Hello and welcome to the "Magic of Mathematics"
channel. My name is Manohar Moorthy and you
are watching one amongst a series of videos
exploring the world of Mathematics.
In this video, I intend to explain how to
perform addition and subtraction of Integers
on the number line. If you are interested
in understanding the rules for performing
addition and subtraction of Integers on the
number line in an intuitive way, then you
will find this video useful.
I assume here that you are familiar with Integers
and how to represent them on the number line.
I also assume that you are familiar with addition
and subtraction of natural numbers on the
number line.
I would like to mention here that I will use
only simple cases involving a single addition
or subtraction operator. However, the logic
that I present in this video can be easily
extended to deal with more complex arithmetic
expressions.
Now that we know what we will be dealing with
in this video, let us get down to the task at hand.
Before I go ahead with presenting the intuitive
rules for performing addition and subtraction
of Integers on the number line, I would like
to state here a well known fact, namely that
conceptually positive and negative Integers
are opposites of each other. In other words,
a negative integer is conceptually the opposite
of the positive integer of the same magnitude.
If you are not familiar with this concept,
you should watch my video where I deal with
the question "What are Integers and why do
we need them?"
With this background, we can go ahead and
look at how we can intuitively perform addition
and subtraction of Integers on the number
line.
For this, let us first draw a number line
and mark zero on the number line. As we know,
we represent positive Integers to the right
of zero and negative Integers to the left of zero.
We will make markings on the number line when
we look at specific examples. But first, I
will present the rules for performing addition
and subtraction of Integers on the number
line, in a generic way. And for that, we don't
need markings representing specific numbers.
To present the intuitive rules for performing
addition and subtraction of Integers on the
number line, I will perform a number line
walk. Before I do this, I would like to mention
that we are dealing with a lot of different
cases here as the operator could be the addition
operator or the subtraction operator and each
of the operands involved could be a positive
Integer or a negative Integer.
To make the explanation of the rules simple,
I will be using a specific case. Once the
rules are clear in the context of one case,
I will consider a few examples to show that
the rules are general enough to work on all
cases.
Now we are ready for our number line walk.
For starting the number line walk, I look
at the first operand to determine the starting
point. The starting point could be on the
positive part of the number line or on the
negative part of the number line and is determined
by the first operand.
For now, let us look at the case where the
first operand is a positive Integer, which
means that I will stand at a point on the
positive part of the number line as shown.
Next, I look at the operator. If it is an
addition operator, I turn to walk towards
the right, as shown and if it is a subtraction
operator I turn to walk towards the left.
Notice here that I have used the well known
fact that subtraction is in a sense the opposite
of addition.
Again, for explaining the rules, let us look
at the case where our operator is an addition
operator, in which case I will be ready to
walk towards the right side of the number
line as shown.
Now, I look at the second operand to determine
how I walk. If the second operand is a positive
Integer, I will walk forwards by a number
of unit steps equal to the magnitude of the
operand. However, if the second operand is
a negative Integer, I will walk backwards
by a number of unit steps equal to the magnitude
of the operand. Again, this makes intuitive
sense as negative Integers are conceptually
the opposite of positive Integers.
Now that we have looked at the rules for performing
addition and subtraction of Integers on the
number line, let us apply the rules in the
context of a few examples.
The first example that I will take up is negative
3 plus negative 2.
As before, I draw a number line, but this
time with markings representing the positive
and negative Integers.
In this example, the first operand is negative
3. So, I will start my journey from negative
3. Now, since we have the addition operator,
I turn to walk towards the right. Next, I
look at the second operand which happens to
be a negative Integer of magnitude 2. This
means I need to walk two steps backwards,
thereby ending up at negative 5, which is
what our expression evaluates to.
The second example that I will take up is
negative 3 minus negative 2.
Here, the first operand is again negative
3. So, I stand at negative 3 again. However,
the operator in this expression is the subtraction
operator and hence, I turn to walk towards
the left. Next, let us look at the second
operand. This is negative 2, which means I
need to walk two steps backwards, thereby
ending up at negative 1, which is the result
of evaluating the given expression.
The third example that I will take up is negative
2 plus 2.
Here, the first operand is negative 2 and
hence, I begin my journey from negative 2.
The operator being the addition operator,
I turn to walk towards the right. The second
operand is 2, which means I walk two steps
forward, thereby ending up at 0 and thus,
the given expression evaluates to 0.
The last example that I will take up is 2
minus 4.
In this example, the first operand is 2 and
hence, I begin my journey from 2. Since the
operator here is the subtraction operator,
I turn to walk towards the left. The second
operand is 4, which means I walk four steps
forward, thereby ending up at negative 2 and
hence, the given expression evaluates to negative
2.
In this video, we looked at an intuitive way
to perform addition and subtraction of Integers
on the number line. We also looked at four
examples to illustrate the rules that I presented.
I hope you found the video useful. Thanks
for watching.
