Why Squaring a Sum is Not Equal to the Sum of Two Squares

It is a common misconception to assume that squaring a sum is equivalent to the sum of the squares of the individual terms. However, this is not the case. Using algebraic identities, we can demonstrate that squaring a sum introduces an additional term which makes it distinct from the sum of the squares of its individual components.

The Algebraic Identity of the Square of a Sum

The formula for the square of a sum is:

(a b)2 a2 2ab b2

This identity reveals that when you square the sum of two numbers (a and b), the result includes the sum of the squares of a and b, plus an additional term (2ab).

Consider the following steps to verify this:

Expand the left side of the equation: (a b)2 (a b)(a b) Distribute the terms: (a b)(a b) a2 ab ba b2 Simplify and combine like terms: a2 ab ba b2 a2 2ab b2

Therefore, (a b)2 a2 2ab b2.

When Are They Equal?

For the square of a sum to equal the sum of two squares, the additional term (2ab) must be zero. This occurs under the following conditions:

If either a or b is zero (i.e., a 0 or b 0) If both a and b are zero (i.e., a 0 and b 0)

Otherwise, the square of a sum will always include the extra term (2ab). This demonstrates that the square of a sum is generally not equal to the sum of the squares of the terms.

Finding Prime Numbers as the Sum of Squares

Interestingly, certain prime numbers can be expressed as the sum of two squares. This is based on a theorem in number theory, which states that a prime number p can be written as the sum of two squares if and only if p ≡ 1 mod 4. Here are some examples:

17: 17 ≡ 1 mod 4
17 42 12
(sqrt{17})2 42 12 29: 29 ≡ 1 mod 4
29 52 22
(sqrt{29})2 52 22

By choosing the right pairs of numbers, we can represent these prime numbers as the sum of two squares. For instance, we can express 17 and 29 as various pairs of squares:

17: (4 1)2 42 12 29: (5 2)2 52 22

Alternatively, we can choose other pairs of squares that sum to the same result:

17: (3 4)2 32 42 29: (5 2)2 32 42 10: (3 8)2 32 82

In conclusion, the square of a sum is not generally equal to the sum of the squares of its terms, unless a or b is zero. However, some special cases, such as certain prime numbers, can indeed be expressed as the sum of two squares. This highlights the interesting interplay between algebraic identities and number theory.