klauber triangle and the Witch’s spiral.

# Exploring a fascinating idea from special relativity – Penrose-Terrell rotation – with kids

I saw some really neat tweets from John Carlos Baez and Greg Egan on Penrose-Terrell rotation last week:

Even though even the most basic ideas from relativity are far outside of what kids can grasp, I thought it would be fun to share these animations with my younger son. The animations in the above tweets are definitely something that kids can appreciate, and I was excited to hear what my son would have to say.

So, I started out the project today asking my son to describe anything he knew about relativity and then what he thought he’d see if a cube passed by him really fast:

Next we talked about some simple ideas from relativity and what impact those ideas might have on a cube passing by. Also, since he’s just starting to learn about square roots and quadratics in school, I showed him the Lorentz contraction formula and we did one simple calculation:

Finally, we went to the computer to look at the tweets and animations from John Carlos Baez and Greg Egan that I linked above. As always, it is really fun to hear a kid react to and describe ideas from advanced math (and physics!):

# Playing Mapmaker -> the Gerrymandering Game

Saw a great tweet from Jordan Ellenberg over the weekend:

Though it would be fun to try . . . and the game arrived today!

Tonight I had the chance to play with my younger son – he didn’t know what gerrymandering was, but we talked a little bit about the idea as we opened the box:

Here’s a quick peek at the game set up:

Finally – here’s what the game looked like at the end and then a short discussion of some of the mathematical ideas for kids that come through in the game. What ideas can help you win? What can you do to cause your opponent some problems?

I really like this game a lot. It is easy to play right out of the box – the directions probably took under 5 min to read and understand. The game is a great way to teach important ideas about gerrymandering and also has great math ideas in it for kids. I highly recommend it – thanks to Jordan Ellenberg for the recommendation.

# Revisiting a connection between arithmetic and geometry

I saw a really great thread on twitter this week and wanted to share some of the ideas with the boys for our Family Math project today:

We started off looking at the sum 1 + 2 + 3 + . . . .

Next we looked at the sum of squares and searched for a geometric connection:

Now I showed them the fantastic way of looking at the sum of squares in the Jeremy Kun blog post. This method is a terrific way to share an advanced idea in math with kids – it is totally accessible to them and gives them a chance to talk through a fairly complicated idea:

Finally, I showed how the ideas we were just talking about extend to some of the basic ideas is calculus. It was neat to hear my younger son talking through the ideas here, too:

Definitely a neat morning – it is always amazing to see the connections between arithmetic and geometry.

# Using some temperature graphs from Deke Arndt to talk about probability distributions with kids

Via my friend Ed Adlerman, I saw some amazing temperature graphs made by Deke Arndt:

I thought these graphs could be used for a great introductory statistics talk for kids, so I decided to used them to talk about probability distributions today.

We started by looking a probability distributions in a relatively simple situation -> dice:

Next we moved to talking about a probability distribution in a more complicated situation -> a “Galton board”:

Now we moved on to discussing Arndt’s graphs. The conversation about these graphs went on for 11 minutes. As always it is fascinating to hear what kids see when they look at advanced math.

Here’s the first part of that 11 min discussion – here you’ll hear their initial observations and a bit about how to interpret the distributions on display here.

Here’s the second part of our conversation about the graphs. Here we talk about what all of these pictures are telling us about the temperature in Sydney during the last 60 years.

We wrap up by talking about whether or not they liked this presentation and different ways to present the data that would have been made the presentation harder to understand (I though this would be an easier question for kids than finding ways to make it better).

I think Arndt’s work here is amazing all by itself, but is also something that I think can be used really effectively to talk about probability and statistics with kids. Thanks to him for sharing this great work.

# Celebrating Pi day with a fun calculus paper from Susan Jane Colley

About a week ago I saw a great tweet from Dave Richeson about some journal articles that were being made freely available for the month of March:

The paper from the The College Mathematics Journal by Susan Jane Colley caught my attention for being both an really neat result and being explained at a level a student taking calculus could understand.

So, this morning to celebrate Pi day I decided to use the paper to talk a bit of calculus with my son. Pulling all of the different ideas together was challenging for him, so we went slow but still made it through the main points in about 30 min.

We took a quick look at the paper and then started digging into the math by looking at the famous alternating series for $\pi / 4$.

I should say for clarification that I forgot to look up Susan Jane Colley’s current position before we started the project and wasn’t sure if she was still at Oberlin or had moved to a different university since the paper was published in 2003. But to be clear, she is the Andrew & Pauline Delaney Professor of Mathematics at Oberlin.

Next we dove in to the connection between the alternating series and $\pi$. I thought I’d try to introduce the connection in a sneaky way, but it was sort of a dead end. Eventually, though, he thought about arctangent.

At the end of the last video the formula for an infinite geometric series came up, but that formula wasn’t quite at the top of his head. So we took a little detour to re-derive that formula. Once we had that formula we could see that the alternating series we were looking at converged to $\pi/4$:

Now we looked at the main result of the paper – a different series for $\pi$ that converges really fast.

