The 11 Rule

You likely all know the 10 rule (to multiply by 10, just add a 0 behind the number) but do you know the 11 rule? It is as easy! You should be able to do this one in you head for any two digit number. Practice it on paper first!

To multiply any two digit number by 11:

• For this example we will use 54.

• Separate the two digits in you mind (5__4).

• Notice the hole between them!

• Add the 5 and the 4 together (5+4=9)

• Put the resulting 9 in the hole 594. That's it! 11 x 54=594

The only thing tricky to remember is that if the result of the addition is greater than 9, you only put the "ones" digit in the hole and carry the "tens" digit from the addition. For example 11 x 57 ... 5__7 ... 5+7=12 ... put the 2 in the hole and add the 1 from the 12 to the 5 in to get 6 for a result of 627 ... 11 x 57 = 627
Practice it on paper first!

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Squaring a 2-digit number ending in 8

1. Choose a 2-digit number ending in 8.
2. The last digit of the answer is always 4: _ _ _ 4
3. Multiply the first digit by 6 and add 6 (keep the
   carry): _ _ X _
4. Multiply the first digit by the next consecutive
   number and add the carry: the product is the first
   two digits: XX _ _.

Example:

1. If the number is 78:
2. The last digit of the answer is 4: _ _ _ 4
3. Multiply the first digit (7) by 6 and add 6 (keep the
   carry): 7 × 6 = 42, 42 + 6 = 48; the next digit of the
   answer is 8 (keep carry 4): _ _ 8 4
4. Multiply the first digit (7) by the next number (8)
   and add the carry (4):
   7 × 8 = 56, 56 + 4 = 60 (the first two digits): 6 0 _ _
5. So 78 × 78 = 6084.

See the pattern?

1. For 38 × 38
2. The last digit of the answer is 4: _ _ _ 4
3. Multiply the first digit (3) by 6 and add 6 (keep the
   carry): 3 × 6 = 18, 18 + 6 = 24; the next digit of the
   answer is 4 (keep carry 2): _ _ 4 4
4. Multiply the first digit (3) by the next number (4)
   and add the carry (2):
   3 × 4 = 12, 12 + 2 = 14 (the first two digits): 1 4 _ _
5. So 38 × 38 = 1444

Learn the pattern, practice other examples, and you will be a whiz at giving these squares.

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Squaring a 2-digit number ending in 7

1. Choose a 2-digit number ending in 7.
2. The last digit of the answer is always 9: _ _ _ 9
3. Multiply the first digit by 4 and add 4
   (keep the carry): _ _ X _
4. Multiply the first digit by the next consecutive number and
   add the carry: the product is the first two digits:
   XX _ _.

Example:

1. If the number is 47:
2. The last digit of the answer is 9: _ _ _ 9
3. Multiply the first digit (4) by 4 and add 4
   (keep the carry): 4 × 4 = 16, 16 + 4 = 20; the next
   digit of the answer is 0 (keep carry 2): _ _ 0 9
4. Multiply the first digit (4) by the next number (5)
   and add the carry (2):
   4 × 5 = 20, 20 + 2 = 22 (the first two digits): 2 2 _ _
5. So 47 × 47 = 2209.

See the pattern?

1. For 67 × 67
2. The last digit of the answer is 9: _ _ _ 9
3. Multiply the first digit (6) by 4 and add 4 (keep the
   carry): 4 × 6 = 24, 24 + 4 = 28; the next digit of the
   answer is 0 (keep carry 2): _ _ 8 9
4. Multiply the first digit (6) by the next number (7)
   and add the carry (2):
   6 × 7 = 42, 42 + 2 = 44 (the first two digits): 4 4 _ _
5. So 67 × 67 = 4489.
   

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Squaring a 2-digit number ending in 6

1. Choose a 2-digit number ending in 6.
2. Square the second digit (keep the carry): the last digit
   of the answer is always 6: _ _ _ 6
3. Multiply the first digit by 2 and add the carry
   (keep the carry): _ _ X _
4. Multiply the first digit by the next consecutive number and
   add the carry: the product is the first two digits:
   XX _ _.

Example:

1. If the number is 46, square the second digit :
   6 × 6 = 36; the last digit of the answer is 6
   (keep carry 3): _ _ _ 6
2. Multiply the first digit (4) by 2 and add the carry
   (keep the carry): 2 × 4 = 8, 8 + 3 = 11; the next digit
   of the answer is 1: _ _ 1 6
3. Multiply the first digit (4) by the next number (5)
   and add the carry: 4 × 5 = 20, 20 + 1 = 21
   (the first two digits): 2 1 _ _
4. So 46 × 46 = 2116.

See the pattern?

