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# An interesting explanation about steppers voltage rating

Discussion in 'Motors' started by dddman, Oct 29, 2014.

1. ### dddman Master Builder

Joined:
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Took from this website: http://www.selene.co/Blog/2011/07/stepper-motor-voltages-explained/

Stepper Motor Voltages Explained
Posted on July 8, 2011 by Collin Stoner
Stepper motor ‘voltage’ is an often confusing subject. Many motor drivers use widely varying bus voltages to operate stepper motors. In fact, identical stepper motors can be (and commonly are) operated at greatly different voltages in different systems. How can the same motor be run at different voltages? How does the bus voltage impact the stepper motor’s performance?

Current and torque – Some Background
Stepper motors come with a nameplate current rating, winding resistance, output torque and voltage rating. The important value is the current rating. Torque is generated proportionally to the winding current. The current rating on the nameplate is the winding current required to produce the rated torque output of the motor. So where does the nameplate voltage ‘rating’ come from?

The voltage ‘rating’ of the motor is simply derived from the motor’s rated current and winding resistance. An example motor shown above is rated for 5 [A] (per winding), with a winding resistance of .5 [Ω]. The motor’s nameplate voltage is therefore listed as 2.5 [V].

This is the voltage that produces 5 [A] of current into the winding in steady state. However, a stepper motor must change the current in its windings rapidly and will not always run in steady state. Unfortunately, inductance is associated with the winding (since it is a large coil of wire). The winding inductance prevents current from changing instantaneously and will require time to increase the current flowing through the winding after a voltage is applied to it.

Some Analysis
A simplified, but instructive model of the motor winding is the familiar R-L circuit:

if $[IMG]$ is the applied input voltage, this circuit can be modeled by the familiar equation:

$[IMG]$
which generates the Laplace Domain Transfer Function:

$[IMG]$
Using this transfer function, the dynamics of the stepper motor’s winding can analyzed. By applying a step input of the motor’s rated voltage (that is, applying 2.5 [V] at t = 0[sec]) the problem becomes clearer:

The inductive nature of the motor winding is now obvious. While the winding eventually reaches 5 [A], it takes nearly 10 milliseconds to reach 90% and 16 milliseconds to settle at 98% of its final value!

While this might sound fast, consider this: A common stepper motor (including our example motor) requires 200 steps to complete a single revolution. If the driver waits 16 milliseconds for each step to reach full torque, it will take a full 3.2 seconds to complete a single revolution! This is unacceptably slow at 18.75 rpm. Attempting to step faster than this with only 2.5 [V] available means that the motor will not reach full torque output.

By applying closed loop control, the rise time can be improved dramatically. PI control is commonly employed for this purpose (see AN1307 at Microchip) by bipolar chopper drives like the SE-100.

By applying control, the rise time is greatly improved. What is important here is not the control, but the control effort – the voltage that is applied to the motor winding. It starts at 5 [V] – that is twice the nameplate voltage! In time, the applied voltage decays to the steady state ‘rated’ voltage of the motor but it arrives much faster than the natural response did. That is the purpose of a chopper drive and it is the reason that large bus voltages are required. Increasing the available voltage to the drive decreases the time it takes to reach steady state, increasing the motor’s maximum possible operating speed:

Even with only 5 [V], twice the rated voltage, a significant improvement is made over the 16 millisecond rise time of the natural response. Further increasing the voltage creates even faster responses.

Benefits of Higher Bus Voltage
The primary benefit, as discussed in the last section is the improvement in rise time, which translates to a higher possible operating speed.

Other benefits of an increased rise time include higher operating torque and higher mechanical power output. As the rise time improves, the winding will spend longer periods of time at its rated current, therefore producing a higher time averaged torque.

Since torque output is now higher, the mechanical output power will be higher per rpm (Since $[IMG]$). Caution: this is not to say that all higher torque motors are more powerful as they often trade speed for torque – a discussion beyond the scope of this writing.

Undesirable Side Effects
As with any engineering decision, a compromise must be made. Running the motor with higher voltage will cause it to run stiffer. Since the torque output is increased, the motor will tend to snap into position, causing it to vibrate. This can become a problem if the motor vibrates too much – It could jump out of step or begin to resonate. Because of this vibration, the motor will also produce more audible noise.

Secondly, with increased current resulting from the higher applied voltage comes an increase winding power loss. This power loss increases the temperature of the motor, which can damage the motor if it is to great.

Closing Thoughts
Stepper motor drive bus voltages are a tool that can be used to tune the performance of your system. By increasing the bus voltage the driver runs on, the speed and torque and power output of the motor all increase. Because of these benefits, stepper motors are operated with bipolar ‘chopper’ drives at voltages many times their nameplate voltages. Generally, 12 [V] is the smallest voltage used to drive actuator motors, with higher voltages at 24 [V], 48 [V] and even 80[V] being used for motion control systems. A good rule of thumb is to use between 10 and 24 times the motor’s nameplate voltage for the system bus voltage.

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# Empty Set (Null or Void Set) in Math – Symbol and Examples

Here you will learn what is an empty set in math, its symbol and definition with examples.

Let’s begin –

## What is the Empty Set in Math ?

Definition : A set is said to be empty set if it has no element. It is also called null or void set.

It is denoted by the symbol $$\phi$$.

In Roster method, $$\phi$$ is denoted by { }.

It follows from the definition that a set A is an empty set if the statement x $$\in$$ A is not true for any x.

Example 1 : { x $$\in$$ R : $$x^2$$ = -1 } = $$\phi$$

Example 2 : The set A given by A = { x : x is an even number greater than 2 } is an empty set because 2 is the only even prime number.

A set consisting of atleast one element is called a non-empty or non-void set.

Note : If A and B are any two empty sets, then x $$\in$$ A iff (if and only if) x $$\in$$ B is satisfied because there is no element x in either A or B to which the condition may be applied. Thus A = B. Hence, there is only one empty set and we denote it by $$\phi$$.

Note : The power set of an empty set has only one element i.e. P(A) = {$$\phi$$}.

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2019
Том 71
№ 11

# Linear groups with minimality condition for some infinite-dimensional subgroups

Abstract

Let $F$ be a field, let $A$ be a vector space over $F$, and let $GL(F, A)$ be the group of all automorphisms of the space $A$. If $H$ is a subgroup of $GL(F, A)$, then we set aug $\dim_F (H) = \dim_F (A(ωFH))$, where $ωFH$ is the augmentation ideal of the group ring $FH$. The number ${\rm{aug} \dim}_F (H)$ is called the augmentation dimension of the subgroup $H$. In the present paper, we study locally solvable linear groups with minimality condition for subgroups of infinite augmentation dimension.

English version (Springer): Ukrainian Mathematical Journal 57 (2005), no. 11, pp 1726-1740.

Citation Example: Dixon M. R., Evans M. J., Kurdachenko L. A. Linear groups with minimality condition for some infinite-dimensional subgroups // Ukr. Mat. Zh. - 2005. - 57, № 11. - pp. 1476–1489.

Full text


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# Tween

Tween class

Interpolates between its begin and end values.

Just defines how to interpolate between two values.

Many types have specific Tween subclasses that provide type-specific interpolation.

You can define your own interpolations by creating your own subclass of Tween and overriding its lerp function.

Tween on api.flutter.dev

### Combine a tween with an animation

To get a concrete value for the current frame of an animation, you also need an animation to determine the current state.

Tween

## Tween — Structure map

Clickable & Draggable!


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### Overview

We are all home and bored, so it feels like a good time to host a tournament. This is going to be a combination of a mapping and jumping tournament. A general outline of the tournament is below.

• Mapping contest maps are made
• Mapping contest maps are judged
• Sign-ups are on our Discord
• If you are new, and want to try to get involved, ask us questions on discord to get set up.

Current Prizepool: Total: $500 — Ace:$200, Goblin: $100, Opkez:$100, Koala: $50, Caboom:$50

Twenty minute time limits for each map, played live. All the maps will be brand new. It will start with two heats, with the top four from each moving directly to the finals. The remaining people will go to a last chance qualifier, where the top two will move to the finals. In the finals, the top five will get a prize. The points distribution is below.

Place Points Place Points
1 15 6 5
2 12 7 4
3 10 8 3
4 8 9 2
5 6 10 1

### Results

Here are the results of the tournament. The people who advanced have a * next to their name in the heats and LCQ. In the finals we had Goblin take 1st, Killa take 2nd, ace take 3rd, slip take 4th, and yogurt take 5th. APKIS, prd, bossmies, neox, and opkez rounded out the remaining finalists. We had an incredible turn out for this being our first tournament in years with some good games and great times. Thanks to everyone old and new who showed up to make this a success.

##### Heat 1
pos player cc_neox1 cc_ace1 cc_mehis cc_mako total
1 goblin* 1 1 1 2 57
2 APKIS* 3 2 2 1 49
3 prd* 4 3 3 3 38
4 754* 2 4 4 5 34
5 bossmies 8 5 6 4 22
6 barry_allen 6 6 9 6 17
7 mazze 7 9 5 7 16
8 zerosignal 5 8 7 9 15
9 direktor 9 7 8 8 12
10 gouki 10 10 10 3
11 Woeste Wessel 11 0

##### Heat 2
pos name cc_slip2 cc_zero1 cc_play cc_direktor1 total
1 opkez* 4 6 1 1 43
2 killa* 3 2 4 2 42
3 yogurt* 2 1 5 5 39
4 ace* 1 3 7 3 39
5 mehis 8 4 3 8 24
6 kallu 6 9 2 7 23
7 zod 9 5 8 4 19
8 caboom 7 8 9 6 14
9 neox 5 6 11 11
10 sluggy 10 7 11 9 7
11 mako 11 10 10 10 3

##### LCQ
pos name cc_slip1 cc_opkez cc_mako cc_phx1 cc_ace2 cc_slip2 total
1 neox* 4 1 2 5 6 3 56
2 bossmies* 8 4 1 6 1 6 51
3 mazze 6 3 3 3 2 47
4 barry_allen 1 10 5 7 4 9 36
5 zod 3 12 4 3 8 11 31
6 caboom 8 6 9 5 1 31
7 mehis 5 7 8 4 7 5 31
8 direktor 9 2 12 1 13 12 29
9 kallu 11 7 2 4 24
10 zerosignal 10 5 11 2 11 7 23
11 play 2 9 10 10 8 19
12 mako 7 6 9 8 9 10 17
13 gouki 13 13 12 13 0
14 Woeste Wessel 14 0

##### Final
pos name cc_mako2 cc_direk2 cc_zero2 cc_neox2 cc_slip3 cc_barry1 total prize
1 goblin 1 1 4 1 1 1 83 $100 2 killa 5 2 1 10 5 2 52$63
3 ace 8 5 2 3 3 5 47 $50 4 754 10 6 3 5 2 6 39$25
5 yogurt 3 4 6 7 4 9 37 \$13
6 APKIS 2 7 5 6 8 10 31
7 prd 9 3 10 8 6 3 31
8 bossmies 4 8 9 9 7 4 27
9 neox 7 10 7 2 8 24
10 opkez 6 9 8 4 9 7 24

### Mapping Contest

The mapping contest is very open ended with only a few limitations. These maps will be used to fill the mappool for the tournament following it. They should be designed around the idea of playing them for 20-30 minutes. Mappers can submit as many suitable maps as they want. The constraints are:

• The approximate perfect time is between 20 and 60 seconds
• The difficulty should be anywhere from easy to medium, nothing super hard
• Any style of map is fine, ie. ice, tech, strafe, tower maps
• Judging will be based on gl_modulate 3, intensity 1.85

#### Judging:

The judging will be done via a survey of players who participated in the cup and people who have played the maps on the main server. You can find a link to previews of the maps here.