Here we look briefly at the formula for this series (sorry for the reading typos by me – trying to read the paper and stay behind the tripod and not cast a shadow was hard . . . . )

Finally, we went to Mathematica to evaluate the integral and look at the speed of convergence of the two series we’ve been studying:

I think that Colley’s paper is absolutely terrific and a great resource to use to show calculus students some advanced math. It is an extra terrific resource to use on Pi day 🙂

# A useful integral for kids learning calculus

My son is in the middle of a long review for the BC Calculus exam in May. Right now he’s doing some integration practice and came across this integral today:

We didn’t get a chance to talk about it too much this morning, but we did review the problem when he got home tonight.

First we I asked him to approach the problem as he did this morning -> by trying to evaluate the integral. The practice integrating rational functions turned out to be useful:

Next we took a more geometric approach:

Finally, I wanted to show him how you could see that the integral was zero from the u-substitutions that he made in the first video. Even though this method wasn’t really the best one for this particular integral, I still wanted him to see how the algebra worked out:

# Bob Lochel’s random “Matt” problem

Saw a fun tweet from Bob Lochel last week:

For today’s Family Math problem I thought I’d talk through the math behind this situation with my younger son.

First I showed him what had happened and asked for a few ideas about how to approach finding the probability that all of the Matt’s would end up in group d. It has been a while since we’ve done a probability and counting project, but the ideas came back to him as we talked:

Next we talked about how to calculation the number of different ways to put the students into the the groups of 5 and 4. Then I asked my son to estimate what the number was without calculating. His estimate was off and we returned to why in part 4 of the project:

Now we calculated the number of ways to arrange the students with all 3 Matt’s landing in group D. Once we had that number we found the probability of the outcome that happened. I also asked my son if he thought the outcome happened by random chance, or was Bob possibly playing a little joke:

Finally, we revisited the calculation from part 2 of the project. It turned out that his estimate actually was nearly right – investigating where it went wrong was a good use of time. Once we had the exact value by hand, we also computed the exact probability for the original problem by hand.

Thanks to Bob Lochel for sharing this fun outcome. If you’d like to see a similar probability / counting problem check out our exploration of the “Snapchat problem” from a few years ago:

Exploring the “Shapchat Problem” with my kids

# Using an integral problem from the 2012 BC Calculus exam to practice with numerical integration

We had a snow day today and I was able to use some extra time with my son to talk about numerical integration techniques. I’d guess that I hadn’t seen the topic in at least 25 years, so it was fun to teach and review at the same time!

This evening I had my son review the ideas using an integration problem from the 2012 BC Calculus exam. I started by having him evaluate the integral exactly with no numerical integration techniques:

Next we used the trapezoidal rule and compared the answer to the exact answer –

The one thing I learned today was that the midpoint rule was roughly twice as accurate as the trapezoidal rule. Here my son used the midpoint rule and we do find that the answer is closer to the true value than the answer we found in the last video:

Finally, we used Simpson’s rule. Despite one little mistake in the middle, we ended up finding an answer that was surprisingly close to the exact asnwer.

# Sharing James Tanton’s area models with my younger son

I saw a really neat tweet from James Tanton yesterday:

The video in the “setting the scene” link is terrific and I had my younger son watch it this morning. I can’t figure out how to get the video to embed, so here’s a direct link to the page it is on:

James Tanton’s area model lecture

One reason that I was extra excited to see Tanton’s video is that by total coincidence I’d used essentially the same idea with my older son to explain why a negative times a negative is a positive:

But I’d never gone through the same ideas with my younger son – at least not that I can remember. So, for our Family Math project today I had him watch Tanton’s video and then we talked about it.

Here’s what he thought about the video and area models in general – I was really happy to hear that my son liked Tanton’s area models and thought they were a really great way to think about multiplication and quadratics:

Next I had my son walk through the negative / positive area ideas that Tanton used to talk about multiplication. He did a really nice job replicating Tanton’s process. I think this is a great way for kids to think about multiplication:


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# Does it classify as Machine Learning?

I have a gaussian distributed time series ($$X_t$$) with some parameters in my experiment. Suppose I want to know the mean $$\mu$$. If I define another time series $$Y_t$$ such that $$Y_t=X_t-a$$ for all $$t$$. Now say I vary this parameter $$a$$ and generate altogether different time series for each $$a$$, say $$Y_t(a)$$. I look at the mean of $$Y_t$$ for each $$a$$. The value of a, where I get the mean of $$Y_t$$ closest to $$0$$, will be my estimate of $$\mu$$. Say I will eventually use this learnt value of $$\mu$$ to generate $$Y_t$$ as my final goal. Can this be called ML? I am using some training data of $$X_t$$ to learn about its parameter and then using test data of $$X_t$$ to generate $$Y_t$$.

Now why am I working so hard on this simple problem? Well, actually I am not. I am doing something else, which will have lots of parameters in the time series and will be used to generate other time series after similar parameter extraction. That will be too complicated to discuss here. I just wanted to clear my basics using an over-simplified example. I am a new to ML, and help is appreciated.

