Squaring Numbers

Squaring numbers in the 50s

1. Square the last digit (keep the carry)   _ _ _ X
2. Multiply the last digit by 10, add the carry   _ _ X _
3. The first digits will be 25 plus the carry:   X X _ _

   Example:

     If the number to be squared is 53:

1. Square the last digit (keep the carry):
       3 × 3 = 9 (keep 3)     _ _ _ 9
2. Multiply the last digit by 10, add the carry:
       10 × 3 = 30 (keep 3)    _ _ 0 _
3. The first digits will be 25 plus the carry:
       25 (+ carry): 25 + 3 = 28    2 8 _ _
4. So 53 × 53 = 2809.

   See the pattern?
     If the number to be squared is 56:

1. Square the last digit (keep the carry):
       6 × 6 = 36 (keep 3)     _ _ _ 6
2. Multiply the last digit by 10, add the carry:
       10 × 6 = 60, 60 + 3 = 63    _ _ 3 _
3. The first digits will be 25 plus the carry:
       25 (+ carry): 25 + 6 = 31  3 1 _ _
4. So 53 × 53 = 3136.

Practice and you will soon be producing these products quickly and accurately.

Squaring numbers in the hundreds

1. Choose a number over 100 (keep it low for practice,
   then go higher when expert).
2. The last two places will be the square of
   the last two digits (keep any carry) _ _ _ X X.
3. The first three places will be the number plus
   the last two digits plus any carry: X X X _ _.

   Example:

1. If the number to be squared is 106:
2. Square the last two digits (no carry): 6 × 6 = 36: _ _ _ 3 6
3. Add the last two digits (06) to the number: 106 + 6 = 112: 1 1 2 _ _
4. So 106 × 106 = 11236.

   See the pattern?

1. If the number to be squared is 112:
2. Square the last two digits (keep carry 1): 12 × 12 = 144: _ _ _ 4 4
3. Add the last two digits (12) plus the carry (1) to the number:
   112 + 12 + 1 = 125: 1 2 5 _ _
4. So 112 × 112 = 12544.

With a little practice your only limit will be your ability to square the last two digits!

Squaring numbers in the 200s

1. Choose a number in the 200s (practice with numbers under 210, then progress to larger ones).
2. The first digit of the square is 4: 4 _ _ _ _
3. The next two digits will be 4 times the last 2 digits: _ X X _ _
4. The last two places will be the square of the last digit: _ _ _ X X

   Example:

1. If the number to be squared is 206:
2. The first digit is 4: 4 _ _ _ _
3. The next two digits are 4 times the last digit:
   4 × 6 = 24: _ 2 4 _ _
4. Square the last digit: 6 × 6 = 36: _ _ _ 3 6
5. So 206 × 206 = 42436.

For larger numbers work right to left:
1. Square the last two digits (keep the carry): _ _ _ X X
2. 4 times the last two digits + carry: _ X X _ _
3. Square the first digit + carry: X _ _ _ _

   See the pattern?

1. If the number to be squared is 225:
2. Square last two digits (keep carry):
   25x25 = 625 (keep 6): _ _ _ 2 5
3. 4 times the last two digits + carry:
   4x25 = 100; 100+6 = 106 (keep 1): _ 0 6 _ _
4. Square the first digit + carry:
   2x2 = 4; 4+1 = 5: 5 _ _ _ _
5. So 225 × 225 = 50625.

Squaring numbers in the 300s

1. Choose a number in the 300s (practice with numbers under 310, then progress to larger ones).
2. The first digit of the square is 9: 9 _ _ _ _
3. The next two digits will be 6 times the last 2 digits: _ X X _ _
4. The last two places will be the square of the last digit: _ _ _ X X

   Example:

1. If the number to be squared is 309:
2. The first digit is 9: 9 _ _ _ _
3. The next two digits are 6 times the last digit:
   6 × 9 = 54: _ 5 4 _ _
4. Square the last digit: 9 × 9 = 81: _ _ _ 8 1
5. So 309 × 309 = 95481.