1. For 76 × 76, square 6 and keep the carry (3):
   6 × 6 = 36; the last digit of the answer is 6: _ _ _ 6
2. Multiply the first digit (7) by 2 and add the carry:
   2 × 7 = 14, 14 + 3 = 17; the next digit of the answer
   is 7 (keep carry 1): _ _ 7 6
3. Multiply the first digit (7) by the next number (8)
   and add the carry: 7 × 8 = 56, 56 + 1 = 57
   (the first two digits: 5 7 _ _
4. So 76 × 76 = 5776.
   

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Squaring a 2-digit number ending in 5

1. Choose a 2-digit number ending in 5.
2. Multiply the first digit by the next consecutive number.
3. The product is the first two digits: XX _ _.
4. The last part of the answer is always 25: _ _ 2 5.

Example:

1. If the number is 35, 3 × 4 = 12 (first digit
   times next number). 1 2 _ _
2. The last part of the answer is always 25: _ _ 2 5.
3. So 35 × 35 = 1225.

See the pattern?

1. For 65 × 65, 6 × 7 = 42 (first digit
   times next number): 4 2 _ _.
2. The last part of the answer is always 25: _ _ 2 5.
3. So 65 × 65 = 4225.

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Square of a rational number

Is 2ⁿ + 3ⁿ (where n is an integer) ever the square of a rational number?

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Squaring a 2-digit number ending in 5

1. Choose a 2-digit number ending in 5.
2. Multiply the first digit by the next consecutive number.
3. The product is the first two digits: XX _ _.
4. The last part of the answer is always 25: _ _ 2 5.

Example:

1. If the number is 35, 3 × 4 = 12 (first digit
   times next number). 1 2 _ _
2. The last part of the answer is always 25: _ _ 2 5.
3. So 35 × 35 = 1225.

See the pattern?

1. For 65 × 65, 6 × 7 = 42 (first digit
   times next number): 4 2 _ _.
2. The last part of the answer is always 25: _ _ 2 5.
3. So 65 × 65 = 4225.

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Squaring a 2-digit number ending in 4

1. Take a 2-digit number ending in 4.
2. Square the 4; the last digit is 6: _ _ _ 6
   (keep carry, 1.)
3. Multiply the first digit by 8 and add the carry (1);
   the 2nd number will be the next to the last digit:
   _ _ X 6 (keep carry).
4. Square the first digit and add the carry: X X _ _.

Example:

1. If the number is 34, 4 × 4 = 16 (keep carry, 1);
   the last digit is _ _ _ 6.
2. 8 × 3 = 24 (multiply the first digit by 8), 24 + 1 = 25
   (add the carry):
   the next digit is 5: _ _ 5 6. (Keep carry, 2.)
3. Square the first digit and add the carry, 2: 1 1 5 6.
4. So 34 × 34 = 1156.

See the pattern?

1. For 84 × 84, 4 × 4 = 16 (keep carry, 1);
   the last digit is _ _ _ 6.
2. 8 × 8 = 64 (multiply the first digit by 8),
   64 + 1 = 65 (add the carry):
   the next digit is 5: _ _ 5 6. (Keep carry, 6.)
3. Square the first digit and add the carry, 6: 7 0 5 6.
4. So 84 × 84 = 7056.

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Compute this series?

Compute the infinite product

[sin(x) cos(x/2)]^1/2 · [sin(x/2) cos(x/4)]^1/4 · [sin(x/4) cos(x/8)]^1/8 · ... ,

where 0 x 2.

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Squaring a 2-digit number ending in 4

1. Take a 2-digit number ending in 4.
2. Square the 4; the last digit is 6: _ _ _ 6
   (keep carry, 1.)
3. Multiply the first digit by 8 and add the carry (1);
   the 2nd number will be the next to the last digit:
   _ _ X 6 (keep carry).
4. Square the first digit and add the carry: X X _ _.

Example:

1. If the number is 34, 4 × 4 = 16 (keep carry, 1);
   the last digit is _ _ _ 6.
2. 8 × 3 = 24 (multiply the first digit by 8), 24 + 1 = 25
   (add the carry):
   the next digit is 5: _ _ 5 6. (Keep carry, 2.)
3. Square the first digit and add the carry, 2: 1 1 5 6.
4. So 34 × 34 = 1156.

See the pattern?

1. For 84 × 84, 4 × 4 = 16 (keep carry, 1);
   the last digit is _ _ _ 6.
2. 8 × 8 = 64 (multiply the first digit by 8),
   64 + 1 = 65 (add the carry):
   the next digit is 5: _ _ 5 6. (Keep carry, 6.)
3. Square the first digit and add the carry, 6: 7 0 5 6.
4. So 84 × 84 = 7056.