• I don't understand what you mean by "gaussian distributed time series". Can you precisely define it? Are you saying that each element of the time series is sampled from 1. the same or 2. possibly different Gaussian? In general, I don't fully understand your problem. Do you have the original time series $X_t$? If not, do you want to estimate it from "noisy" similar time series? I don't understand how subtracting $a$ from $X_t$ gives you any insight. Also, I don't understand how you are going to generate $Y_t$ if you don't have $X_t$. Are you assuming that $Y_t$ is $X_t -a$, for some $a$? – nbro Apr 25 at 23:44
• @nbro Thanks for your response. The time series $X_t$ is given, and we know it has a Gaussian distribution, (histogram: Gaussian). So each point in $X_t$ belongs to a gaussian distribution with the same mean and variance. Now if you subtract a from each point in $X_t$ to construct a different series $Y_t$ the mean of the gaussian shifts. For some $a$, the constructed $Y_t$ has 0 mean. What I am asking if you vary $a$ and try to see for what $a$ this is achieved, then use that correctly found a, to generate more data points of $Y_t$, from $X_t$ can I call it as ML? – Raunak Dey Apr 27 at 17:40


text:
# Difference between revisions of "2006 AIME II Problems/Problem 7"

## Problem

Find the number of ordered pairs of positive integers $(a,b)$ such that $a+b=1000$ and neither $a$ nor $b$ has a zero digit.

## Solution

### Solution 1

There are $\left\lfloor\frac{999}{10}\right\rfloor = 99$ numbers up to 1000 that have 0 as their units digit. All of the other excluded possibilities are when $a$ or $b$ have a 0 in the tens digit, and since the equation is symmetric, we will just count when $a$ has a 0 in the tens digit and multiply by 2 (notice that the only time both $a$ and $b$ can have a 0 in the tens digit is when they are divisible by 100, which falls into the above category, so we do not have to worry about overcounting).

Excluding the numbers divisible by 100, which were counted already, there are $9$ numbers in every hundred numbers that have a tens digit of 0 (this is true from 100 to 900), totaling $9 \cdot 9 = 81$ such numbers; considering $b$ also and we have $81 \cdot 2 = 162$. Therefore, there are $999 - (99 + 162) = \boxed{738}$ such ordered pairs.

### Solution 2

Let $a = \overline{cde}$ and $b = \overline{fgh}$ be 3 digit numbers:

cde
+fgh
----
1000

$e$ and $h$ must add up to $10$, $d$ and $g$ must add up to $9$, and $c$ and $f$ must add up to $9$. Since none of the digits can be 0, there are $9 \times 8 \times 8=576$ possibilites if both numbers are three digits.

There are two other scenarios. $a$ and $b$ can be a three digit number and a two digit number, or a three digit number and a one digit number. For the first scenario, there are $9 \times 8 \times 2=144$ possibilities (the two accounting for whether $a$ or $b$ has three digits) and for the second case there are $9 \times 2=18$ possibilities. Thus, thus total possibilities for $(a,b)$ is $576+144+18=738$.

### Solution 3

We first must notice that we can find all the possible values of $a$ between $1$ and $500$ and then double that result.

When $1 < a < 100$ there are $9\times9 = 81$ possible solution for $a$ so that neither $a$ nor $b$ has a zero in it, counting $1$ through $9$, $11$ through $19$, ..., $81$ through $89$. When $100 < a < 200$ there are $9\times8 =72$ possible solution for a so that neither a nor b has a zero in it, counting $111$ through $119$, $121$ through $129$, ..., $181$ through $189$. This can clearly be extended to $100k < a < 100(k+1)$ where $k$ is an integer and $0 . Thus for $100 < a < 500$ there are $72\times4$ = $288$ possible values of $a$.

Thus when $1 < a < 500$ there are $288 + 81 =369$ possible values of $a$ and $b$.

Doubling this yields $369\times2= 738$.

### Solution 4 (Similar to Solution 2)

We proceed by casework on the number of digits of $a.$

Case 1: Both $a$ and $b$ have three digits

We now use constructive counting. For the hundreds digit of $a,$ we see that there are $8$ options - the numbers $1$ through $8.$ (If $a = 9,$ that means that $b$ will be a two digit number, and if $a = 0,$ $a$ will have two digits). Similarly, the tens digit can be $1-8$ as well because a tens digit of $0$ is obviously prohibited and a tens digit of $9$ will lead to a tens digit of $0$ in the other number. The units digit can be anything from $1-9.$ Hence, there are $8 \cdot 8 \cdot 9 = 576$ possible values in this case.

Case 2: $a$ (or $b$) has two digits

If $a$ has two digits, the only restrictions are that the units digit must not be $0$ and the tens digit must not be $9$ (because then that would lead to $b$ beginning with $90...$). There thus are $8 \cdot 9 = 72$ possibilities for $a,$ and we have to multiply by $2$ because there are the same number of possibilities for $b.$ Thus, there are $72 \cdot 2 = 144$ possible values in this case.

