# Squaring numbers ending in 9 or 1.

In our previous posts, we have discussed squaring numbers ending in 5, squaring numbers near powers of 10 and numbers near 50, 500 etc. We will now discuss squaring numbers ending in 9 or 1.

If **a** is any number ending in zero, a number ending in 9 can be represented by **(a – 1)** while a number ending in 1 can be expressed as **(a + 1)**.

Using a common identity, we have for numbers ending in 1,

(a + 1)^{2} = a^{2} + 2(1)a + 1^{2} = a^{2} + 2a + 1

Thus 31^{2} = 30^{2} + 2(30) + 1 = 900 + 60 + 1 = 961

and 71^{2} = 70^{2} + 2(70) + 1 = 4900 + 140 + 1 = 5041

Some would prefer to use this variation which is easier to remember

(a + 1)^{2} = a^{2} + 2a + 1 = **a ^{2} + a + (a + 1)**

Now we have: 41^{2} = 40^{2} + 40 + 41 = 1681

and 61^{2} = 60^{2} + 60 + 61 = 3721

You can check your answers to these last two example using the technique for squaring numbers near 50. ( 41 = 50 – 9 and 61 = 50 + 11).

For numbers ending in 9, we can use a similar technique,

(a – 1)^{2} = a^{2} – 2(1)a + 1^{2} = a^{2} – 2a + 1

Using this result, 29^{2} = 30^{2} – 2(30) + 1 = 900 – 60 + 1 = 841

and 69^{2} = 70^{2} – 2(70) + 1 = 4900 – 140 + 1 = 4761

Similarly, we can use this easier to remember variation shown earlier:

39^{2} = 40^{2} – 40 – 39 = 1521

59^{2} = 60^{2} – 60 – 59 = 3481

For our given example, we use any of our variations;

3999^{2 }= 4000^{2} – 2(4000) + 1 = 16,000,000 – 8,000 + 1 = 15, 992,001

Or 3999^{2 }= 4000^{2} – 4000 – 3999 = 15,996,000 – 3,999 = 15, 992,001

Some may prefer to use digit sums to pick the right answer:

We can cast out the three 9s in 3,999 and 3^{2} = 9. The only choice with a repeated digital sum of 9 (or 0 when casting out 9s is used) is letter **b**.

Exercises: find the square of the following numbers:

- 21
- 91
- 79
- 89
- 101
- 121
- 109
- 199
- 4001
- 7999

Answers to previous exercises: Find the square of the following numbers near 50, 500, etc

- 53 → 25+3 |3^2 = 2809
- 57 → 25+7 |7^2 = 3249
- 63 → 25+13 |13^2 = 3969
- 46 → 25-4 |4^2 = 2116
- 44 → 25-6 |6^2 = 1936
- 36 → 25-14 |14^2 = 1296
- 507 → 250+7 |7^2 = 257,049
- 497 → 250-3 |3^2 = 247,009
- 5021 → 2500 + 21 |21^2 = 25, 210, 441
- 4989 → 2500 – 11 |11^2 = 24, 890, 121