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Squaring a 2-digit number ending in 3

1. Take a 2-digit number ending in 3.
2. The last digit will be _ _ _ 9.
3. Multiply the first digit by 6: the 2nd number will be
   the next to the last digit: _ _ X 9.
4. Square the first digit and add the number carried from
   the previous step: X X _ _.

Example:

1. If the number is 43, the last digit is _ _ _ 9.
2. 6 × 4 = 24 (six times the first digit): _ _ 4 9.
3. 4 × 4 = 16 (square the first digit), 16 + 2 = 18
   (add carry): 1 8 4 9.
4. So 43 × 43 = 1849.

See the pattern?

1. For 83 × 83, the last digit is _ _ _ 9.
2. 6 × 8 = 48 (six times the first digit): _ _ 8 9.
3. 8 × 8 = 64 (square the first digit), 64 + 4 = 68
   (add carry): 6 8 8 9.
4. So 83 × 83 = 6889.

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Squaring a 2-digit number ending in 2

1. Take a 2-digit number ending in 2.
2. The last digit will be _ _ _ 4.
3. Multiply the first digit by 4: the 2nd number will be
   the next to the last digit: _ _ X 4.
4. Square the first digit and add the number carried from
   the previous step: X X _ _.

Example:

1. If the number is 52, the last digit is _ _ _ 4.
2. 4 × 5 = 20 (four times the first digit): _ _ 0 4.
3. 5 × 5 = 25 (square the first digit), 25 + 2 = 27 (add carry): 2 7 0 4.
4. So 52 × 52 = 2704.

See the pattern?

1. For 82 × 82, the last digit is _ _ _ 4.
2. 4 × 8 = 32 (four times the first digit): _ _ 2 4.
3. 8 × 8 = 64 (square the first digit), 64 + 3 = 67 (add carry): 6 7 2 4.
4. So 82 × 82 = 6724.

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Squaring a 2-digit number ending in 1

1. Take a 2-digit number ending in 1.
2. Subtract 1 from the number.
3. Square the difference.
4. Add the difference twice to its square.
5. Add 1.

Example:

1. If the number is 41, subtract 1: 41 - 1 = 40.
2. 40 × 40 = 1600 (square the difference).
3. 1600 + 40 + 40 = 1680 (add the difference twice
   to its square).
4. 1680 + 1 = 1681 (add 1).
5. So 41 × 41 = 1681.

See the pattern?

1. For 71 × 71, subtract 1: 71 - 1 = 70.
2. 70 × 70 = 4900 (square the difference).
3. 4900 + 70 + 70 = 5040 (add the difference twice
   to its square).
4. 5040 + 1 = 5041 (add 1).
5. So 71 × 71 = 5041.

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Squaring a 2-digit number beginning with 9

1. Take a 2-digit number beginning with 9.
2. Subtract it from 100.
3. Subtract the difference from the original number:
   this is the first part of the answer.
4. Square the difference: this is the last part of the answer.

Example:

1. If the number is 96, subtract: 100 - 96 = 4, 96 - 4 = 92.
2. The first part of the answer is 92 _ _ .
3. Take the first difference (4) and square it: 4 × 4 = 16.
4. The last part of the answer is _ _ 16.
5. So 96 × 96 = 9216.

See the pattern?

1. For 98 × 98, subtract: 100 - 98 = 2, 98 - 2 = 96.
2. The first part of the answer is 96 _ _.
3. Take the first difference (2) and square it: 2 × 2 = 4.
4. The last part of the answer is _ _ 04.
5. So 98 × 98 = 9604.

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Squaring a 2-digit number beginning with 5

1. Take a 2-digit number beginning with 5.
2. Square the first digit.
3. Add this number to the second number to find the first part of the answer.
4. Square the second digit: this is the last part of the answer.

Example:

1. If the number is 58, multiply 5 × 5 = 25 (square the first digit).
2. 25 + 8 = 33 (25 plus second digit).
3. The first part of the answer is 33 3 3 _ _
4. 8 × 8 = 64 (square second digit).
5. The last part of the answer is 64 _ _ 6 4
6. So 58 × 58 = 3364.

See the pattern?

1. For 53 × 53, multiply 5 × 5 = 25 (square the first digit).
2. 25 + 3 = 28 (25 plus second digit).
3. The first part of the answer is 28 2 8 _ _
4. 3 × 3 = 9 (square second digit).
5. The last part of the answer is 09 _ _ 0 9
6. So 53 × 53 = 2809.

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Play with four

Using four 4's and any operations, try to write equations that have the numbers from 0 to 100 as the answer.
Example: 49 = 4! * (sqrt 4) + (4/4)

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