Square it in your head

The idea

A number can be squared by writing it as a sum or difference of convenient numbers and applying the identity for (a + b)², (a – b)² or (a + b + c)².

A round number

How would you find 43² without a calculator? Here is a trick: 43 is 40 + 3, and 40 is a round number that is easy to square.

Which split of 43 makes 43² easiest?

• 40 + 3
• 20 + 23
• 21 + 22

Choose the split

Pick a split a + b for each number. Any split gives the right answer, but only one of them is easy by hand.

Round numbers

To square a number, write it as a sum or difference of convenient numbers and use the identity: (a + b)² = a² + 2ab + b².

43² = (40 + 3)² = 40² + 2 × 40 × 3 + 3² = 1600 + 240 + 9 = 1849 For a number just below a round one, let b be negative: 29 = 30 + (−1). The identity for a number split into three parts, (a + b + c)², comes later in this chapter.

Notes

Write a number as a sum or difference of convenient numbers, then square it with the identity: 43² = (40 + 3)² = 1849.

Check yourself

Find 52² using (50 + 2)².

Answer: 2704

2500 + 200 + 4 = 2704.

Which split makes 98² easiest?

Find 105² using (100 + 5)².

Answer: 11025

10000 + 1000 + 25 = 11025.

Find 99². Write 99 = 100 + (−1). The middle term 2ab is negative.

Answer: 9801

10000 − 200 + 1 = 9801.

• 90 + 8. It works, but 90², 2 × 90 × 8 and 8² are all harder.
• 100 − 2 — correct. Yes. 98 = 100 + (−2), so 98² = 10000 − 400 + 4 = 9604.
• 49 + 49. It works, but 49² is not easier than 98².