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Arithmetic operations

Posted in Mathematics, Methinks by ytelotus on June 24, 2013

Why is minus into minus a plus?

Why is minus into plus a minus?

For answers, read on…

Introduction

Arithmetic is mathematics in its most nascent form.  The operations of addition, subtraction, multiplication and division that are useful to man in day to day life come under this elementary branch of mathematics.

Fig. 1 Number Line

Fig. 1 Number Line

The algorithm underlying the arithmetic operations is  best understood using the Number line.  The relative arrangement of negative numbers, zero and positive numbers on the number line matters in this context more than any other.  With the help of  a series of lateral movements and flipping actions on the number line, we can actually calculate the result of some of the otherwise inconceivable arithmetic operations involving negative numbers.

For the ease of understanding, in this article, we discuss the arithmetic operations involving integers only.  However, the concept is easily extendible to the real numbers.

Also, the various techniques and rules of arithmetic operations have been well laid out already.  This article does not aim at finding new avenues.  Rather, it is an attempt to reinvent the wheel with the goal to understand the logic behind the whole process.  Be prepared – it may get messy at times!

Addition and Subtraction

The operations of addition and subtraction become easy when we pictorially project the number line as a series of infinite stairs with negative numbers at the lower end and positive numbers at the higher end.

Fig. 2 Number Stairs

Fig. 2 Number Stairs

We can afford to paint such a picture of the numbers because the number line has its markings gradated in the ascending order of numerical value from left to right.  An analogy with stairs is therefore easier to relate to than a flat number line.  The ascent takes you to higher numbers and the descent takes you to lower numbers relative to your current position on the stairs.

Addition
Say, you’re at step 3.  You climb up by 5 stairs.  You’ll find yourself at step 8.  What you inadvertently did here is addition.   3 + 5 = 8

Subtraction
Say, you’re at step 3.  You climb down by 5 stairs.  You’ll find yourself at step -2.  This in effect is subtraction.  3 – 5 = -2

Fig. 3 Addition and Subtraction

Fig. 3 Addition and Subtraction

Does it mean, we can declare that addition is about ascent (rightward movement on the number line) and subtraction is about descent (leftward movement)?

Not exactly.  Here’s why.

The signs of the numbers involved also matter, not just the operations.  The signs determine the direction of travel – upward or downward.

Say, we’re at step 3.  We’re asked to climb up by -2 steps.  How do we do it?  Physically, it does not mean anything to climb up by -2 steps.  But conceptually, it implies moving in the opposite direction i.e. relatively lower by 2 steps.  When we do so, we find ourselves at step 1.  In effect, what we did here is subtraction by 2 i.e. 3 + (-2) = 3 – 2 = 1.
Climbing up by -x steps = Climbing down by +x steps
In other words, addition by -x is equivalent to subtraction by x, where x is a positive number.

Now, say we’re at step -1.  We’re asked to climb down by -3 steps.  As before, it doesn’t make sense to go down by -3 steps.  It means moving in the relatively opposite direction by 3 steps.  So, we actually climb up by 3 steps and find ourselves at step 2.  In effect, what we did is addition by 3.   i.e. -1 – (-3) = -1 + 3 = 2.
Climbing down by -x steps = Climbing up by +x steps
In other words, subtraction by -x is equivalent to addition by x, where x is a positive number.

Effectually, we can say, ascent is an addition process and descent is a subtraction process.

The table in Fig. 4 shows how the addition and subtraction operations work for different combinations of positive and negative integers :

Fig. 4 Addition and Subtraction

Fig. 4 Addition and Subtraction

Multiplication and Division

Multiplication

Multiplication is equivalent to repetitive addition process.  When we multiply ‘a’ by ‘b’, we actually add ‘a’ repeatedly ‘b’ number of times, considering ‘b’ is positive.  Physically, multiplication can be perceived as being given a method to reach some destination.  Here’s how it is so :

Multiplication along the positive axis
(2 x 3) = 2 + 2 + 2 = (3 x 2) = 3 + 3 = 6

Fig. 5 Multiplication along the positive axis

Fig. 5 Multiplication along the positive axis

(2 x 3) is about making either 3 hops of length 2 OR two hops of length 3, to find yourself at 6.  The same holds for the hops along the negative axis.

Multiplication along the negative axis
(-2 x 3) =  (-2) + (-2) + (-2) = (-3 x 2) = (-3) + (-3) = -6

Fig. 6 Multiplication along the negative axis

Fig. 6 Multiplication along the negative axis

Visually, it tells you how fast and far you have moved away from 0 along either axes.  This amounts to scaling.  The term ‘x 2’ implies twice the scale, ‘x 3’ is thrice the scale and so on.  i.e. if you multiply some entity with any number, the effect is that of proportionate scaling of that entity.

Also, observe from Fig. 5 and Fig. 6 that, when you multiply a positive number by another positive number, you go along the positive direction away from 0 and when you multiply a negative number by a positive number, you go along the negative direction that many times far away from 0.  This is an empirical proof that (plus into plus) will always remain plus and (minus x plus) will always stay minus.