Case 3: $a$ (or $b$) has one digit

This is easy -- $a$ can be anything from $1$ to $9,$ for a total of $9$ possible values. We multiply this by $2$ to account for the single digit $b$ values, so we have $9 \cdot 2 = 18$ possible values for this case.

Adding them all up, we get $576 + 144 + 18 = \boxed{738},$ and we're done.

Solution by Ilikeapos

### Solution 5

For every $a \in [1, 999]$, $(a, 1000 - a)$ is a potential candidate for a solution, barring the cases where $a$ or $1000 -a$ has zero digits.

First, let's consider all viable $a$ that do not have a zero digit. As there are 9 non-zero digits, we have:

• $9^1$ 1-digit numbers without a zero digit
• $9^2$ 2-digit numbers without a zero digit
• $9^3$ 3-digit numbers without a zero digit

However, we have still overlooked the cases where $1000 - a$ contains zero digits:

• Case 1: If the one's digit of $1000 - a$ is zero, then $a$ will trivially end in a zero, which we've already excluded.
• Case 2: If the ten's digit of $1000 - a$ is zero, with digital representation $\overline{X0Y}$, then $a$ has the digital representation $\overline{[9-X]9[10-Y]}$. $X$ and $Y$ can each take on any value in $[1, 9]$ to produce a value in our set of potential $a$. There are thus $9^2$ cases that we overlooked, where $a$ had no zero digits, but $1000 - a$ did.

Adding up the cases with $a \in [1, 999]$ with no zero digits and removing the cases with $1000 - a$ with zero digits gives us $$(9^1 + 9^2 + 9^3) - 9^2 = \boxed{738}$$

~SaifHakim


text:
## Diffraction patterns

$\lambda=\frac{h}{p}$

ramiro_romero
Posts: 49
Joined: Sat Sep 07, 2019 12:16 am

### Diffraction patterns

I undertstand that diffraction patterns result from constructive and destructive interference but what is the difference between the 2 and how do I identify them?

SarahCoufal_1k
Posts: 55
Joined: Thu Jul 25, 2019 12:17 am

### Re: Diffraction patterns

I'm not sure how to identify them. But constructive interference occurs when two light sourced are in phase with each other, such as peak on peak. This can result in bigger peaks. Destructive interference occurs when the light waves are not in phase with each other and as a result are smaller peaks

Cavalli_1H
Posts: 49
Joined: Thu Sep 19, 2019 12:15 am

### Re: Diffraction patterns

constructive interference is when the maxima of two waves add together and the waves are in phase w each other and destructive interference is when the waves are out of phase and cancel each other out rather than adding together.

Sreyes_1C
Posts: 45
Joined: Fri Sep 28, 2018 12:19 am

### Re: Diffraction patterns

I like to remember that two in phase waves leads to constructive and two out of phase lead to destructive. What helps me is drawing out what in phase/out of phase waves would look like to visualize how it would lead to destructive or constructive.

EricZhao3G
Posts: 51
Joined: Tue Oct 08, 2019 12:16 am

### Re: Diffraction patterns

Constructive interference occurs when two waves are in phase and "add" to each other. This happens when both waves are positive or negative at the same time so their amplitudes will increase after they collide. Destructive interference is the opposite and occurs when two waves are out of phase and "cancel" each other out, resulting in smaller amplitudes.


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# Monthly Archives: April 2012

## A variance campaign that failed

April 23, 2012
By

they ought at least be allowed to state why they didn’t do anything and also to explain the process by which they didn’t do anything. First blush One of the nice things about R is that new statistical techniques fall into it.  One such is the glasso (related to the statistical lasso) which converts degenerate … Continue reading...

## Visualising the Path of a Genetic Algorithm

April 23, 2012
By

We quite regularly use genetic algorithms to optimise over the ad-hoc functions we develop when trying to solve problems in applied mathematics. However it’s a bit disconcerting to have your algorithm roam through a high dimensional solution space while not being able to picture what it’s doing or how close one solution is to another. … Continue reading...

April 22, 2012
By

I came across a free source of Intraday Forex data while reading Forex Trading with R : Part 1 post. You can download either Daily or Hourly historical Forex data from the FXHISTORICALDATA.COM. The outline of this post: Download and Import Forex data Reference and Plot Intraday data Daily Backtest Intraday Backtest First,I created a

## 118 years of US State Weather Data

April 22, 2012
By

A recent post on the Junkcharts blog looked at US weather dataand the importance of explaining scales (which in this case went up to 118). Ultimately, it turns out that 118 is the rank of the data compared to the previous 117 years of data (in ascending order, so that 118 is the highest). At … Continue reading...

April 22, 2012
By

## Machine learning for identification of cars

April 22, 2012
By

There are plenty of data on internet, however it is raw data. Think for a second about public surveillance cameras - useful to check the traffic on the route or busy place, but anything else? What if you want to know how many cars are on the route? How many car were yesterday at the same time?

