The term perfect square is used in mathematics in two meanings:

• an integer which is the square of some other integer, i.e. can be written in the form n² for some integer n (and because of this a square is always nonnegative)
  • Examples: 0, 1, 4, 9, 16, 25, 36, 49, ... See square number.
• an algebraic expression that can be factored as the square of some other expression, e.g. a² ± 2ab + b² = (a ± b)². (see Square (algebra)).

This is not the same as a magic square.

Using differences of squares as multiplication

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Integer multiplication can be done entirely by a difference of two squares.

Examples:

• $10\backslash times\; 10\; =\; (10-0)\backslash times\; (10+0)\; =\; 10^2\; -\; 0^2\; =\; 100\; -\; 0\; =\; 100$
• $9\backslash times\; 11\; =\; (10-1)\backslash times\; (10+1)\; =\; 10^2\; -\; 1^2\; =\; 100\; -\; 1\; =\; 99$
• $8\backslash times\; 12\; =\; (10-2)\backslash times\; (10+2)\; =\; 10^2\; -\; 2^2\; =\; 100\; -\; 4\; =\; 96$
• $7\backslash times\; 13\; =\; (10-3)\backslash times\; (10+3)\; =\; 10^2\; -\; 3^2\; =\; 100\; -\; 9\; =\; 91$

In general, the product of two numbers is equal to the square of their average minus their difference from the average squared.

• $A\backslash times\; B\; =-\; [(A-B)/2^2$

A geometric constructive "proof" of this relation is shown the following animation:

The starting rectangle is A by B. The resulting large square is length (A+B)/2, and the smaller gray square (remainder being subtracted) is length |A-B|/2.

Using this relation, you can multiply relatively large nearly equal numbers more quickly if you memorize a relatively small list of squares.

If you're multiplying an even by an odd, you can avoid "halves" by adjust one number, by requiring one more addition at the end

• $A\backslash times\; B\; =\; A\backslash times\; (B-1)\; +\; A$

Example:

• $27\backslash times\; 34\; =33+\; 27\; =-\; 3^2+\; 27\; =\; 900\; -\; 9\; +\; 27\; =\; 918$

Last Digits of Perfect Squares in Base 10

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In base 10, the last digit of every perfect square is either 0 or 1 or 4 or 5 or 6 or 9. If you know a number is a perfect square then you can determine the last digit of its square root as follows:

• If the last digit of the perfect square is 0, then the last digit of the square root is 0.
• If the last digit of the perfect square is 1, then the last digit of the square root is either 1 or 9.
• If the last digit of the perfect square is 4, then the last digit of the square root is either 2 or 8.
• If the last digit of the perfect square is 5, then the last digit of the square root is 5.
• If the last digit of the perfect square is 6, then the last digit of the square root is either 4 or 6.
• If the last digit of the perfect square is 9, then the last digit of the square root is either 3 or 7.

Trivia: Determining the Square Root of a Perfect Square Less Than 10000

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Suppose you are given a perfect square that is greater than or equal 100 but less than 10,000. All of these perfect squares have two digit square roots. A simple party trick algorithm exists for determining its square root with one or two guesses.

First, separate the digits into pairs of digits starting from the right and moving to the left. For examples, consider the perfect squares 4624, 729, and 1600. Break 4624 into 46|24, or break 729 into 7|29, or 1600 into 16|00.

Second, we find the first digit of the square root. If the leftmost pair is a perfect square, find its square root. If the leftmost pair is not a perfect square, find the square root of the largest perfect square smaller than the leftmost pair. That square root will be the leftmost digit of the square root. For examples, for 46|24, the leftmost pair is 46, which is not a perfect square; but the largest perfect square less than 46 is 36 which has a square root of 6. For 7|29, 7 is not a perfect square, but 4 is the largest square less than 7, and its square root is 2. For 16|00, the square root of 16 is 4.

Third, as mentioned in an earlier section, if we know the last digit of a perfect square in base ten, then we can also know the last digit of its square root in two guesses. Thus, we know the last digit of the square root of 4624 is either 2 or 8; for 729 it is either 3 or 7; for 1600 it is 0.

Fourth, we can guess the square root by conjuncting the second and third steps. The square root of 4624 is either 62 or 68; the square root of 729 is either 23 or 27; the square root of 1600 is 40.

If you use a calculator, you will find that the square roots of 4624, 729, and 1600 are actually 68, 27, and 40.

See also

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• Perfect power

External links

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• List of perfect square between 1-10,000
• JavaScript code for finding ten-digit numbers, consisting of distinct digits, that are perfect squares

Algebra | Integer sequences

Cuadrado perfecto | Carré parfait | Quadrado perfeito | Popolni kvadrat | 完全平方