It is fairly easy to multiply so long as we multiply an entity (positive or negative) by another positive entity.  This positive entity tells you how many times you ought to add the given entity to get the final product.  So, we can say :
Multiplication of a number ‘a’ by ‘b’ is equivalent to adding ‘a’ repetitively ‘b’ number of times OR hopping in strides of ‘a’, ‘b’ number of times, where ‘b’ is a positive number.

So far, we’ve been able to visualize the arithmetic operations and translate them into movements this way or that way on the number line.  The complication is about to begin.

What if both the numbers are negative?  What is -2 x -3 = ?  How do we add -2 some negative number of times?

It is impossible to calculate this unless we convert it into a form where one of the numbers in the equation is a positive number.  The way to do this is by expressing  a negative number in terms of its relative position with respect to any other number on the scale.  As an example, -3 = (0-3) = (1-4) = (2-5) = …  For convenience, we shall rewrite (-2 x -3) as :
-2 x -3 = -2 x (1 – 4)
By distributive property,
-2 x (1 – 4) = (-2 x 1) – (-2 x 4)
Now, we have transformed one multiplication operation into two, however with each having a positive number to multiply with.
-2 x -3 = -2 – (-8) = 6

By this very approach, we can also prove, -1 x -1 = +1 :
-1 x (1 – 2) = (-1 x 1) – (-1 x 2) = -1 – (-2) = +1

Again, an empirical proof that minus into minus is always a plus!

Division

Division is the reverse process of multiplication.  With multiplication, it is about finding the destination that we should reach, given the methods.  With division, it is about finding the method to reach the given destination with a partial knowledge of the method.  This involves repetitive subtraction process.

Division along the positive axis
In the equation 2 x y = 8, finding y is the goal of division.  Here, we know the destination we want to reach i.e. 8.  We know what we’re equipped with i.e. 2.  Now how many times should we add 2 to itself to reach 8?  That gives us ‘y’.

Fig. 7 Division along the positive axis

Fig. 7 Division along the positive axis

The way to find ‘y’ is by starting at 8 and hopping towards 0 in steps of 2.  The number of times you need to do this to reach 0 from 8 gives you ‘y’.
8 – 2 – 2 – 2 – 2 = 0
Here, the number 2 has been subtracted 4 times.  So, y = 4.  Another way to write the equation 2 x 4 = 8 without changing its meaning is, 4 x 2 = 8.  Here, we hop from 8 in steps of 4.  It takes 2 such hops to reach 0.

Division along the negative axis
Given -2 x y = -6, find y.

Fig. 8 Division along the negative axis

Fig. 8 Division along the negative axis

By the same approach as in the previous case,
-6- (-2) – (-2) – (-2)  = 0
We get y = 3.

Let’s step up the challenge :  Given, 4 x y = -20, find y.
-20 – 4 – 4 – …?   Can we ever reach 0 by this method?  No.
Here, we need to approach with a different interpretation of the question : Which number when subtracted 4 times from -20 gives us 0?
-20 – (-5) – (-5) – (-5) – (-5) = 0
We get y = -5.

Now for the real challenge : Given -4 x y = 20, find y.
20 – (-4) – (-4) – (-4) …?
This is kind of an impossible case.  It’s as good as traversing in the negative direction to reach a destination on the positive side.  Looks like we have hit the limitation at last!  However, we need to admit that not everything about mathematics is conceivable.  Sometimes, we need to work our way around to find a solution if one exists.  That’s what we’re going to do now.  We flip our destination point and place ourselves on the negative side.  How do we do it?  By multiplying by -1.
-1 x (-4 x y) = -1 x 20
By associative property, we have
(-1 x -4) x y = -1 x 20
4 x y = -20
Now this equation is as good as the previous one that we solved.  Hence, y = -5.

The table in Fig. 9 shows how the multiplication and division operations work for different combinations of positive and negative integers :

Fig. 9 Multiplication and Division

Fig. 9 Multiplication and Division

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7 Responses

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  1. Pankaj Saraf said, on May 10, 2014 at 8:15 am

    Nice Article… It is something which answers a big “WHY” in mathematics.

  2. robsblog234 said, on April 5, 2015 at 7:33 am

    Hi – nice article – however a small problem with the negative by negative multiplication… You used the distributive property. Normally that would be fine – but not when the laws in question are as fundamental as the distributive law itself. Do you have a proof of the distributive property itself without recourse to any of the multiplicative sign laws? I think you will find that they depend on one another for proof. It’s a circular argument…

    • ytelotus said, on April 6, 2015 at 4:25 am

      Thank you for your comment. I agree with you. The use of one fundamental law to arrive at another related fundamental law would only make it circular.
      Let me put it this way:
      The intent here is to see if I can visualize the arithmetic operations in a tangible way. It’s easy to do so with positive numbers. With negative numbers, there’s a challenge because they don’t exist naturally. The only way is to convert the operations involving negative numbers into those involving positive numbers. This is where the use of distributive property helps. It would certainly be good to do without it, if there’s some other way.

  3. Vignesh said, on June 21, 2016 at 1:52 pm

    Good

  4. Felix Erkinheimo said, on April 29, 2017 at 10:47 am

    Nice and clear article! One note: division can be derived from multiplication and any division process can be presented as multiplication in the following way: a/b = a x b^-1. This gives us the opportunity to use the same principles for explaining negative-positive co-operations, that are used for multiplication, which seems simpler for me. Thank you for the article anyway!


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