## Fancy HTML5 Slides with knitr and pandoc

April 22, 2012
By

Karthik Ram gave an Introduction to R a couple of weeks ago, and I strongly recommend you to take a look at his cool HTML5 slides. I started trying HTML5 slides last year, and now it is difficult for me to go back to beamer, which I have used for a few...

## Phase space plot of the kicked rotor

April 21, 2012
By
$Phase space plot of the kicked rotor$

In the idealized physical world, a rotor is simply a mass attached to an axis of length , free to move in the plane. Gravity and friction are absent. Such a rotor becomes a kicked rotor if it is periodically hit with a hammer. Every kick transfers momentum to the rotor and the time between

## David Olive’s median confidence interval

As I have discussed in a number of previous posts, the median represents a well-known and widely-used estimate of the “center” of a data sequence.  Relative to the better-known mean, the primary advantage of the median is its much reduced outlier sensitivity.  This post briefly describes a simple confidence interval for the median that is discussed in a paper...


text:
Physics: Principles with Applications (7th Edition)

18 or 72$^o$.
Use the level horizontal range formula that is derived in the text, with a range of 2.5 meters. You will find $sin(2 \theta) = 0.5799$ which can be solved for the given angles. These two angles give the same range. The 18$^o$ solution results in a fast, low trajectory, while the higher angle results in the water spending more time in the air, and arcing down onto the target.


text:
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# Creating a Homing Missile in Unity UNDER REVIEW

By Preet Kamal Minhas | Published Mar 05 2014 01:53 PM in Game Programming

Unity homing missile slerp

Homing missiles have been my favorite in all types of games. There is nothing more fun than a fire & forget weapon. Homing missiles will generally lock on to a target & keep following it till a collision occurs OR the missile might auto-destruct after a certain period of time.

This article will talk about creating homing missiles in Unity using a kinematic rigidbody in order to keep things simple. A kinematic rigidbody’s transform can be manipulated directly.

Also, I would like to introduce you to my game, Hawks, which is a WIP at the moment with 90% of the functionality in place.

## A basic homing missile will operate based on three properties: Speed, Auto Destruct time & target.

Important: The following code should be attached to a gameobject having a Rigidbody component with isKinematic set to true.

1. Speed: Since we are using a kinematic rigidbody, the speed of the projectile can be controlled by translating in the Update() function:

transform.Translate(0,0,speed * Time.deltaTime,Space.Self);

2. Auto Destruct: This functionality can be implemented by calling a coroutine which self-destructs after a set amount of time:

function Start () {StartCoroutine(AutoExplode());}private function ExplodeSelf() {Instantiate(explosion,transform.position,Quaternion.identity);Destroy(gameObject);}private function AutoExplode() {yield WaitForSeconds(autoDestroyAfter);ExplodeSelf();}

3. Target: The missile needs to follow the target. We can achieve this by rotating the missile towards the target transform using the wonderful Slerp function:

var relativePos : Vector3 = target.position - transform.position;var rotation : Quaternion = Quaternion.LookRotation(relativePos);transform.rotation = Quaternion.Slerp(transform.rotation, rotation, homingSensitivity);

Notice the homingSensitivity variable that is used as the last parameter to the Quaternion.Slerp(from, to, t) function. To understand its usage, we need to look at what the Slerp actually does. Slerp == Spherical Interpolation. Mathematical details available at: http://en.wikipedia.org/wiki/Slerp. For those not so mathematically inclined, given two values, the Slerp function will calculate an intermediate position between from & to based on the third parameter which lies in the range [0,1]. If the third param is 0, the from value will be returned & 1 will return the to value. Try experimenting with different values to achieve the desired effect in your game.

Below is the complete code for the HomingMissile.js script:

#pragma strict
var target : Transform;
var speed : float;

var autoDestroyAfter : float;
//prefab for explosion
var explosion : GameObject;

/* Represents the homing sensitivity of the missile.
Range [0.0,1.0] where 0 will disable homing and 1 will make it follow the target like crazy.
This param is fed into the Slerp (defines the interpolation point to pick) */
var homingSensitivity : float = 0.1;

function Start () {
StartCoroutine(AutoExplode());
}

function Update () {
if(target != null) {
var relativePos : Vector3 = target.position - transform.position;
var rotation : Quaternion = Quaternion.LookRotation(relativePos);

transform.rotation = Quaternion.Slerp(transform.rotation, rotation, homingSensitivity);
}

transform.Translate(0,0,speed * Time.deltaTime,Space.Self);
}

function OnTriggerEnter(other: Collider) {
//Damage the other gameobject & then destroy self
ExplodeSelf();
}

private function ExplodeSelf() {
Instantiate(explosion,transform.position,Quaternion.identity);
Destroy(gameObject);
}

private function AutoExplode() {
yield WaitForSeconds(autoDestroyAfter);
ExplodeSelf();
}

## Hawks: Guardians of the Skies

I began development on Hawks around 6 months back. The game has come a long way since then & I would like to give back to the community with more articles & insights into the game dev process.