Category Points Description
Design 10 How does it look?
Gameplay 10 How does it play competitively?
Total 20


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# PCA leads to some highly Correlated Principal Components

I have a dataset of 64 highly-correlated variables (with many being interactions of others for example) plus a date and the target variable for a total of 66. I want to derive the principal components of the 64 independent variables and choose the final predictors among those (I don't want to use principal components regression directly as it must choose the first n components instead of choosing the statistically significant ones).

Code in R:

PC=prcomp(mydata[,3:66], center=TRUE, scale=TRUE)
PrincipalComponents=as.data.frame(PC$x)  I would expect Principal Components to be orthogonal and essentially uncorrelated (as apparently these aren't quite as interchangeable as I had thought). cor(PrincipalComponents$PC1, PrincipalComponents$PC2)  returns a number very close to 0, as expected and so do most pair-wise correlations. However, cor(PrincipalComponents$PC1, PrincipalComponents$PC63) cor(PrincipalComponents$PC1, PrincipalComponents$PC64)  return 0.61 and 0.41. These high correlations are only observed for the last two principal components. Running a regression using the principal components as independent variables, I get a VIF>10 on nearly 30 of them. Running a regression but excluding the last two principal components, I get VIF=1 for all predictors. Am I missing something? If that is something that is expected, doesn't that defeat the purpose of PCA? I'd like to understand what is going on here before I make the decision to exclude those 2 components for example. • Non-linear associations? – Alexis May 30 at 17:41 ## 1 Answer You're right that the principal components should all be mutually orthogonal, so this is not expected. I think you probably have columns in your data matrix which are linearly dependent. If the column rank of your data matrix is < 64, it is not possible to find 64 mutually orthogonal vectors in its column space. It might be better behavior for the package to return an error, or maybe fewer than 64 columns, in this case, but that is not what prcomp does. Example: m <- matrix(c(1,0,0,0,0,0,1,0,0), nrow=3) cor(prcomp(m)$x)

Claims there are three principal components with pairwise correlation +/- 1. Something similar is probably happening in your data.


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# Explore Parametric Equation, Calculus and more!

### Explore related topics

Slope, length, area for polar curves in calculus is one of the most interesting and bit complex topic of Calculus. In this section, we will look at areas enclosed by polar curves. Here, we used the word “enclosed” instead of “under”.

At last, a practical use for calculus.

statest: @pickover: Sometimes, the exotic formulas of Indian mathematician Ramanujan (1887-1920) make me shiver a little. You?

Limit Practice Problems

Limits at Infinity Rules

n calculus, antiderivative is an operation which perform opposite operation on derivatives or we can say that antiderivative performs integration operation because integration is opposite form of differentiation .

The definition of a limit is not concerned with value of f(x) when, x=c. So, we care about the values of f(x) when x is close to c, on either the left side or right side.Now, we move on to rules regarding limits. Limits have some rules which are useful when we solve different limit problems. First Rule: First rule is called as the constant rule. In this rule we state- if we have f(x) =b (where f is constant for all x) then, the limits as x approaches c must be equal to b.

Sometimes I try to picture what everything would look like if the whole spectrum were compressed into the visible spectrum. Also sometimes I try to picture your sister naked.


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# Spindle bearings

## Description

Spindle bearings are a special type of single row angular contact ball bearings. They are often used in the manufacturing of machine tools, as well as in ranges of application with high to extremely high demands in terms of accuracy or permissible speed required by the mounting/ bearing installation.
In comparison with standard angular contact ball bearings, spindle bearings are distinguished by the contact angle, the tolerance classes, and the cage design. Spindle bearings can be delivered with a contact angle of 15° or 25°.

Spindle bearing BS series
The BS719 and BS70 series use a larger number of smaller diameter balls than the B719 and B70 series with identical external dimensions. As a result, higher speeds can be achieved compared to the standard series. The BS series represents a high-performance bearing concept for demanding spindle units that is economical for the customer.

In their main dimensions, high-speed spindle bearings are identical to spindle bearings in the B series. They are particularly characterized by their suitability for higher speeds, as well as by lower friction and heat build-up, and are manufactured in the HS719 and HS70 series.

## Dimensions/standards

In spindle bearings, the main dimensions are standardized according to DIN 628-6 (Radial-angular contact ball bearings, single row, contact angle 15° and 25°), ISO 15 (Radial bearings – boundary dimensions) or DIN 616 (Roller bearings – boundary dimensions).

## Tolerances

SLF normally produces spindle bearings in P4S according to DIN 620-2 (Roller bearing tolerances – tolerances for radial bearings) and ISO 492 (Radial bearings – dimensions and tolerances).
For P4S, the main directions are defined according to ISO tolerance class 4, while all form and sliding tolerances are set according to ISO tolerance class 2.
Spindle bearings with limited diameter tolerance can be delivered in P4S.K5 upon request.

## Bearing design types

Spindle bearings are produced in the B719, B70, B72 and A73 series. They can also be manufactured in various material combinations and designs.
To make it possible to assemble the cage and lubricate the bearing optimally during operation, at least one ring has a stepped shoulder diameter. The bearings are not dismountable.
SLF produces spindle bearings using vacuum- degassed 100Cr6 chromium steel or an equivalent material. For special applications, SLF produces the spindle bearings with ceramic balls and roller bearing rings made of Cronidur® 30.
SLF normally manufactures the spindle bearings in the universally used UL, UM and US designs, which can be mounted in any arrangement. But we can also deliver matched bearing sets with defined preload of the bearing.

### Bearing arrangements

#### O-arrangement (designation suffix DB)

The thrust lines diverge in the bearing axis direction. This results in a large support width H on the bearing axis. This arrangement provides a very stiff mounting that withstands the tilt moment, and the mounting carries axial forces in two directions.

#### X arrangement (designation suffix DF)

The thrust lines converge in the bearing axis direction. This results in a short support width H on the bearing axis. Tilting stiffness is lower than in the O arrangement. The arrangement better withstands alignment errors. Load-suspension and bearing spring resistance are analogous to the O arrangement.

#### Tandem arrangement (designation suffix DT)

Both paired bearings are arranged in parallel in the load direction. As a result of this arrangement, in the load direction, a higher axial load is possible than with a single bearing. Each of the two bearings carries the same amount of the axial load. However, it must be noted that the tandem pair must be set against a third bearing in each case.

#### Multiple bearing arrangements

At higher loads or if high stiffness values are required, then 3, 4, and sometimes even 5 bearings are put together into sets and installed. At SLF, bearings of this pairing type are manufactured, marked, and packed in pairs or sets.

### Distance washers

For spindle bearings in universal design, distance washers may be used in all arrangements. When installing distance washers between the paired bearings, it is possible to, for instance, increase support width H, enhance lubrication, reduce or eliminate frictional heat, and vary the preload. Information about the difference sizes required is available upon request. When manufacturing the distance washers, ensure that the outer and inner ring have exactly the same width and are plane-parallel.

Tolerances for the outer and inner distance washers

 Tolerance class P4 P4S, P2 Difference in width between inner and outer distance washer 3 2 Variation of width 2.5 1.3 Axial runout 2.5 1.3

maximally permissible values in µm

### Hybrid spindle bearings

Hybrid spindle bearings are bearings whose raceways are made of antifriction bearing steel, and whose balls are made of a ceramic material (silicon nitride Si3N4) with the highest homogeneity and hardness. At the same time, ceramic balls weigh less than steel bearings. These characteristics result in lower centrifugal forces and less friction. They are electrically insulating and are non-magnetic. Moreover, they withstand corrosion. These bearings are  specifically engineered to serve as high-performance bearings for machine tool spindles and provide the prerequisites for top performance.

Originally, hybrid bearings were primarily used in the high-speed range. However, thanks to their lifetime being the same or longer than that of steel bearings, they are also in use in all other bearing design series.

Due to the outstanding properties of ceramic, hybrid bearings are characterized by significantly reduced friction during operation, and consequently have the following advantages over steel balls:

• withstand about 20% higher speeds
• better emergency run properties in case of poor lubrication
• higher bearing stiffness
• lower vibrations
• less noise level
• better adapted acceleration and deceleration characteristics

## Preload

The predefined axial forces are: low (L), medium (M) and high (S).
The preloads specified in the bearing tables are applied to the axial preload of bearing pairs (in O or X arrangement). When combining more than 2 bearings, multiply the preloads as given below:

 Pair (Matching) Factor DB, DF 1 TBT, TFT 1.35 QBT, QFT 1.6 QBC, QFC 2.0

## Cage

SLF normally equips spindle bearings with a one-piece, outer shoulder-guided solid window cage made of laminated fabric (a cotton fabric saturated with phenolic resin). The cages made of this high-performance material withstand high centrifugal and acceleration forces, as well as working temperatures up to 100 °C. This material has excellent tribologic characteristics; when lubricant is insufficient, it is characterized by outstanding emergency running properties.

