Squaring can be defined as ‘multiplying a number by itself.’

There are many different ways of squaring numbers. Many of these techniques have their roots in multiplication as squaring is simply a process of multiplication.

Examples: 3^{2} is 3 multiplied by 3 which equals 9

4^{2} is four multiplied by 4 which equals 16 The techniques that we will study are:

- Squaring of numbers using the Criss-Cross system
- Squaring of number using formulae

# (A) SQUARING OF NUMBERS USING CRISS- CROSS SYSTEM

The Urdhva-Tiryak Sutra (the Criss-Cross system) is by far the most popular system of squaring numbers among practitioners of Vedic Mathematics. The reason for its popularity is that it can be used for any type of numbers.

Ex: Find the square of 23. Ans:

(a) * *

* *

(b) X * *

* *

(c) | * *

* *

2 3

X 2 3

(a) First, we multiply 3 by 3 and get the answer as 9. (Answer at this stage is 9).

(b)Next, we cross multiply (2 * 3) and add it with (2 ´ 3). The final answer is 12. We write down 2 and carry over 1.(Answer at this stage is 29).

(c)Thirdly, we multiply (2 * 2) and add the 1 to it. The answer is 5.

# The final answer is 529.

Similarly, numbers of higher orders can be squared. Refer to the chapter on Criss-Cross system for further reference.

# (B) FORMULA METHOD

There are various formulae used in general mathematics to square numbers instantly. Let us discuss them one by one.

(i) (a + b)^{2} = a^{2} + 2ab + b^{2}

This method is generally to square numbers which are near multiples of 10. In this method, a given number is expanded in such a manner that the value of ‘a’ is a number which can be easily squared and the value of ‘b’ is a small number which too can be easily squared.

**(Q) Find the square of 1009.**

We represent the number 1009 as 1000 + 9. Thus, we have converted it into a form of (a + b) where the value of a is 1000 and the value of b is 9.

(1000 + 9)^{2} = (1000)^{2} + 2 (1000) (9) + (9)^{2}

= 1000000 + 18000 + 81

= 1018081

**(Q) Find the square of 511.**

The number 511 will be written as 500 + 11 (500 + 11)^{2} = (500)^{2} + 2 (500) (11) + (11)^{2}

= 250000 + 11000 + 121

= 261121

The second formula that we will discuss is also very well known to the students. It is a part of the regular school curriculum. This formula is used to square numbers which can be easily expressed as a difference of two numbers ‘a’ and ‘b’ in such a way that the number ‘a’ is one which can be easily squared and the number ‘b’ is a small number which too can be easily squared. The formula is,

(ii) (a – b)^{2} = a^{2} – 2ab + b^{2}

This formula is very much like the first one. The only difference is that the middle term carries a negative sign in this formula.

**(Q) Find the square of 995.**

We will express the number 995 as (1000 – 5) (1000 – 5)^{2} = (1000)^{2} – 2 (1000) (5) + (5)^{2}

= 1000000 – 10000 + 25

= 990025

**(Q) Find the square of 698.**

We will express the number 698 as (700 – 2) (700 – 2)^{2} = (700)^{2} – 2 (700) (2) + (2)^{2}

= 490000 – 2800 + 4

= 487204

* * * * * *

Thus, we see that the two formulae can help us find the squares of any number above and below a round figure respectively. There is another formula which is used to find the square of numbers, but it is not so popular. I discuss it below.

We know that:

a^{2} – b^{2} = (a + b) (a – b) (Therefore) a^{2} = (a + b) (a – b) + b^{2}

This is the formula that we will be using: **a2 = (a + b) (a –b) + b2**

# METHOD

Suppose we are asked to find the square of a number. Let us call this number ‘a.’ Now in this case we will use another number ‘b’ in such a way that either (a + b) or (a – b) can be easily squared.

**(Q) Find the square of 72.**

Ans: In this case, the value of ‘a’ is 72. Now, we know that a^{2} = (a + b) (a – b) + b^{2}

Substituting the value of ‘a’ as 72, we can write the above formula as:

(72)^{2} = (72 + b) (72 – b) + b^{2}

We have substituted the value of ‘a’ as 72. However, we cannot solve this equation because a variable ‘b’ is still present. Now, we have to substitute the value of ‘b’ with such a number that the whole equation becomes easy to solve.

Let us suppose I take the value of ‘b’ as 2. Then the equation becomes,

(72)^{2} = (72 + 2) (72 – 2) + (2)^{2}

= (74) (70) + 4

In this case we can find the answer by multiplying 74 by 70 and adding 4 to it. However, if one finds multiplying 74 by 70 difficult, we can simplify it still further. First, multiply 70 by 70 and then multiply 4 by 70 and add both for the answer.

Let us continue the example given above:

= (70 * 70) + (4 * 70) + 4

= 4900 + 280 + 4

= **5184**

Thus, the square of 72 is 5184.

In this example we have taken the value of ‘b’ as 2. Because of this, the value of (a – b) became 70 and the multiplication procedure became easy (as the number 70 ends with a zero).

**(Q) Find the square of 53.**

Ans: Uing the formula a^{2} = (a + b) (a – b) + b^{2} and taking the value of ‘a’ as 53, we have:

(53)^{2} = (53 + b) (53 – b) + (b)^{2}

Now we have to find a suitable value for ‘b’. If we take the value of ‘b’ as 3, the expression (53 – 3) will be 50 and hence it will simplify the multiplication procedure. So we will take the value of ‘b’ as 3 and the equation will become:

(53)^{2} = (53 + 3) (53 – 3) + (3)^{2}

= (56) (50) + 9

= (50 * 50) + (6 ´ 50) + 9

= 2500 + 300 + 9

= **2809**

**(Q) Find the square of 67.**

Ans: In this case the value of ‘a’ is 67. Next, we will substitute ‘b’ with a suitable value. In this case, let us take the value of ‘b’ as 3 so that the value of (a + b) will become (67 + 3) which equals 70.

Thus: (67)^{2} = (67 + 3) (67 – 3) + (3)^{2}

= (70) (64) + 9

= (70 * 60) + (70 * 4) + 9

= 4200 + 280 + 9

= **4489**

**(Q) Find the square of 107.**

Ans: In this case, we will take the value of ‘a’ as 107 and take the value of ‘b’ as 7. The equation becomes:

(107)^{2} = (107 + 7) (107 – 7) + (7)^{2}

= (114) (100) + 49

= 11400 + 49

= **11449**

**(Q) Find the square of 94.**

Ans: In this example we will take the value of ‘a’ as 94. Next, we will take the value of ‘b’ as 6 so that the value of (a + b) becomes 100.

(94)^{2} = (94 + 6) (94 – 6) + (6)^{2}

= (100) (88) + 36

= **8836**

# EXERCISE

**PART A**

**Q. (1) Find the squares of the following numbers using the Criss-Cross System.**

(1) 42

(2) 33

(3) 115

**PART B**

**Q. (2) Find the squares of the numbers using the formula for (a + b) ^{2}.**

(1) 205

(2) 2005

(3) 4050

**Q. (3) Find the squares of the numbers using the formula for (a – b) ^{2}.**

(4) 9991

(5) 9800

(6) 1090

**PART C**

**Q. (4) Find the squares of the following numbers using the formula: a ^{2} = (a + b) (a – b) + b^{2}.**

(1) 82

(2) 49

(3) 109

(4) 97