Check out Hawks on IndieDB

## Article Update Log

28 Feb 2014: Initial release

Well this applies to anything meant to move towards a target, here http://unitygems.com/basic-ai-character/ I apply the same principle to a character moving either towards a waypoint or towards the player.

Also, you should add the fact that at fast speed, you may miss collision and then should add a raycast system to make sure your rocket is not one frame on one side of the wall and next frame on the other side.

You mention the homingSensitivity should be within 0 and 1, why not adding an attribute to make sure?

At 1 it won't follow the target like crazy it will hit instantly regardless of any obstacles, so you may want to prevent that value as well.

What is set to IsTrigger? The rocket collider?

You should also tell to check for what is colliding, if you have an invisible trigger box meant to be used for level design like a waypoint, then your rocket will explode for no reason (at least to the player).

Well this applies to anything meant to move towards a target, here http://unitygems.com/basic-ai-character/ I apply the same principle to a character moving either towards a waypoint or towards the player.

Also, you should add the fact that at fast speed, you may miss collision and then should add a raycast system to make sure your rocket is not one frame on one side of the wall and next frame on the other side.

You mention the homingSensitivity should be within 0 and 1, why not adding an attribute to make sure?

At 1 it won't follow the target like crazy it will hit instantly regardless of any obstacles, so you may want to prevent that value as well.

What is set to IsTrigger? The rocket collider?

You should also tell to check for what is colliding, if you have an invisible trigger box meant to be used for level design like a waypoint, then your rocket will explode for no reason (at least to the player).

"At fast speed you may miss collision". This can be mitigated by setting the appropriate collision detection mode for the rigidbody: https://docs.unity3d.com/Documentation/ScriptReference/Rigidbody-collisionDetectionMode.html

"Homing sensitivity 0,1..." The homing sensitivity is used to control the rotation & not the position. Hence, setting it to 0,1 will change the rotation & not the position. It is safe to set it to 1 in this script. Try it out.

This value is already exposed in the inspector since it is a public attribute. (Check the complete code at the end of the article)

"OnTriggerEnter" This part can be elaborated further. I will update it along with one of my upcoming articles which talks about trigger based collision detection.

It would be better to teach new games programmers the basics of things like motion in a more general way. This tutorial just adds to the huge list of tutorials (usually written to gather ad revenue) where people can copy the code they need without understanding what they're doing, producing developers that make shoddy games by copy and pasting tutorials and getting stuck at every point where they might have to solve a problem.

There's no explanation of why anything is done, or how it works in more general cases. People will learn how to copy and paste your code into their project and nothing else.

Another one for the "sir what are the codes plz" crowd.

It would be better to teach new games programmers the basics of things like motion in a more general way. This tutorial just adds to the huge list of tutorials (usually written to gather ad revenue) where people can copy the code they need without understanding what they're doing, producing developers that make shoddy games by copy and pasting tutorials and getting stuck at every point where they might have to solve a problem.

There's no explanation of why anything is done, or how it works in more general cases. People will learn how to copy and paste your code into their project and nothing else.

Another one for the "sir what are the codes plz" crowd.

I appreciate your point of view. However, this article was not meant to explain the basics to beginners. This is more of a hands on article with a clear purpose of teaching a certain kind of technique.

Hei,

homingSensitivity case is solved, I was wrong on that one.

Concerning the collision detection, setting to continuous dynamic I think Unity used to tell it will help but not solve the issue on every case.

It would be better to teach new games programmers the basics of things like motion in a more general way. This tutorial just adds to the huge list of tutorials (usually written to gather ad revenue) where people can copy the code they need without understanding what they're doing, producing developers that make shoddy games by copy and pasting tutorials and getting stuck at every point where they might have to solve a problem.

There's no explanation of why anything is done, or how it works in more general cases. People will learn how to copy and paste your code into their project and nothing else.

Another one for the "sir what are the codes plz" crowd.

Just logged in to up vote you sir.

I'm sorry, but I have a few negative points:

• This code is really specific, as said it could be more general.
• The code is trivial to write for somebody with reasonable programming experience and/or access to other tutorials.
• The biggest point: this article feels like advertisement to me. You have a hole heading dedicated to your game, even though it doesn't have much content, while I don't really see any relationship aside from what you mentioned.

The article is however practical and may be useful. But maybe try to put more content in it next time. ;)

Edit: why is my mark-up not saved?

I'm so dumb haha. I was making my own vector3 function, and forgot there is one built into the engine I am using. Thanks for this! The process will come in handy, and I might be able to use it for my AI script. Nice simple code too.

I would also add a RayCast on the tip of it to check if it is about to collide with something and then auto destroy it on act.

Lets say the player uses an obstacle to block the incoming missile.

The translate function ignores colliders, so the missile would be passing through walls and etc.

I would also add a RayCast on the tip of it to check if it is about to collide with something and then auto destroy it on act.

Lets say the player uses an obstacle to block the incoming missile.

The translate function ignores colliders, so the missile would be passing through walls and etc.

This can be accomplished using various techniques. Raycasting is one.

