_{ }operations do, but over the years we forget the main idea, the core of the operation and just remember the execution. So, to ensure that these fundamentals are revisited, I will first discuss a few interesting things about these basic mathematical operators.

Addition:

Adding two numbers is probably the simplest mathematical operation, still you will be surprised at the amount of time people take to add 126 to 94. If you are unfortunate enough, you may even get to see the following:

1 2 6

+ 9 4

____________

2 2 0

Let’s look at a few examples, which I promise, I will type at random:

95 + 48 = 100 + 43 = 143

64 +73 = 70 + 60 ( = 130) + 7 = 137

739 + 182 = 740 + 181 = 840 + 81 = 920 + 1 = 921

45 + 39 + 18 + 26 = 40 + 30 + 10 + 20 (= 100) + 5 + 9 + 8 + 6 = 100 + 14 + 14 = 128

Subtraction:

Most people find subtraction a little harder than addition. Again, subtract in parts. If you want to subtract 27 out of 132, subtract 20 first, to get 112 and then 7 to get 105. Or, subtract 30 out of 132 to get 102 and then add 3 back because you only had to subtract 27. You get 105. Though, in subtraction, many a times, looking at the numbers and imagining their place in the typical subtraction diagram may turn out to be quite effective. For example if I want to subtract 36 from 83, I look at the unit’s digits and say that 3 – 6 will be 7 and 1 borrowed and then after subtracting the 1 from the tens digit, 7 – 3 will be 4 so the answer will be 47. Essentially, I am doing what eight year olds do, that is

8 3

- 3 6

_________________

4 7

But I am doing it in my mind, which makes it quicker.

^{th}number; I want all the numbers starting from 11 and ending at 25. We need to add 1 to the result i.e. 14 so ensure that the 11

^{th}number that we threw out, is retained. Consequently, the number of numbers from 11 to 25, both inclusive is 1 + 14 = 15 numbers.

Multiplication:

It is a basic mathematical operator but, essentially, multiplication is still addition. When I ask, “What is 3 × 5?” I am actually asking, “What is 3 + 3+ 3 + 3 + 3?” i.e. addition of 3s written 5 times or addition of 5s written 3 times (because 5 × 3 is also the same as 3 × 5). This fact is very useful when performing multiplication orally. Now, if you want me to show you how to perform 139 ×463 orally, I am not that talented. But, in your everyday life and in exams that test your core concepts rather than calculation skills, you will rarely need to do such a multiplication. More often than not, the tougher multiplication required will be 16 × 12 or 23 × 18 etc. This is quite manageable.

When I ask for 16 × 12, I am asking for addition of twelve 16s i.e. 16 + 16 + 16 + … 12 times. Can I easily add ten 16s? Sure. It is 160. What about two 16s? That’s 32. So twelve 16s would add up to give 160 + 32 = 192.

Let’s take another example: 19 × 26. Easiest way to do this would be to find 20 times 26 which is 520 and then subtract 26 out of it because we wanted 26, 19 times, not 20 times. How do we subtract 26 from 520? By subtracting 20 first, to get 500, and then subtracting 6, to get 494.

^{4}× 4

^{2}. Is it really a cause of worry? 3

^{4}is just a notation for 3 × 3 × 3 × 3 i.e. Four 3s multiplied together. Then, 4

^{2}must be just 4 × 4. This means that 3

^{4}× 4

^{2}= 3 ×3 × 3 × 3 × 4 × 4 which is again just a multiplication, a pretty long one, but nevertheless, just a multiplication. More on indices, later.

Division:

Intuition says that if multiplication is actually addition, division should be subtraction. It is! When I divide 6 by 2, what I am doing is from 6, making groups of 2 and subtracting them.

6 – 2 = 4 (One group of 2 subtracted.)

4 – 2 = 2 (Another group of 2 subtracted.)

2 – 2 = 0 (Third group of 2 subtracted.)

I subtracted three groups so I get 3 as my quotient. Nothing is leftover so remainder is 0.

If I want to divide 11 by 3, I would be able to subtract 3 groups of 3 giving me a quotient of 3, and 2 will be leftover. Then 2 will be the remainder.

So if I ask you, is 82 divisible by 4, my question is, can I subtract groups of 4 from 82 such that nothing is leftover? When I subtract 20 groups of 4 (=80) from 82, I am left with 2. Hence 82 is not divisible by 4.

That's all for now. I will refer to these concepts again and again in other discussions so ensure that you are comfortable with them.

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