For working temperatures above (>) 100 °C, we recommend the use of brass cages or PEEK cages. The maximal working temperature of PEEK cages is 150 °C.

## Working temperature

SLF spindle bearings with an outer diameter up to 240 mm are standard stabilized in dimensions by S0, meaning that they are heat treated to make them usable up to a working temperature of 150 °C. At greater than 240 mm outer diameter, the deep groove ball bearings are standard stabilized in dimensions with S1, meaning that they are heat treated to make them usable up to a working temperature of 200 °C. However, the maximal operating temperature is not limited by the dimensional stability of the bearing rings and balls. In many cases, it is limited by the cage, the seals, or the lubricant.

 Component Upper range of use in terms of temperature (Upper working temperatures) Roller bearing rings 150°C Laminated fabric cage (standard) 100°C Brass cage 150°C PEEK cage ca. 260°C (150°C without limitation in performance) Packing disks made of NBR (2RSD) 110°C Lubricant grease L252 (standard) 120°C

If you are uncertain or have specific questions regarding the temperature limits of our bearings, the SLF team is ready to support you.

## Greasing/lubrication & sealing

Spindle bearings with sealed design (2RSD) are – as specified in the bearing tables – part of the SLF manufacturing program. Greasing/ lubrication of these maintenance-free bearings will allow the bearings to run at the highest speeds at low temperatures over a long period of time.
The low outlay necessary for assembly, lubrication, and maintenance offers an optimal solution for a long period of service for the user.

Unsealed ungreased spindle bearings must be lubricated with oil or grease by the user. Here, the bearing position must be sealed around / on the surrounding components.
Bearings of the same size and series in unsealed designs are interchangeable with the same bearings in sealed designs.

## Dimensioning

Spindle bearings must guide machine parts very precisely and simultaneously transfer forces even at very high speeds in a reliable manner. The bearings are primarily chosen according to precision, stiffness, and running characteristics. In many applications, the bearings must be fatigue endurable.

To check the fatigue limit, the load ratio S0* is calculated by means of the equation below:

$$S_0 = \frac{C_0}{P_0}$$
 S0* Load ratio for fatigue limit (dynamic load safety factor) [-] C0 Static load rating [N] P0* Equivalent load P0* is calculated by means of the forces of the dynamic load according to the equation of the static equivalent load. [-]

The minimal value for the load ratio S0* depends on the bearing design type and the component material.
To provide sufficient fatigue strength, the load ratio S0* should achieve the following values from the table:

 $$\frac{F_{0a}}{F_{0r}} \leq 1.09$$ $$\frac{F_{0a}}{F_{0r}} > 1.09$$ $$\frac{F_{0a}}{F_{0r}} \leq 1.20$$ $$\frac{F_{0a}}{F_{0r}} > 1.20$$ $$\frac{F_{0a}}{F_{0r}} \leq 1.30$$ $$\frac{F_{0a}}{F_{0r}} > 1.30$$ Bearing design type Contact angle 15° Contact angle 20° Contact angle 25° B / HCB 8 12 - - 8 10 HS / HC 8 12 - - 8 10 BS / HCBS - - 8 11 - - XC 3 4 - - 3 4

### Static equivalent load

Static equivalent bearing load P0 is calculated from the loads affecting the bearing in the axial and radial directions, according to the formula below.

 $$P_0 = F_{0r}$$ for $$\frac{F_{0a}}{F_{0r}} \leq e$$ $$P_0 = 0.5 * F_{0r} + Y * F_{0a}$$ for $$\frac{F_{0a}}{F_{0r}} > e$$
 Contact angle e Y 15° 1,09 0,46 20° 1,20 0,42 25° 1,30 0.38
 P0 static equivalent load [kN] F0r radial static load [kN] F0a axial static load [kN] e, Y factor [-]

### Static load safety factor

To use the high accuracy of spindle bearings, the static load safety factor S0 must be >2.

$$S_{0} = \frac{C_0}{P_0}$$
 S0 static load safety factor [-] C0 static load rating [kN] P0 static equivalent load [kN]

### Speed limits in bearing arrangements

The speed limits listed in the product table are reduced in the case of rigidly or more highly preloaded bearings, as well as for bearing pairs or groups. The specified speed must be multiplied by the reduction factor fr.

Drehzahlreduktion für Spindellagersätze $$n * f_r$$

$$Factor f_r$$
bearing preload
low medium high
L M S
inter-bearing distance large
0,85 0,75 0,5
0,8 0,7 0,5
0,75 0,65 0,45
Fixed bearing Floating bearing
0,75 0,6 0,35
0,65 0,5 0,3
0,65 0,5 0,3
0,72 0,57 0,37
0,54 0,4 0,37


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## More axis support in the joystick plugin

Requests or suggestions for new features
darius
Posts: 5
Joined: 15 Apr 2014, 13:15

### More axis support in the joystick plugin

Hi,

I am developing control for a robot using the Logitech RumblePad 2 and the existing windows joystick plugin. This works great with one exception: it only supports 3 axis. And I need very much the 4th axis for my complex control. I hope very much that is not too much effort to add the support for the 4th axis and that it could be added before not too long. It looks like there is no need to change the existing joystick API.

The d-pad also is not supported, but this is not a big issue. For not the 10 buttons that work is a good solution.

Did I understood correctly that I have lots of trouble if I want to run ROS on Windows? If yes, then would be your suggestion for an alternative path to create my own full support for the RumblePad? I would like to continue using the Windows platform.

Best regards,
Darius

coppelia
Posts: 7857
Joined: 14 Dec 2012, 00:25

### Re: More axis support in the joystick plugin

Hello Darius,

the current implementation of the joystick plugin returns only 3 axis and the button states (i.e. 2 data sets). We have modified the plugin so that it returns more infos (i.e. 5 data sets):

Code: Select all

table_3 axes, number buttons, table_3 rotAxis, table_2 sliders, table_4 PovValues = simExtJoyGetData(number joystickIndex)

The returned values correspond to the first 9 entries in this page. Not sure if your 4th axis will be mapped onto one of those.

The updated plugin, with source code can be downloaded here.

If you want to recompile the plugin yourself, you will have to download the DirectX SDK.

Cheers

darius
Posts: 5
Joined: 15 Apr 2014, 13:15

### Re: More axis support in the joystick plugin

Thank you very much for the fix and advices.
I will try to recompile my self.

darius
Posts: 5
Joined: 15 Apr 2014, 13:15

### Re: More axis support in the joystick plugin

OK, I found that the compiled DLL was already included in the zip, so I did not try to compile it myself.

Everything seems to work as expected.

darekm
Posts: 2
Joined: 10 Mar 2015, 14:31

### Re: More axis support in the joystick plugin

It seems to be still an issue in the latest version (lack of axis[4]), can you provide the code for the plugin again ?

coppelia
Posts: 7857
Joined: 14 Dec 2012, 00:25

### Re: More axis support in the joystick plugin

Hello,

we haven't implemented any axis[4]. Just as described in our previous post above.
The source code can be found in programming/windowsOnlyProjects/v_repExtJoystick

Cheers

darekm
Posts: 2
Joined: 10 Mar 2015, 14:31

### Re: More axis support in the joystick plugin

Is there a manual somewhere which will tell me how to compile it?

coppelia
Posts: 7857
Joined: 14 Dec 2012, 00:25

### Re: More axis support in the joystick plugin

There is a VS2005 project file. But you will also need to install the DirectX SDK.

Cheers

skitano
Posts: 1
Joined: 14 Feb 2018, 10:39

### Re: More axis support in the joystick plugin

Hi,

I know this thread was closed long time ago, but I also wanted to read 4th axis of the joystick.
But In my case which uses Logicool gamepad F310, rotAxes[3] is the 4th axis of the joystick!

With this plugins, it became lot easier to control v-rep model, thanks!

coppelia


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Friday, March 4, 2022 — 4:00 PM EST

Nicole Kitt, Pure Math Department, University of Waterloo

"Buffon's Needle Problem"

In 1777, Georges Louis Leclerc de Buffon posed the following problem: If one were to drop a needle on ruled paper with equally spaced lines, what is the probability that the needle lands in a position where it crosses at least one of the lines? This answer will of course depend on many things, such as the length of the needle, the distance between two successive lines, etc. In the case that the length of the needle is at most the distance between two successive lines on the paper, Buffon found an elegant answer to this problem which happens to involve $\pi$. This is neat because it means that one can perform an experiment, dropping a bunch of needles onto ruled paper, to get an approximate value for $\pi$. In this talk, we will discuss Buffon's problem in more detail, provide some interesting history, and sketch a proof for the specific case mentioned above.

Location: Online

### January 2023

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text:
A can do as much work as B and C together can do. A and B can together do a piece of work in 9 hours 36 minutes and C can do it in 48 hours. The time that B needs to do the work alone, is :
[A] 12 hours
[B] 18 hours
[C] 24 hours
[D] 30 hours


text:
• Selective control of HOD photodissociation using CW lasers

• # Fulltext

https://www.ias.ac.in/article/fulltext/jcsc/119/05/0377-0384

• # Keywords

Selective bond-breaking; photodynamic control; photodissociation; CW laser; field optimized initial state (FOIST).

• # Abstract

Selective control of HOD photodissociation (H-O + D $\leftarrow$ HOD → H + O-D) has been theoretically investigated using CW lasers with appropriate carrier frequency and |0, 0$\rangle$, |0, 1$\rangle$ and |0, 2$\rangle$ with zero quantum of excitation in the O-H bond and zero, one and two quanta of excitation in the O-D bond as the initial states. Results indicate that the O-H bond in HOD can be selectively dissociated with a maximum flux of 87% in the H + O-D channel from the ground vibrational state |0, 0$\rangle$. For the O-D bond dissociation, it requires two quanta of excitation (|0, 2$\rangle$) in the O-D mode to obtain 83% flux in the H-O + D channel. Use of a two colour laser set-up in conjunction with the field optimized initial state (FOIST) scheme to obtain an optimal linear combination of |0, 0$\rangle$ and |0, 1$\rangle$ vibrational states as the initial state provides an additional 7% improvement to flux in the H-O + D channel as compared to that from the pure |0, 1$\rangle$ state.

