Squaring 2 digit numbers – I

Squaring two digit numbers is canonical in mathematics and being able to do it fast and mentally is a useful skill. The most famous method can be reviewed here. Additionally, mastering this skill opens up pathways for mastering yet more impressive stuff like squaring three digit numbers which I’ll talk about in the next post.

The method in this entry is based on the following algebraic manipulation: $(10x + y)^2=100(x)(x+1)+20(x+1)(y-5) +(10-y)^2, \ \ \ 5 \leq y \leq 9$

Using the above formula, we reduce a multiplication problem to that of lighter multiplications followed by simple addition. The simplicity of the formula lies in the fact that we do not have to plug in values and calculate every time. There is a pattern based on this formula and once the pattern is understood, this method becomes a much easier scenario of adding three numbers together.

From the above identity, let $A=100(x)(x+1) \\ B=20(x+1)(y-5) \\ C=(10-y)^2$

After studying the formula, we infer that for A and C, no heavy calculations are needed and can be figured out almost instantly. For A, you multiply the first digit by its successor and append two zeroes. For C, you figure out its distance from 10 and square it. For B, it will take some to practice and get the hang of it but after a while one can see that all we are doing is first taking $x+1$ and multiplying it with 10 and doubling it. We then figure out how far $10x+y$ is from $10x+5$ and multiply it with that distance.

This will become clear through the following two examples:

Example 1 $68^2$

Step 1: Find A: Multiply 6 and 7 and append two 0’s to get 4200. Find C: The unit’s digit is 8, so squaring the distance from 10 that is 2 gives you 4.

Step 2:  Find B: Take 7 and multiply it first by 10 and then by 2 to get 140. 68 is 3 units awayfrom 65. So multiply 140 by 3 to get 420.

Step 3: Add A, B and C to get the final answer. 4200 + 420 + 4 = 4624.

Example 2 $46^2$

Step 1: Find A: Multiply 4 and 5 and append two 0’s to get 2000. Find C: The unit’s digit is 6 which is 4 units away from 10. So $4^2$ gives you 16

Step 2: Find B: Take 5 and multiply it first by 10 and then by 2 to get 100. 46 is 1 unit away from 45. So multiply 100 by 1 to get 100.

Step 3: Add A, B and C to get the final answer. 2000 + 100 + 16 = 2116.

This method works when $5 \leq y \leq 9$. For $y < 5$, we simply use the deviation method which can be found here.