Squaring numbers in the 400s

1. Choose a number in the 400s (keep the numbers low at first; then progress to larger ones).
2. The first two digits of the square are 16: 1 6 _ _ _ _
3. The next two digits will be 8 times the last 2 digits: _ _ X X _ _
4. The last two places will be the square of the last two digits: _ _ _ _ X X

   Example:

1. If the number to be squared is 407:
2. The first two digits are 16: 1 6 _ _ _ _
3. The next two digits are 8 times the last 2 digits:
   8 × 7 = 56: _ _ 5 6 _ _
4. Square the last digit: 7 × 7 = 49: _ _ _ 4 9
5. So 407 × 407 = 165,649.

For larger numbers reverse the steps:

1. Square the last two digits (keep the carry): _ _ _ _ X X
2. 8 times the last two digits + carry: _ _ X X _ _
3. 16 + carry: X X _ _ _ _

Squaring numbers in the 500s

1. Choose a number in the 500s (start with low numbers at first; then graduate to larger ones).
2. The first two digits of the square are 25: 2 5 _ _ _ _
3. The next two digits will be 10 times the last 2 digits: _ _ X X _ _
4. The last two places will be the square of the last two digits: _ _ _ _ X X

   Example:

1. If the number to be squared is 508:
2. The first two digits are 25: 2 5 _ _ _ _
3. The next two digits are 10 times the last 2 digits:
   10 × 8 = 80: _ _ 8 0 _ _
4. Square the last digit: 8 × 8 = 64: _ _ _ 6 4
5. So 508 × 508 = 258,064.

Squaring numbers in the 600s

1. Choose a number in the 600s (practice with smaller numbers, then progress to larger ones).
2. The first two digits of the square are 36: 3 6 _ _ _ _
3. The next two digits will be 12 times the last 2 digits: _ _ X X _ _
4. The last two places will be the square of the last two digits: _ _ _ _ X X

   Example:

1. If the number to be squared is 607:
2. The first two digits are 36: 3 6 _ _ _ _
3. The next two digits are 12 times the last 2 digits:
   12 × 07 = 84: _ _ 8 4 _ _
4. Square the last 2 digits: 7 × 7 = 49: _ _ _ _ 4 9
5. So 607 × 607 = 368,449.

For larger numbers reverse the steps:

1. If the number to be squared is 625:
2. Square the last two digits (keep carry):
   25x25 = 625 (keep 6): _ _ _ _ 2 5
3. 12 times the last 2 digits + carry:
   12x25 = 250 + 50 = 300 + 6 = 306: _ _ 0 6 _ _
4. 36 + carry: 36 + 3 = 39: 3 9 _ _ _ _
5. So 625 × 625 = 390,625.

Squaring numbers in the 700s

1. Choose a number in the 700s (practice with smaller numbers, then progress to larger ones).
2. Square the last two digits (keep the carry): _ _ _ _ X X
3. Multiply the last two digits by 14 and
   add the carry: _ _ X X _ _
4. The first two digits will be 49 plus the carry: X X _ _ _ _

   Example:

1. If the number to be squared is 704:
2. Square the last two digits (keep the carry):
   4 × 4 = 16: _ _ _ _ 1 6
3. Multiply the last two digits by 14 and
   add the carry: 14 × 4 = 56: _ _ 5 6 _ _
4. The first two digits will be 49 plus the carry: 4 9 _ _ _ _
5. So 704 × 704 = 495,616.

   See the pattern?

1. If the number to be squared is 725:
2. Square the last two digits (keep the carry):
   25 × 25 = 625: _ _ _ _ 2 5
3. Multiply the last two digits by 14 and
   add the carry: 14 × 25 = 10 × 25 + 4 × 25
   = 250 + 100 = 350. 350 + 6 = 356: 56: _ _ 5 6 _ _
4. The first two digits will be 49 plus the carry: 49 + 3 = 52: 5 2 _ _ _ _
5. So 725 × 725 = 525,625.

Squaring numbers between 800 and 810

1. Choose a number between 800 and 810.
2. Square the last two digits:
   _ _ _ _ X X
3. Multiply the last two digits by 16
   (keep the carry): _ _ X X _ _
4. Square 8, add the carry: X X _ _ _ _

   Example:

1. If the number to be squared is 802:
2. Square the last two digits:
   2 × 2 = 4: _ _ _ _ 0 4
3. Multiply the last two digits by 16:
   16 × 2 = 32: _ _ 3 2 _ _
4. Square 8: 6 4 _ _ _ _
5. So 802 × 802 = 643,204.