• # Author Affiliations

1. Department of Chemistry, Indian Institute of Technology, Powai, Mumbai 400 076
2. Department of Chemistry, Indian Institute of Technology, Guwahati, North Guwahati 781 039

• # Journal of Chemical Sciences

Volume 133, 2021
All articles
Continuous Article Publishing mode

• # Editorial Note on Continuous Article Publication

Posted on July 25, 2019


text:
Challenges

+3
−0

# Challenge

Given an array in any suitable format, create a frequency table for it. i.e: Pair each unique element with the number of times it appears in the array.

You can return the frequency table as a list of pairs, hashmap/dictionary, output the pairs directly, etc.

# Tests

{ 1 1 2 3 5 6 6 }           -> { { 1 2 } { 2 1 } { 3 1 } { 5 1 } { 6 2 } }
{ }                         -> { }
{ 1 6 7 1 2 6 6 1 53 4 10 } -> { { 1 3 } { 2 1 } { 4 1 } { 53 1 } { 6 3 } { 7 1 } { 10 1 } }

Why does this post require moderator attention?
Why should this post be closed?

Empty arrays (1 comment)

+1
−0

# J, 8 bytes

~.,:#/.~

Tacit function, this is the de facto method for this problem in J.

Attempt it online!

Why does this post require moderator attention?

+1
−0

# Factor, 46 bytes

USE: math.statistics
IN: h
ALIAS: h histogram

Why does this post require moderator attention?

+1
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# Python 3, 3836 33 bytes

lambda x:{n:x.count(n)for n in x}

-2 bytes thanks to @Razetime

Another -3 bytes thanks to @orthoplex

Attempt it online!

Why does this post require moderator attention?

+0
−0

# Lua, 98 bytes

function(t)r={}for k,v in pairs(t)do r[v]=0 end for k,v in pairs(t)do r[v]=r[v]+1 end return r end

Attempt This Online!

Why does this post require moderator attention?

golf (1 comment)
+0
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# JavaScript (Node.js), 38 bytes

(a,d={})=>(a.map(v=>d[v]=1+d[v]||1),d)

Try it online!

Basically just this but returning the result

Why does this post require moderator attention?

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−0

# Ruby, 12 bytes

->{_1.tally}

Attempt it online!

Why does this post require moderator attention?

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# C (gcc), 45, 42 bytes

t[9999];f(s,a)int*a;{for(;s;)t[a[--s]]++;}

Try it online!

Assumptions:

• Passing the array size to a function can be used as a means to deal with empty arrays (not supported in C).
• A table of integers can be regarded as a hashtable with an integer value as hash key and the value's frequency as data. Or if you will: a key-value pair.
• Maximum integer value 9999.
• The global variable t may have its contents printed outside the function.
Why does this post require moderator attention?

+0
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# Vyxal, 1 byte

Ċ

Try it Online!

Yes, there really is a built-in for this.

Why does this post require moderator attention?

+0
−0

# APL(Dyalog Unicode), 4 bytes SBCS

Anonymous tacit prefix function.

,∘≢⌸

Try it on APLgolf!

⌸ between each unique element and its indices, apply:

,∘≢ concatenate the unique element to the tally of indices

Why does this post require moderator attention?


text:
• 24-hour customer service +34 91 524 24 00
• Help

# Flights to Tel Aviv

Find cheap flights

## Flights to Tel Aviv

Flights to Tel Aviv (Israel)

Fly to Tel Aviv (Israel) with Destinia.com. We offer you flights at economical prices or in the class you choose. To select your flights Tel Aviv, type the name of the city you wish to depart from in the Departure box and write Tel Aviv in the Arrival field. Then select the departure and return dates. You can buy several airline tickets to Tel Aviv at the same time by indicating the number of adults, children and babies travelling. You can filter the type of flight if you are only interested in direct flights to Tel Aviv (Israel), multiple destinations or flight stopovers.

## Airports near Tel Aviv

### Tel Aviv-Yafo Sde Dov (SDV):

The distance from Tel Aviv-Yafo Sde Dov airport to the centre of Tel Aviv is 6.95 Km

### Tel Aviv-Ben Gurion (TLV):

The distance from Tel Aviv-Ben Gurion airport to the centre of Tel Aviv is 12.118 Km

### Flights to Tel Aviv

Flights to Tel Aviv (Israel) most sought after by users, in the following chart we show the flights from some important cities to Tel Aviv (Israel) .

### Flights from Tel Aviv

Book your flights now from Tel Aviv (Israel) to the most important cities in the world. The box below contains flights from Tel Aviv (Israel) to the most important cities.

## Top destinations in Israel

Book your flights now from Tel Aviv (Israel) to the most important cities in the world. The box below contains flights from Tel Aviv (Israel) to the most important cities.


text:
# What am I missing in this integration?

Trying to solve diff.eq.has led me to:

{a, b} = {6, 7}
Do[With[{solve =
NSolve[{0.8 p (y^0.2/x^0.2) + x (x - 2 a)/(x - a)^2 == 0,
0.2 p (x^0.8/y^0.8) + y (y - 2 b)/(y - b)^2 == 0,
p^-2 == x^0.8 y^0.2, 0 <= x <= a, 0 <= y <= b}, {p, x, y}]},
k = Extract[solve, {1}]; l = Extract[solve, {2}]]
a = a + 0.001 (k + l) a;
b = b + 0.001 b k l, {100}]
{a, b}

but don't understand, why it fails?

• When you have a problem like this, try to construct a minimal example that shows the problem. Break the code into pieces and check what each piece gives separately. You will likely find the problem on your own. This example is far from minimal; as a start, do only one iteration instead of 100 and check the values of all your variables. – Szabolcs Aug 11 '15 at 17:53
• Your expression l = Extract[solve, {2}] does not work: there is no second part to the solution that is returned by NSolve. You also have syntax errors in the code you posted: you probably need a ; after the closing square bracket for With. – MarcoB Aug 11 '15 at 18:10


text:
Python rate limiting library using Redis for shared state.

Red Bucket

Python rate limiting library using Redis for shared state.

Installation

To install the latest released version:

pip install redbucket

To install the current development version from the master branch on GitHub:

pip install -U git+https://github.com/nickgaya/redbucket.git

Usage

The following snippet configures a rate limiter with two rate limiting zones:

from redbucket import RedisRateLimiter, RateLimit, Zone
from redis import Redis

# Accept up to 5 requests per user per second.
# Allow bursts of up to 10 requests.
user_zone = Zone('user', rate=5)
user_limit = RateLimit(user_zone, burst=10)

# Accept up to 20 requests per IP per second.
# Allow up to 10 excess requests with a delay.
ip_zone = Zone('ip', rate=20)
ip_limit = RateLimit(ip_zone, delay=10)

redis = Redis()
rate_limiter = RedisRateLimiter(redis)
rate_limiter.configure(user=user_limit, ip=ip_limit)

We can now use the rate limiter as follows:

def example_operation(user, ip_address):
if not response.accepted:
raise Exception("Rate limit exceeded")  # Reject request
if response.delay > 0:
sleep(response.delay)  # Wait for delay seconds
...  # Perform operation

Note that we don't have to specify a key for every zone. For example, we could exclude certain IP addresses from IP rate limiting while still applying the user rate limit, as follows:

def example_operation(user, ip_address):
response = rate_limiter.acquire(user=user)
else:
...

Implementations

The default RedisRateLimiter uses Redis's Lua script engine to atomically update rate limiter state. This implementation requires Redis 3.2 or greater. For older Redis versions, you can use the RedisTransactionalRateLimiter.

Where supported, the script-based implementation is recommended as it handles each rate limiting request in a single round-trip to the Redis server, whereas the transactional implementation performs several consecutive Redis commands per request.

State encoding

By default, rate limiter state is stored in Redis using a packed binary representation. You can switch to JSON for a less efficient but more human-readable encoding.

rate_limiter = RedisRateLimiter(redis, codec='json')

Rate limiting model

Red Bucket uses a rate limiting model inspired by Nginx. Rate limiting state is stored in one or more zones. Each zone has a name and a base rate. A zone represents a namespace of rate limiting keys. A rate limit references a zone along with two parameters, burst and delay. A rate limiter references one or more rate limits.

To apply rate limiting to an operation, the application makes a request to the rate limiter by specifying a key for each desired rate limit. The rate limiter evaluates each rate limit to determine whether to reject, delay, or immediately accept the request. If using multiple rate limits, the most restrictive outcome will be applied.

For a rate limit with base rate r, burst value b, and delay value d, the rate limiter maintains a virtual counter for each key that continuously increases at a rate of r requests per second up to a maximum value of b + 1. When a request arrives, the rate limiter checks the current value of the counter v. If v − 1 ≥ 0, the rate limiter decrements the counter by 1 and accepts the request immediately. If v − 1 ≥ −d, the rate limiter decrements the counter by 1 and accepts the request after a delay of (v − 1) / r seconds. Otherwise, i.e. if v − 1 < −d, the rate limiter rejects the request.

In practice, this means that from an initial idle state for a given key, the rate limiter will allow a burst of b + 1 requests immediately, throttle the next d requests to the desired rate, and reject any further requests past the limit.

Comparison with Nginx

As noted above, Red Bucket's rate limiting model is inspired by Nginx and uses a similar algorithm. However, the way burst and delay are specified is a little different. The following configuration examples illustrate the difference.

• Basic rate limiting

# nginx
limit_req zone=mylimit;

# redbucket
RateLimit(zone=mylimit)

• Burst with delay

# nginx
limit_req zone=mylimit burst=20;

# redbucket
RateLimit(zone=mylimit, delay=20)

• Burst with no delay

# nginx
limit_req zone=mylimit burst=20 nodelay;

# redbucket
RateLimit(zone=mylimit, burst=20)

• Two-stage rate limiting

# nginx
limit_req zone=ip burst=12 delay=8;

# redbucket
RateLimit(zone=mylimit, burst=8, delay=4)

Development

This project uses Tox to manage virtual environments for unit tests and other checks. Unit tests are written using the Pytest framework.

The unit tests require a Redis instance. By default, the tests attempt to connect to port 6379 of localhost. This can be overridden by setting the REDIS_URL environment variable.

export REDIS_URL=redis://localhost:6379

To start a Redis Docker container for running the tests, you can use the docker_redis.sh script. For example, to run the tests against a Docker redis instance, you can run:

./docker_redis.sh tox

You can also run the script without a command and execute output to set environment variables in your current shell.

eval "\$(./docker_redis.sh)"

By default, the script uses the redis:alpine image. You can supply a different tag for the redis image with the -t flag, or a different image name with -i.