For my current game, I attach a specific script to objects which can take damage/ act as obstacles. OnTriggerEnter, check whether the other game object has this script. If yes, call the method which reduces other's health & then ExplodeSelf().

Firstly, I'd like to say that I value contributions. I'm sure there are people that could get value from that script.

In my opinion articles should either have some general applicability to the reader's game development abilities, or go deeply into an esoteric topic. The scope of this article seems quite narrow, and largely consists of the code itself.

I feel that either this could be part of an article on a related set of technologies from your game, or may be better suited to one of the variety of sites devoted to short unity scripts.

Note: Please offer only positive, constructive comments - we are looking to promote a positive atmosphere where collaboration is valued above all else.

PARTNERS


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• 7

## Learning Opportunities

This puzzle can be solved using the following concepts. Practice using these concepts and improve your skills.

## Goal

If you have 4 bells (say 1, 2, 3 and 4), you can ring them in 24 ways: for example, 1, 2, 4 then 3: 1 2 4 3 or 3, 1, 2 then 4: 3 1 2 4. You can ring the 24 melodies one by one; but how can someone (a monk or a priest) learn the full pattern of 24 melodies? They have developed mnemotechnical ways to remember them all and not to forget even one with what we call mathematical permutations.
Plain Bob Minimus is such an algorithm. Here is its beginning:
1 2 3 4→2 1 4 3→2 4 1 3→4 2 3 1→4 3 2 1→3 4 1 2

A permutation of a set is a bijection (a one-to-one function) from the set onto itself, like a shuffle of the elements of the set.
We can describe Plain Bob Minimus’ permutations as (1 2)(3 4)(1 4)(2 4)(1 3)(2 3)(4 3)(2 1).
First, (1 2)(3 4) exchanges the bells 1 and 2, 3 and 4.
Then (1 4) exchanges the bells 1 and 4.
Then (2 4)(1 3) exchanges the bells 2 and 4, 1 and 3. And so on.
In blue, it’s a list of bells, a melody with four bells. In yellow, it’s the permutation that rearranges the bells from a melody to the next melody.

A permutation can move more than two bells. (1 4 5) is the permutation that sends 14, 45 and 51 and does not move neither 2 nor 3 nor any other number (it’s a 3-cycle).

Beware, because permutations are functions, you must compute them from right to left.

You have to print the resulting permutation from a succession of permutations, that is a product of disjoint cycles (no number may appear in more than one cycle).

Note that (1 4)(2 3) = (4 1)(2 3) = (3 2)(4 1) and so on. Write down the first in the lexicographic order where (1 4)(2 3)<(3 2)(4 1)<(4 1)(2 3).
Don’t print the 1-cycle permutations like (1) or (3) that move nothing in the decomposition even if they might be present as input. There might be only one permutation and the result might be the original. If the permutation moves nothing, print () (two parentheses).
Look at the examples below.

Example, (1 3)(1 2 3)(1 3):
1→3 then 3→1 then 1→3
2→2 then 2→3 then 3→1
3→1 then 1→2 then 2→2
Conclusion, (1 3)(1 2 3)(1 3) = (1 3 2) (a 3-cycle).

The last example, (1 2 3)(1 4 3):
1→4 then 4→4
2→2 then 2→3
3→1 then 1→2
4→3 then 3→1
Conclusion, (1 2 3)(1 4 3) = (1 4)(2 3) (two disjoint 2-cycle).
Input
Single line The succession of permutations.
Output
Single line The result of their product.
Constraints
The input is properly formatted (no unpaired parentheses, etc.)
1≤input≤5
Example
Input
(1 2)(2 3)
Output
(1 2 3)

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1. ## simplify

hi i need to simplify this but i don't know how to, can someone explain how to please?!

2. $\sqrt{75}=\sqrt{3\cdot25}=5\sqrt{3}$

$\sqrt{27}=\sqrt{3\cdot 9}=3\sqrt{3}$

Then:

$\sqrt{75}+\sqrt{27}=5\sqrt{3}+3\sqrt{3}=8\sqrt{3}$

3. Originally Posted by felper
$\sqrt{75}=\sqrt{3\cdot25}=5\sqrt{3}$

$\sqrt{27}=\sqrt{3\cdot 9}=3\sqrt{3}$

Then:

$\sqrt{75}+\sqrt{25}=5\sqrt{3}+3\sqrt{3}=8\sqrt{3}$
Interesting typo in your last line

4. Originally Posted by e^(i*pi)
Interesting typo in your last line

?

5. Originally Posted by e^(i*pi)
Interesting typo in your last line
Oh, sorry, now it is correct (i think)


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# Which of the following solvents has he largest value of $K_t$?

$\begin{array}{1 1} water \\ Benzene \\ Camphor \\ Hexane \end{array}$


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# Profiling

Profile.@profileMacro
@profile

@profile <expression> runs your expression while taking periodic backtraces. These are appended to an internal buffer of backtraces.

source

The methods in Profile are not exported and need to be called e.g. as Profile.print().

Profile.printFunction
print([io::IO = stdout,] [data::Vector]; kwargs...)