   See the pattern?

1. If the number to be squared is 807:
2. Square the last two digits:
   7 × 7 = 49: _ _ _ _ 4 9
3. Multiply the last two digits by 16
   (keep the carry): 16 × 7 = 112: _ _ 1 2 _ _
4. Square 8, add the carry (1): 6 5 _ _ _ _
5. So 807 × 807 = 651, 249.

Squaring numbers in the 900s

1. Choose a number in the 900s - start out easy with numbers near 1000; then go lower when expert.
2. Subtract the number from 1000 to get the difference.
3. The first three places will be the number minus the difference: X X X _ _ _.
4. The last three places will be the square of the difference: _ _ _ X X X
   (if 4 digits, add the first digit as carry).

   Example:

1. If the number to be squared is 985:
2. Subtract 1000 - 985 = 15 (difference)
3. Number - difference: 985 - 15 = 970: 9 7 0 _ _ _
4. Square the difference: 15 × 15 = 225: _ _ _ 2 2 5
5. So 985 × 985 = 970225.

   See the pattern?

1. If the number to be squared is 920:
2. Subtract 1000 - 920 = 80 (difference)
3. Number - difference: 920 - 80 = 840: 8 4 0 _ _ _
4. Square the difference: 80 × 80 = 6400: _ _ _ 4 0 0
5. Carry first digit when four digits: 8 4 6 _ _ _
6. So 920 × 920 = 846400.

Squaring numbers between 1000 and 1100

1. Choose a number between 1000 and 1100.
2. The first two digits are: 1,0 _ _, _ _ _
3. Find the difference between your number and 1000.
4. Multiply the difference by 2: 1,0 X X, _ _ _
5. Square the difference: 1,0 _ _, X X X

   Example:

1. If the number to be squared is 1007:
2. The first two digits are: 1,0 _ _, _ _ _
3. Find the difference: 1007 - 1000 = 7
4. Two times the difference: 2 × 7 = 14:
   1,0 1 4, _ _ _
5. Square the difference: 7 × 7 = 49:
   1,0 1 4, 0 4 9
6. So 1007 × 1007 = 1,014,049.

   See the pattern?

1. If the number to be squared is 1012:
2. The first two digits are: 1,0 _ _, _ _ _
3. Find the difference: 1012 - 1000 = 12
4. Two times the difference: 2 × 12 = 24:
   1,0 2 4, _ _ _
5. Square the difference: 12 × 12 = 144:
   1,0 2 4, 1 4 4
6. So 1012 × 1012 = 1,024,144.

Start with lower numbers and then extend your expertise to all the numbers between 1000 and 1100. Remember to add the first digit as carry when the square of the difference is four digits.

Squaring numbers between 3000 and 3099

1. Choose a number between 3000 and 3099. (Start with numbers below 3025 to begin with, then graduate to larger numbers.)
2. The first two digits are: 9 0 _ _ _ _ _
3. The next two digits are 6 times the last two digits:
   9 0 X X _ _ _
4. For the last three digits, square the last two digits in the number chosen (insert zeros when needed):
   9 0 _ _ X X X

   Example:

1. If the number to be squared is 3004:
2. The first two digits are: 9 0 _ _ _ _ _
3. The next two digits are 6 times the last two:
   6 × 4 = 24: _ _ 2 4 _ _ _
4. For the last three digits, square the last two:
   4 × 4 = 16: _ _ _ _ 0 1 6
5. So 3004 × 3004 = 9,024,016.

   See the pattern?

For larger numbers, reverse the order:

1. If the number to be squared is 3025:
2. For the last three digits, square the last two:
   25 × 25 = 625: _ _ _ _ 6 2 5
3. The middle two digits are 6 times the last two (keep the carry):
   6 × 25 = 150 (keep carry of 1): _ _ 5 0 _ _ _
4. The first two digits are 90 + the carry:
   90 + 1 = 91: 9 1 _ _ _ _ _
5. So 3025 × 3025 = 9,150,625.