Project details

Uploaded source
Uploaded py3


text:
# Faster growing busy beaver function

Standard busy beaver function draws attention to final count of nonzero symbols on tape. We could instead look at largest amount of nonzero symbols appearing on tape at any point of computation. This function's lower bound would be $\Sigma(n)$ and upper bound would be $S(n)$ (max shifts function). Was there made any research on such functions? If so, are there any known values for this?

-
its likely just as hard as BB fn which is uncomputable. one could also count the total # of 1's ever written, etc... – vzn Apr 11 '14 at 16:36

Suppose there is a machine using $n$ states which, at some point, has $X$ nonzero symbols on the tape. One can build a machine with $O(n)$ states which simulates a run of the original machine with a tape alphabet of three symbols $\{0,1,2\}$, with the following small change: whenever the original machine changes $1$ to $0$, the new machine changes it to $2$; whenever the tape is read, $2$ is interpreted as $0$. The new machine ends up with $Y \in [X,2X]$ nonzero symbols on the tape (we get $\Theta(X)$ instead of $X$ since the simulation requires using two symbols for each original symbol). So your new function, let's call it $F(n)$, satisfies $B(n) \leq F(n) \leq B(O(n))$.

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Is there any way to sharpen the $O()$ and $\Theta()$? Just curiosity (probably best left to the OP) – vonbrand Jan 27 '13 at 3:32
I have no idea. My guess is that on the scale from $B(n)$ to $B(O(n))$, $F(n)$ is closer to the latter. It may be possible to prove a lower bound on $F(n)$ somehow. Give it a try! – Yuval Filmus Jan 27 '13 at 6:34

Your function (call it $F$) obviously satisfies $$\Sigma(n) \le F(n) \le space(n)$$ where $space(n)$ is the maximum number of tape squares visited by any $n$-state halting TM in the same class as those considered in the definition of $\Sigma$. Furthermore, $$space(n) \le \Sigma(3n-1)$$ as proved in the following paper:

Ben-Amram, A. M., B. A. Julstrom, and K. Zwick. "A Note on Busy Beavers and Other Creatures." Mathematical Systems Theory 29.4 (1996).

Therefore $$\Sigma(n) \le F(n) \le \Sigma(3n-1)$$

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text:
# Account for radiation energy in the statement "total energy of universe is zero"

I've read in Hawking's book The Theory of Everything that total energy of the Universe is zero. He explained this by saying that if there are two bodies separated by a distance and having some mass, they possess potential energy and the same amount of negative energy with the gravitational field. Thus he concluded that total energy of the Universe is zero. But here I've a question: how do we account for energy which is in the form of radiation?

The total energy of the universe consists of the mass energy of all the matter (both normal and dark), the mass-energy of the radiation and of the dark energy plus the gravitational potential energy of the universe (which is negative). The hypotheses is that this sums to zero. As far as I know, there is no experimental evidence for this.

But there is good evidence for the size of the various mass-energy terms. The best current estimates are that the mass-energy of the universe is:

• 73% -- Dark energy
• 23% -- Dark matter
• 3.6% -- Interstellar gas
• 0.4% -- Stars (includes black holes)
• v. small -- Relativistic neutrinos

This balance changes. As the universe expands, the density of matter (dark and normal) varies as the inverse 3rd power of the size of the universe. (Matter stays the same and the volume expands as the cube of the radius.)

But the density of dark energy stays constant, since it is a property of empty space.

The density of radiation is proportional to the inverse 4th power of the radius, because not only does the increased volume dilute the radiation like it dilutes the matter, but also the radiation gets red-shifted, contributing an extra power of the radius.

(Incidentally, that means that the fraction of the total mass-energy of the universe in radiation decreases with time, and is estimated presently to be 1000:1 matter:radiation.)

So radiation seems simply to be too small to matter.

• Good answer. The different scalings with size of the Universe means that before the Universe was ~50,000 years old, radiation dominated the evolution, then matter, and finally after ~10 billion years dark energy.
– pela
May 29 '18 at 12:42
• Yet this implies that at the universe's origin, the energy equation was dominated by radiation. This is quite important, because the issue is, how can a universe appear out of nothing if energy is required? Hawking's goal was to allow us a way where we need zero net energy to have a universe, but that certainly means we must address the radiation. It's mostly what we must address, in fact. To get light out of nothing, Hawking was claiming you can regard the energy for the light as coming from gravitational potential energy-- of the light. May 29 '18 at 14:33
• @ Ken G : I know of no situation wherein conservation of energy does not apply. Nor does anybody else. May 29 '18 at 15:45
• @John Duffield: Conservation of energy is conserved perfectly only in systems which are perfectly time symmetric. (Noether's Theorem.) This is true to a very high accuracy locally (for some pretty big locales, BTW), but not true globally since spacetime is expanding. (Some people try to make energy be conserved globally by adding in terms for universal gravitational energy, but these are a bit iffy.) No one can say presently anything about either symmetries or energy conservation "before" the universe existed. May 29 '18 at 16:11
• Sure: Where does the energy of redshifted light from distant galaxies go? Where does dark energy come from (remember, that as the universe expands dark energy, a property of empty space, must be created.) May 29 '18 at 18:10

I've read in Hawking's book The Theory of Everything that total energy of the Universe is zero.

It isn't true.

He explained this by saying that if there are two bodies separated by a distance and having some mass, they possess potential energy and the same amount of negative energy with the gravitational field.

That isn't true either. If you let these two bodies fall towards each other, their gravitational potential energy is converted into kinetic energy. Then when the bodies collide some of this kinetic energy gets radiated away, and you're left with a mass deficit. But note that it's only a mass deficit. The two bodies don't disappear and end up as some zero-energy nothing. Besides, gravitational field energy isn't negative, it's positive. That's why Einstein said "the energy of the gravitational field shall act gravitatively in the same way as any other kind of energy".

The point to note is that when you throw a 1kg brick up in the air you do work on it. You give it kinetic energy*. Gravity then converts this kinetic energy into gravitational potential energy, and the brick slows down. Note that this gravitational potential energy is in the brick, not in the gravitational field, or anywhere else. You did work on the brick, and when the brick is at the top of its arc, its mass is greater. Hence when the brick falls to Earth you end up with the mass deficit. Hence when you throw the brick upwards at 11.7 km/s, it's got escape velocity. It leaves the system, taking the original 1kg worth of mass-energy away, along with the mass-equivalence of the 11.7 km/s of kinetic energy.

Thus he concluded that total energy of the Universe is zero.

It's popscience rubbish I'm afraid. Hawking was always coming out with things like that, and few would criticise him because of the wheelchair. Those that did tended not to get any publicity.

But here I've a question: how do we account for energy which is in the form of radiation?

By measuring the CMBR. There isn't much radiation compared to other stuff. The universe review dark energy article says radiation comprises only about 0.005% of the mass-energy of the universe.

* Whilst conservation of p=mv momentum means there was an effect on the Earth, it's very slight. The KE=½mv² kinetic energy is not shared equally because the Earth didn't move in any detectable fashion.

• Potential energy is a highly technical topic in GR, and there are ways to make it formal which could be interpreted in the way Hawking does. It's not "popscience rubbish," but its significance is often doubted by experts. In contrast, your own position seems internally inconsistent, as you at once claim it is wrong that the energy is zero, but it is right that the energy is conserved! A better understanding may come from a paper on this at prespacetime.com/index.php/pst/article/view/81, though I am not expert enough to be able to tell if it suffices to put the question to bed. May 30 '18 at 14:17
• @Ken G : it isn't a highly technical topic when you throw a brick up in the air. Energy is conserved. Please note that I didn't say the energy of the universe was ever zero. I've read Philip Gibbs' work on conservation of energy in GR before, it's on viXra: vixra.org/abs/1008.0051. Whilst I concur that energy is conserved, I dislike his reasoning. See the Tamara Davis article in Scientific American: Is the Universe leaking energy?. May 30 '18 at 15:59
• To say "energy is conserved" is nothing more than an exercise in bookkeeping. There's not a "thing" called energy, there's a mathematical effort to create a quantity that is conserved. We succeed when throwing a brick, we have a tougher time in GR but under some conditions we can, with enough effort, succeed there too. It's on us, we do it. Likewise, that's what Hawking is doing-- it was on him to try to find a way to make the energy zero. To say "the energy isn't zero" is to fail to understand the process of inventing energy. May 30 '18 at 20:32
• @Ken G : there is a thing called energy. That's why the mass of a body is a measure of its energy content. Energy can be neither created nor destroyed. There are no perpetual motion machines. We have no evidence of any way by which energy can be invented, so understanding the process is extremely questionable. May 31 '18 at 11:44
• If you have an equation of motion that is integrable to produce a constant, you have a useful bookkeeping result. To say it's a "thing" is just a kind of metaphysical attitude, which is fine, but one must not make the error of thinking that "thing" has now transcended the mathematics that motivated it. When you change the mathematics, like using General Relativity instead of Newton's laws, you should expect to need a new "thing"-- and it's up to you to decide what that new "thing" is. That's what Hawking did when he said the energy was zero. Jun 1 '18 at 14:26

It must be the same situation as if you had bits of mass flying off the surface of the two masses in the simple model. It would not take much kinetic energy, at the huge $mc^2$ scale, to get those bits to fly off to infinity in a Newtonian potential energy, but you'd have a huge $mc^2$ lost from your system. So you must also have a similar loss in the general relativistic gravitational potential energy, if the energy was zero originally.

So the lost radiation must also represent a significant loss of potential energy, not just the tiny gravitational redshift. I don't know how Hawking is counting potential energy to get the energy to be zero, but it can't be the simple Newtonian potential. Remember that the zero in a potential energy scale is not usually of physical consequence, because it's only changes in potential that matter, but to get the total energy to be zero, he must have some way to get a relevant scale. Much more formal mathematical arguments are available, such as http://www.prespacetime.com/index.php/pst/article/view/81, but even these are debated by experts, who each attribute different degrees of significance to the ability to make the total energy zero. It is not at all clear that total energy is a relevant physical quantity in general relativity, but for those who wish to make it so, as Hawking does, it may indeed be possible.