Prints profiling results to io (by default, stdout). If you do not supply a data vector, the internal buffer of accumulated backtraces will be used.

The keyword arguments can be any combination of:

• format – Determines whether backtraces are printed with (default, :tree) or without (:flat) indentation indicating tree structure.

• C – If true, backtraces from C and Fortran code are shown (normally they are excluded).

• combine – If true (default), instruction pointers are merged that correspond to the same line of code.

• maxdepth – Limits the depth higher than maxdepth in the :tree format.

• sortedby – Controls the order in :flat format. :filefuncline (default) sorts by the source line, :count sorts in order of number of collected samples, and :overhead sorts by the number of samples incurred by each function by itself.

• noisefloor – Limits frames that exceed the heuristic noise floor of the sample (only applies to format :tree). A suggested value to try for this is 2.0 (the default is 0). This parameter hides samples for which n <= noisefloor * √N, where n is the number of samples on this line, and N is the number of samples for the callee.

• mincount – Limits the printout to only those lines with at least mincount occurrences.

• recur – Controls the recursion handling in :tree format. :off (default) prints the tree as normal. :flat instead compresses any recursion (by ip), showing the approximate effect of converting any self-recursion into an iterator. :flatc does the same but also includes collapsing of C frames (may do odd things around jl_apply).

source
print([io::IO = stdout,] data::Vector, lidict::LineInfoDict; kwargs...)

Prints profiling results to io. This variant is used to examine results exported by a previous call to retrieve. Supply the vector data of backtraces and a dictionary lidict of line information.

See Profile.print([io], data) for an explanation of the valid keyword arguments.

source
Profile.initFunction
init(; n::Integer, delay::Real))

Configure the delay between backtraces (measured in seconds), and the number n of instruction pointers that may be stored. Each instruction pointer corresponds to a single line of code; backtraces generally consist of a long list of instruction pointers. Default settings can be obtained by calling this function with no arguments, and each can be set independently using keywords or in the order (n, delay).

source
Profile.fetchFunction
fetch() -> data

Returns a copy of the buffer of profile backtraces. Note that the values in data have meaning only on this machine in the current session, because it depends on the exact memory addresses used in JIT-compiling. This function is primarily for internal use; retrieve may be a better choice for most users.

source
Profile.retrieveFunction
retrieve() -> data, lidict

"Exports" profiling results in a portable format, returning the set of all backtraces (data) and a dictionary that maps the (session-specific) instruction pointers in data to LineInfo values that store the file name, function name, and line number. This function allows you to save profiling results for future analysis.

source
Profile.callersFunction
callers(funcname, [data, lidict], [filename=<filename>], [linerange=<start:stop>]) -> Vector{Tuple{count, lineinfo}}

Given a previous profiling run, determine who called a particular function. Supplying the filename (and optionally, range of line numbers over which the function is defined) allows you to disambiguate an overloaded method. The returned value is a vector containing a count of the number of calls and line information about the caller. One can optionally supply backtrace data obtained from retrieve; otherwise, the current internal profile buffer is used.

source
Profile.clear_malloc_dataFunction
clear_malloc_data()

Clears any stored memory allocation data when running julia with --track-allocation. Execute the command(s) you want to test (to force JIT-compilation), then call clear_malloc_data. Then execute your command(s) again, quit Julia, and examine the resulting *.mem files.

source


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Visual C++ Compiler Error

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I recently uninstalled Visual C++ 6 Standard and Installed Visual C++ 6 Professional. The sole purpose was to be able to use SSE2 on my Pentium IV 2.4GHz. After installing Visual Studio I installed Serive Pack 5 then the Processor Pack designated for Service Pack 5. I then went on to making a simple program with inline SSE2 assembly. I found that quite a few opcodes would not compile but I was able to work around them for the time being. My main problem is that I can''t get the STL complex class to work now. The are probably other problem I have yet to run into. I tried to compile the following program: #include <iostream.h> #include <complex> int main() { complex a(3,4); cout << a << endl; return 0; } only to get the errors: complextest.cpp(7) : error C2065: ''complex'' : undeclared identifier complextest.cpp(7) : error C2062: type ''double'' unexpected complextest.cpp(8) : error C2065: ''a'' : undeclared identifier

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std::complex
or
using namespace std;

.lick

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Another error is your use of <iostream.h> (in conjunction with the standard headers even!) rather than the standard <iostream>

Qui fut tout, et qui ne fut rien


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# In Bohr’s hydrogen atomic model, the radius of the stationary

orbit is directly proportional to (n = principal quantum number)

1. n2
2. n-2
3. 2n2
4. 3n3

## In Bohr’s hydrogen atomic model, the radius of the stationary orbit is directly proportional to

In Bohr’s hydrogen atomic model, the radius of the stationary orbit is directly proportional to n2.

$\mathbf{r_{n}&space;=&space;0.53&space;\frac{n^{2}}{z}A^{0}}$
⇒  $\mathbf{r_{n}&space;\propto&space;\frac{n^{2}}{z}}$