Also note that the tiny energy in the radiation today is not really relevant to this question, because the question is about originating a universe. When the universe originated, radiation was extremely important, so the question does need to be addressed for anyone who wishes the energy to be zero. The real issue appears to be, whether or not that is an important goal in the first place.


text:
# Radius of Convergence of Power Series over Factorial

## Theorem

### Real Case

Let $\xi \in \R$ be a real number.

Let $\ds \map f x = \sum_{n \mathop = 0}^\infty \frac {\paren {x - \xi}^n} {n!}$.

Then $\map f x$ converges for all $x \in \R$.

That is, the interval of convergence of the power series $\ds \sum_{n \mathop = 0}^\infty \frac {\paren {x - \xi}^n} {n!}$ is $\R$.

### Complex Case

Let $\xi \in \C$ be a complex number.

Let $\ds \map f z = \sum_{n \mathop = 0}^\infty \dfrac {\paren {z - \xi}^n} {n!}$.

Then $\map f z$ converges absolutely for all $z \in \C$.

That is, the radius of convergence of the power series $\ds \sum_{n \mathop = 0}^\infty \frac {\paren {z - \xi}^n} {n!}$ is infinite.


text:
Regarding the shopping questions!

Are shopping question where a user(Me) has done a bit of research but is noob and doesn't want to stupidly spend money on a High end(> 150$) electronic measurement device and is comparing two products but cannot figure out which one suits his needs valid? • I bet some people here chuckles a bit when you call$150 "high end". You might find a high end passive oscilloscope probe for that amount. :) But everything is relative...
– pipe
Jan 21 '18 at 7:10
• You'll get better results in our EE.SE chat with those questions. Jan 21 '18 at 22:14

Comparing two products would be off-topic, but a more general discussion about how to measure what you need would be fine.

• Ok. Not even comparing two products specifications for the needs of hobby electronics use? Jan 20 '18 at 17:53
• Right. That would be too specific to your particular needs. If it's for"general hobbyist use", that would be too speculative. Jan 20 '18 at 17:54
• Alright. Thanks :) Jan 20 '18 at 17:58
• If you have a highly specific question about the meaning of a particular parameter (and one not readily determined from the data sheet or a web search) that could potentially be on topic, but a question of this sort is going to start out with a very large amount of suspicion against it, so that you would have to very clearly state a technical query rather be seeking a preference or anticipated usefulness opinion. Jan 20 '18 at 18:22


text:
# Prime factorization of 5202

If it's not what You are looking for type in the field below your own integer, and You will get the solution.

Prime factorization of 5202:

By prime factorization of 5202 we follow 5 simple steps:
1. We write number 5202 above a 2-column table
2. We divide 5202 by the smallest possible prime factor
3. We write down on the left side of the table the prime factor and next number to factorize on the ride side
4. We continue to factor in this fashion (we deal with odd numbers by trying small prime factors)
5. We continue until we reach 1 on the ride side of the table

 5202 prime factors number to factorize 2 2601 3 867 3 289 17 17 17 1

Prime factorization of 5202 = 1×2×3×3×17×17= $1 × 2 × 3^2 × 17^2$

## Related pages

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text:
Revision history [back]

Problem: Stupidity.

Solution: RTFM.

Details By pressing q/w/e you are changing default speeds. To actually make it move at those speeds, you need to press the keys in the 'Moving around' matrix that you included in the picture of what you see in the teleop terminal. E.g., 'u, i, and o'.

So, in the printout on the screen when it said that w/x increased the speed, I took that to mean press w/x to make it move. I didn't look at the large matrix of movement instructions about it, highlighted:

Moving around:
u    i    o
j    k    l
m    ,    .

When I started to press u, i, o, and the other keys in this large map, the turtle began to move around in the map. Basically the instructions are a map:

u : rotate ccw and move forward (forward left)
i : move forward
o : rotate cw and move forward (forward right)
j : rotate ccw
k : stop
l : rotate cw
m : rotate cw and move back (backward left)
, : move back
. : rotate ccw and move back (backward right)

Problem: Stupidity.

Solution: RTFM.

Details By pressing q/w/e you are changing default speeds. To actually make it move at those speeds, you need to press the keys in the 'Moving around' matrix that you included in the picture of what you see in the teleop terminal. E.g., 'u, i, and o'.

So, in the printout on the screen when it said that w/x increased the speed, I took that to mean press w/x to make it move. I didn't look at notice the large matrix of movement instructions about it, highlighted:instructions:

Moving around:
u    i    o
j    k    l
m    ,    .

When I started to press u, i, o, and the other keys in this large map, these keys, the turtle began to move around in the map. Basically the instructions are a topographic map:

u : rotate ccw and move forward (forward left)
i : move forward
o : rotate cw and move forward (forward right)
j : rotate ccw
k : stop
l : rotate cw
m : rotate cw and move back (backward left)
, : move back
. : rotate ccw and move back (backward right)

So...oops.


text:
# 8 (number)

(Redirected from ٨)
"8th", "Eight" and "Eighth" redirect here. For other uses see 8 (disambiguation)
 ← 7 8 9 →
Cardinal eight
Ordinal 8th
(eighth)
Factorization 23
Divisors 1, 2, 4, 8
Roman numeral VIII
Roman numeral (unicode) Ⅷ, ⅷ
Greek prefix octa-/oct-
Latin prefix octo-/oct-
Binary 10002
Ternary 223
Quaternary 204
Quinary 135
Senary 126
Octal 108
Duodecimal 812
Vigesimal 820
Base 36 836
Greek η (or Η)
Arabic ٨,8
Urdu
Amharic
Bengali
Chinese numeral 八，捌
Devanāgarī
Telugu
Tamil
Hebrew ח (Het)
Hebrew שמונה (shmoneh)
Khmer
Korean
Thai

8 (eight ) is the natural number following 7 and preceding 9. It is the root word of two other numbers: eighteen (eight and ten) and eighty (eight tens). The word is derived from Middle English eighte.

## In mathematics

8 is a composite number, its proper divisors being 1, 2, and 4. It is twice 4 or four times 2. Eight is a power of two, being $2^3$ (two cubed), and is the first number of the form $p^3$, p being an integer greater than 1. It has an aliquot sum of 7 in the 4 member aliquot sequence (8,7,1,0) being the first member of 7-aliquot tree. It is symbolized by the Arabic numeral (figure)

All powers of 2 ;($2^x$), have an aliquot sum of one less than themselves.

A number is divisible by 8 if its last 3 digits are also divisible by 8.

Eight is the first number to be the aliquot sum of two numbers other than itself; the discrete biprime 10, and the square number 49.

8 is the base of the octal number system, which is mostly used with computers. In octal, one digit represents 3 bits. In modern computers, a byte is a grouping of eight bits, also called an octet.

The number 8 is a Fibonacci number, being 3 plus 5. The next Fibonacci number is 13. 8 is the only positive Fibonacci number, aside from 1, that is a perfect cube.[1]

8 is the only nonzero perfect power that is one less than another perfect power, by Mihăilescu's Theorem.

8 is the order of the smallest non-abelian group all of whose subgroups are normal.

8 and 9 form a Ruth–Aaron pair under the second definition in which repeated prime factors are counted as often as they occur.

There are a total of eight convex deltahedra.

A polygon with eight sides is an octagon. Figurate numbers representing octagons (including eight) are called octagonal numbers.

A polyhedron with eight faces is an octahedron. A cuboctahedron has as faces six equal squares and eight equal regular triangles.

A cube has eight vertices.

Sphenic numbers always have exactly eight divisors.

8 is the dimension of the octonions and is the highest possible dimension of a normed division algebra.

The number 8 is involved with a number of interesting mathematical phenomena related to the notion of Bott periodicity. For example if $O(\infty)$ is the direct limit of the inclusions of real orthogonal groups $O(1)\hookrightarrow O(2)\hookrightarrow\ldots\hookrightarrow O(k)\hookrightarrow\ldots$ then $\pi_{k+8}(O(\infty))\cong\pi_{k}(O(\infty))$. Clifford algebras also display a periodicity of 8. For example the algebra $Cl(p+8,q)$ is isomorphic to the algebra of 16 by 16 matrices with entries in $Cl(p,q)$. We also see a period of 8 in the K-theory of spheres and in the representation theory of the rotation groups, the latter giving rise to the 8 by 8 spinorial chessboard. All of these properties are closely related to the properties of the octonions.

The spin group $Spin(8)$ is the unique such group that exhibits the phenomenon of triality.

The lowest-dimensional even unimodular lattice is the 8-dimensional E8 lattice. Even positive definite unimodular lattice exist only in dimensions divisible by 8.

A figure 8 is the common name of a geometric shape, often used in the context of sports, such as skating. Figure-eight turns of a rope or cable around a cleat, pin, or bitt are used to belay something.

### In numeral systems

Base Numeral system Representation
2 binary 1000
3 ternary 22
4 quaternary 20
5 quinary 13
6 senary 12
7 septenary 11
8 octal 10

## Evolution of the glyph

In the beginning, various groups in India wrote eight more or less in one stroke as a curve that looks like an uppercase H with the bottom half of the left line and the upper half of the right line removed. At one point this glyph came close to looking like our modern five. With the western Ghubar Arabs, the similarity of the glyph to five was banished by connecting the beginning and the end of stroke together, and it was only a matter of the Europeans rounding the glyph that led to our modern eight.[3]

Just as in most modern typefaces, in typefaces with text figures the 8 character usually has an ascender, as, for example, in .

## In science

### Geology

• A disphenoid crystal is bounded by eight scalene triangles arranged in pairs. A ditetragonal prism in the tetragonal crystal system has eight similar faces whose alternate interfacial angles only are equal.

### Biology

• All spiders, and more generally all arachnids, have eight legs. Orb-weaver spiders of the cosmopolitan family Areneidae have eight similar eyes.
• The octopus and its cephalopod relatives in genus Argonauta have eight arms (tentacles).
• Compound coelenterates of the subclass or order Alcyonaria have polyps with eight-branched tentacles and eight septa.
• Sea anemones of genus Edwardsia have eight mesenteries.
• Animals of phylum Ctenophora swim by means of eight meridional bands of transverse ciliated plates, each plate representing a row of large modified cilia.
• The eight-spotted forester (genus Alypia, family Zygaenidae) is a diurnal moth having black wings with brilliant white spots.
• The ascus in fungi of the class Ascomycetes, following nuclear fusion, bears within it typically eight ascospores.
• Herbs of genus Coreopsis (tickseed) have showy flower heads with involucral bracts in two distinct series of eight each.
• Timothy Leary identified a hierarchy of eight levels of consciousness.
• In human adult dentition there are eight teeth in each quadrant. The eighth tooth is the so-called wisdom tooth.
• There are eight cervical nerves on each side in man and most mammals.

## Architecture and engineering

• Various types of buildings are usually eight-sided (octagonal), such as single-roomed gazebos and multi-roomed pagodas (descended from stupas; see religion section below).
• Eight caulicoles rise out of the leafage in a Corinthian capital, ending in leaves that support the volutes.

## In religion

Buddhism
The 8-spoked Dharmacakra represents the Noble Eightfold Path
• In Mahayana Buddhism, the branches of the Eightfold Path are embodied by the Eight Great Bodhisattvas: (Manjusri, Vajrapani, Avalokiteśvara, Maitreya, Ksitigarbha, Nivaranavishkambhi, Akasagarbha, and Samantabhadra). These are later (controversially) associated with the Eight Consciousnesses according to the Yogacara school of thought: consciousness in the five senses, thought-consciousness, self-consciousness, and unconsciousness-'consciousness' (alaya-vijñana). The 'irreversible' state of enlightenment, at which point a Bodhisattva goes on 'autopilot', is the Eight Ground or bhūmi. In general, 'eight' seems to be an auspicious number for Buddhists, e.g., the 'eight auspicious symbols' (the jewel-encrusted parasol; the goldfish (always shown as a pair, e.g., the glyph of Pisces); the self-replenishing amphora; the white kamala lotus-flower; the white conch; the eternal (Celtic-style, infinitely looping) knot; the banner of imperial victory; the eight-spoked wheel that guides the ship of state, or that symbolizes the Buddha's teaching). Similarly, Buddha's birthday falls on the 8th day of the 4th month of the Chinese calendar.
Judaism
• The religious rite of brit milah (commonly known as circumcision) is held on a baby boy's eighth day of life.
• Hanukkah is an eight-day Jewish holiday that starts on the 25th day of Kislev.
• Shemini Atzeret (Hebrew: "Eighth Day of Assembly") is a one-day Jewish holiday immediately following the seven-day holiday of Sukkot.
Christianity
Islam
Other

## In superstition and divination

### As a lucky number

• Eight (; accounting ; pinyin ) is considered a lucky number in Chinese culture because it sounds like the word meaning to generate wealth (發(T) 发(S); Pinyin: ). Property with the number 8 may be valued greatly by Chinese. For example, a Hong Kong number plate with the number 8 was sold for \$640,000.[5] The opening ceremony of the Summer Olympics in Beijing started at 8 seconds and 8 minutes past 8 pm (local time) on 8 August 2008.[6]
• Eight ( hachi, ya?) is also considered a lucky number in Japanese culture, but the reason is different from that in Chinese culture. Eight gives an idea of growing prosperous, because the letter (八) broadens gradually.
• The Japanese thought eight ( ya?) as a holy number in the ancient times. The reason is less well understood, but it is thought that it is related to the fact they used eight to express large numbers vaguely such as manyfold (やえはたえ Yae Hatae?) (literally, eightfold and twentyfold), many clouds (やくも Yakumo?) (literally, eight clouds), millions and millions of Gods (やおよろずのかみ Yaoyorozu no Kami?) (literally, eight millions of Gods), etc. It is also guessed that the ancient Japanese gave importance to pairs, so some researchers guess twice as four ( yo?), which is also guessed to be a holy number in those times because it indicates the world (north, south, east, and west) might be considered a very holy number.
• In numerology, 8 is the number of building, and in some theories, also the number of destruction.

## In music and dance

• A note played for one-eighth the duration of a whole note is called an eighth note, or quaver.
• An octave, the interval between two notes with the same letter name (where one has double the frequency of the other), is so called because there are eight notes between the two on a standard major or minor diatonic scale, including the notes themselves and without chromatic deviation. The ecclesiastical modes are ascending diatonic musical scales of eight notes or tones comprising an octave.
• There are eight notes in the octatonic scale.
• There are eight musicians in a double quartet or an octet. Both terms may also refer to a musical composition for eight voices or instruments.
• Caledonians is a square dance for eight, resembling the quadrille.
• Albums with the number eight in their title include 8 by the Swedish band Arvingarna and The Meaning of 8 by Minnesota indie rock band Cloud Cult
• Dream Theater's eighth album Octavarium contains many different references to the number 8, including the number of songs and various aspects of the music and cover artwork
• "Eight maids a-milking" is the gift on the eighth day of Christmas in the carol "The Twelve Days of Christmas"
• The 8-track cartridge is a musical recording format
• "#8" is the stagename of Slipknot vocalist Corey Taylor
• "Too Many Eights" is a song by Athens, Georgia's Supercluster.
• Eight Seconds, a Canadian musical group popular in the 1980s with their most notable song "Kiss You (When It's Dangerous)"
• "Eight Days a Week" is a #1 single for the music group The Beatles.

## In film and television

8 Guys is a 2003 short film written and directed by Dane Cook

## In foods

• Nestlé sells a brand of chocolates filled with peppermint-flavoured cream called After Eight, referring to the time 8 p.m.
• There are eight vegetables in V8 juice
• In cooking recipes, there are approximately 8 pinches to a teaspoon

## In slang

• An "eighth" is a common measurement of marijuana, meaning an eighth of an ounce.[9] It is also a common unit of sale for psilocybin mushrooms.[10] Also, an eighth of an ounce of cocaine is commonly referred to as an "8-ball."[11]
• Referring to the shape of the numeral, eight was represented in bingo slang, before political correctness, as "One Fat Lady". Eighty-eight was "Two Fat Ladies"
• The numeral "8" is sometimes used in writing to represent the syllable "ate", as in writing "H8" for "hate", or "congratul8ions" for "congratulations". Avril Lavigne's song "Sk8er Boi" uses this convention in the title. The search engine "Targ8"[12] uses it to replace "et" in Target. Often found on vanity plates
• "Section 8" is common U.S. slang for "crazy", based on the U.S. military's Section 8 discharge for mentally unfit personnel
• The Housing Choice Voucher Program, operated by the United States Department of Housing and Urban Development, is commonly referred to as the Section 8 program, as this was the original section of the Act which instituted the program
• In Colombia and Venezuela, "volverse un ocho" (meaning to tie oneself in a figure 8) refers to getting in trouble or contradicting one's self.
• In China, '8' is used in chat speak as a term for parting. This is due to the closeness in pronunciation of '8' (bā) and the English word 'bye'.
• Eight is symbolic for lesbian sexual relations.

## In other fields

Four playing cards showing the "8" of all four suits
• Eight, West Virginia, a community in the United States
• The ordinal adjective is octaval or octavary.
• The distributive adjective is octonary.
• Eight babies delivered in one birth are called octuplets. The first set of eight surviving babies, the Chukwu octuplets, were born in 1998
• October was the eighth month in the Roman calendar; currently August is the eighth month
• A double quatrefoil has eight foils used as the cadency mark of a ninth son.
• A duke's coronet, worn by a British or Irish duke, has eight conventional strawberry leaves upon the rim of the circlet.
• Eight extra facets are sometimes added around the culet at the lower pyramid of a brilliant diamond.
• I-8 is the designation of the US interstate highway that runs from San Diego, California to Casa Grande, Arizona
• Eight is the number of categories the VALS system uses to classify consumer groups, and the number of categories used by Fallon-McElligott's system for teen marketing
• War of the Eight Princes, a war in Chinese history
• "88" is the abbreviated terminology used by the Aryan Brotherhood for the Nazi salute, "Heil Hitler" – "H" being the eighth letter of the alphabet, twice
• The silver piece of eight was coined in the Spanish Empire and moved trade around the world. Examples include the Carolus dollar, cob and cut. A doubloon was worth 8 gold escudos or 16 pieces of eight.
• Eight elementary grades are included in the eight-four plan of school organization (along with four secondary grades).
• 8 is the official number of the Bishop James Madison Society, a secret society at the College of William & Mary.
• There are Eight Principles of Yong in Chinese calligraphy
• 8vo is shorthand for "octavo", a book size.
• The Eight - Eight American painters who exhibited together only once in 1908 in New York City. They joined this exhibition to oppose traditions upheld by the National Academy and help advance modernism in the United States. Five of the eight painters were associated with the Ashcan School: Robert Henri (1865–1929), George Luks (1867–1933), William Glackens (1870–1938), John Sloan (1871–1951), and Everett Shinn (1876–1953), along with Maurice Prendergast (1859–1924), Ernest Lawson (1873–1939), and Arthur Bowen Davies (1862–1928).
• Astadiggajas is the collective title given to the eight Telugu poets in the court of the emperor Sri Krishnadevaraya who ruled the Vijayanagara Empire from 1509 until his death in 1529.
• A stop sign has eight sides.
• In mythology, the horse of the Norse god Odin has eight legs and is called Sleipnir.
• Proposition 8, the ballot measure banning same-sex marriage in California.
• No H8 campaign in protest of Proposition 8 in California.

## References

1. ^ Bryan Bunch, The Kingdom of Infinite Number. New York: W. H. Freeman & Company (2000): 88
2. ^ Ang, Swee Hoon (1997). "Chinese consumers’ perception of alpha-numeric brand names". Journal of Consumer Marketing 14 (3): 220–233. doi:10.1108/07363769710166800.
3. ^ Georges Ifrah, The Universal History of Numbers: From Prehistory to the Invention of the Computer transl. David Bellos et al. London: The Harvill Press (1998): 395, Fig. 24.68
4. ^ "Life Application New Testament Commentary", Bruce B. Barton. Tyndale House Publishers, Inc., 2001. ISBN 0-8423-7066-8, ISBN 978-0-8423-7066-0. p. 1257
5. ^ Steven C. Bourassa, Vincent S. Peng (1999). "Hedonic Prices and House Numbers: The Influence of Feng Shui". International Real Estate Review 2 (1): 79–93.
6. ^ "Patriot games: China makes its point with greatest show" by Richard Williams, The Guardian, published 9 August 2008
7. ^ Barney's burp song
8. ^ A to Z Encyclopaedia of Ice Hockey
9. ^ urban dictionary
10. ^ psilocybin mushrooms
11. ^ http://www.thegooddrugsguide.com/cocaine/faq.htm
12. ^ , Targ8 your search engine


text:
# Probability density of the product of independent identically distributed random variables

Let $$X_1, X_2, \dots , X_n$$ be a random sample from the PDF $$f(x;\theta)=\theta x^{\theta -1},\;\; 0 0$$ What is the probabilty that $$\prod_{i=1}^n X_i > t$$?

The joint pdf is $$f(x_1,\dots,x_n;\theta)=\theta^n \left(\prod_{i=1}^n x_i\right)^{\theta -1},\;\; 0 0$$ So \begin{align}\mathbb{P}\left(\prod_{i=1}^n X_i > t\right) &= \int_{\prod_{i=1}^n x_i > t}f(x_1,\dots,x_n;\theta)dx_1\dots dx_n\\ &= \int_{\prod_{i=1}^n x_i > t\\0 Is evaluating this integral the best way of solving this problem, and if so, what's the fastest way to compute the integral? It seems like we should use the fact that the boundary of the integral is similar to the integrand.

$$Y = -\log\left(\prod\limits_{k=1}^nX_i\right) = \sum\limits_{k=1}^n(-\log(X_i))$$
You can show that $$Y$$ has a Gamma($$n,\theta$$) distribution.


text:
Mathematics » Properties of Real Numbers » Rational and Irrational Numbers

# Identifying Rational Numbers and Irrational Numbers

## Identifying Rational Numbers and Irrational Numbers

Congratulations! You have completed the first six tutorials of this book! It’s time to take stock of what you have done so far in this course and think about what is ahead. You have learned how to add, subtract, multiply, and divide whole numbers, fractions, integers, and decimals. You have become familiar with the language and symbols of algebra, and have simplified and evaluated algebraic expressions. You have solved many different types of applications. You have established a good solid foundation that you need so you can be successful in algebra.

In this tutorial, we’ll make sure your skills are firmly set. We’ll take another look at the kinds of numbers we have worked with in all previous tutorials. We’ll work with properties of numbers that will help you improve your number sense. And we’ll practice using them in ways that we’ll use when we solve equations and complete other procedures in algebra.

We have already described numbers as counting numbers, whole numbers, and integers. Do you remember what the difference is among these types of numbers?

 counting numbers $$1,2,3,\text{4…}$$ whole numbers $$0,1,2,3,\text{4…}$$ integers $$\text{…}-3,-2,-1,0,1,2,3,\text{4…}$$

## Rational Numbers

What type of numbers would you get if you started with all the integers and then included all the fractions? The numbers you would have form the set of rational numbers. A rational number is a number that can be written as a ratio of two integers.

### Definition: Rational Numbers

A rational number is a number that can be written in the form $$\frac{p}{q},$$ where $$p$$ and $$q$$ are integers and $$q\ne o.$$

All fractions, both positive and negative, are rational numbers. A few examples are

$$\frac{4}{5},-\frac{7}{8},\frac{13}{4},\text{and}\phantom{\rule{0.2em}{0ex}}-\frac{20}{3}$$

Each numerator and each denominator is an integer.

We need to look at all the numbers we have used so far and verify that they are rational. The definition of rational numbers tells us that all fractions are rational. We will now look at the counting numbers, whole numbers, integers, and decimals to make sure they are rational.

Are integers rational numbers? To decide if an integer is a rational number, we try to write it as a ratio of two integers. An easy way to do this is to write it as a fraction with denominator one.

$$3=\frac{3}{1}\phantom{\rule{2em}{0ex}}-8=\frac{-8}{1}\phantom{\rule{2em}{0ex}}0=\frac{0}{1}$$

Since any integer can be written as the ratio of two integers, all integers are rational numbers. Remember that all the counting numbers and all the whole numbers are also integers, and so they, too, are rational.

What about decimals? Are they rational? Let’s look at a few to see if we can write each of them as the ratio of two integers. We’ve already seen that integers are rational numbers. The integer $$-8$$ could be written as the decimal $$-8.0.$$ So, clearly, some decimals are rational.

Think about the decimal $$7.3.$$ Can we write it as a ratio of two integers? Because $$7.3$$ means $$7\frac{3}{10},$$ we can write it as an improper fraction, $$\frac{73}{10}.$$ So $$7.3$$ is the ratio of the integers $$73$$ and $$10.$$ It is a rational number.

In general, any decimal that ends after a number of digits (such as $$7.3$$ or $$-1.2684$$) is a rational number. We can use the place value of the last digit as the denominator when writing the decimal as a fraction.

## Example

Write each as the ratio of two integers:

1. $$-15$$
2. $$\phantom{\rule{0.2em}{0ex}}6.81$$
3. $$\phantom{\rule{0.2em}{0ex}}-3\frac{6}{7}.$$

### Solution

 (a) $$-15$$ Write the integer as a fraction with denominator 1. $$\frac{-15}{1}$$
 (b) $$6.81$$ Write the decimal as a mixed number. $$6\frac{81}{100}$$ Then convert it to an improper fraction. $$\frac{681}{100}$$
 (c) $$-3\frac{6}{7}$$ Convert the mixed number to an improper fraction. $$-\frac{27}{7}$$

Let’s look at the decimal form of the numbers we know are rational. We have seen that every integer is a rational number, since $$a=\frac{a}{1}$$ for any integer, $$a.$$ We can also change any integer to a decimal by adding a decimal point and a zero.

$$\begin{array}{cccc}\text{Integer}\hfill & & & -2,-1,0,1,2,3\hfill \\ \text{Decimal}\hfill & & & -2.0,-1.0,0.0,1.0,2.0,3.0\phantom{\rule{1.0em}{0ex}}\text{These decimal numbers stop.}\hfill \end{array}$$

We have also seen that every fraction is a rational number. Look at the decimal form of the fractions we just considered.

$$\begin{array}{cccc}\text{Ratio of Integers}\hfill & & & \frac{4}{5}, & -\frac{7}{8}, & \frac{13}{4}, & -\frac{20}{3}\hfill \\ \text{Decimal Forms}\hfill & & & 0.8, & -0.875, & 3.25, & -6.666… \phantom{\rule{1.0em}{0ex}}\text{These decimals either stop or repeat.}\hfill \end{array}$$

What do these examples tell you? Every rational number can be written both as a ratio of integers and as a decimal that either stops or repeats. The table below shows the numbers we looked at expressed as a ratio of integers and as a decimal.

Rational Numbers
FractionsIntegers
Number$$\frac{4}{5},-\frac{7}{8},\frac{13}{4},\frac{-20}{3}$$$$-2,-1,0,1,2,3$$
Ratio of Integer$$\frac{4}{5},\frac{-7}{8},\frac{13}{4},\frac{-20}{3}$$$$\frac{-2}{1},\frac{-1}{1},\frac{0}{1},\frac{1}{1},\frac{2}{1},\frac{3}{1}$$
Decimal number$$0.8,-0.875,3.25,-6.\stackrel{\text{–}}{\text{6}},$$$$-2.0,-1.0,0.0,1.0,2.0,3.0$$

## Irrational Numbers

Are there any decimals that do not stop or repeat? Yes. The number $$\pi$$ (the Greek letter pi, pronounced ‘pie’), which is very important in describing circles, has a decimal form that does not stop or repeat.

$$\pi =\text{3.141592654…….}$$

Similarly, the decimal representations of square roots of numbers that are not perfect squares never stop and never repeat. For example,

$$\sqrt{5}=\text{2.236067978…..}$$

A decimal that does not stop and does not repeat cannot be written as the ratio of integers. We call this kind of number an irrational number.

### Definition: Irrational Number

An irrational number is a number that cannot be written as the ratio of two integers. Its decimal form does not stop and does not repeat.

Let’s summarize a method we can use to determine whether a number is rational or irrational.

If the decimal form of a number

• stops or repeats, the number is rational.
• does not stop and does not repeat, the number is irrational.

## Example

Identify each of the following as rational or irrational:

1. $$\phantom{\rule{0.2em}{0ex}}0.58\stackrel{\text{–}}{\text{3}}$$
2. $$\phantom{\rule{0.2em}{0ex}}0.475$$
3. $$\phantom{\rule{0.2em}{0ex}}\text{3.605551275…}$$

### Solution

$$0.58\stackrel{\text{–}}{\text{3}}$$

The bar above the $$3$$ indicates that it repeats. Therefore, $$0.58\stackrel{\text{–}}{\text{3}}$$ is a repeating decimal, and is therefore a rational number.

$$0.475$$

This decimal stops after the $$5$$, so it is a rational number.

$$\text{3.605551275…}$$

The ellipsis $$\text{(…)}$$ means that this number does not stop. There is no repeating pattern of digits. Since the number doesn’t stop and doesn’t repeat, it is irrational.

Let’s think about square roots now. Square roots of perfect squares are always whole numbers, so they are rational. But the decimal forms of square roots of numbers that are not perfect squares never stop and never repeat, so these square roots are irrational.

## Example

Identify each of the following as rational or irrational:

1. $$\phantom{\rule{0.2em}{0ex}}\sqrt{36}$$
2. $$\phantom{\rule{0.2em}{0ex}}\sqrt{44}$$

### Solution

The number $$36$$ is a perfect square, since $${6}^{2}=36.$$ So $$\sqrt{36}=6.$$ Therefore $$\sqrt{36}$$ is rational.

Remember that $${6}^{2}=36$$ and $${7}^{2}=49,$$ so $$44$$ is not a perfect square.

This means $$\sqrt{44}$$ is irrational.


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Question:

# Translation A maps (x, y) to (x +r,y+s). Translation B maps (x, y) to (x – a,y+b). Translate a point using Translation B, then Translation A. Write a rule for the final image of the point. (x,y) → (.

Translation A maps (x, y) to (x +r,y+s). Translation B maps (x, y) to (x – a,y+b). Translate a point using Translation B, then Translation A. Write a rule for the final image of the point. (x,y) → (._____________ ___________) a). type x coordinate. Don't type parenthesis. * b). type y coordinate. Don't type parenthesis.