## The Square Root Function

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**The Square Root Function (plaintext version)**

The square root function, for real arguments, is defined in 2 steps: first on the set of non-negative real numbers, and then, using the first step as a stepping stone, on the set of negative real numbers.

Definition 1. If x is a non-negative real number, then (1/2)@x is that non-negative real number u such that 2@u = x.

Example 1. (1/2)@49 = 7, because 2@7 = 49.

Theorem 1. If each of x and y is a non-negative real number, then

(1/2)@x(1/2)@y = (1/2)@(xy).

Proof: 2@((1/2)@x(1/2)@y) = ((1/2)@x(1/2)@y)((1/2)@ x(1/2)@y)

= ((1/2)@x(1/2)@x)((1/2)@ y(1/2)@y) = xy. Since there can be only one number non‑negative number u such that 2@u = xy, the conclusion follows. █

Definition 2. If x < 0, then (1/2)@x = ((1/2)@ (-x))i, where i is the imaginary unit.

Example 2. (1/2)@(-49) = ((1/2)@|-49|)i = ((1/2)@49)i = 7i.

Definition 3. If t is a tuple of real numbers, then sigma(t) is the number of terms in t that are negative.

Example 3. sigma(-3,-7) = 2.

Theorem 2. If each of x and y is a real number, then (1/2)@x(1/2)@y

= ((1/2)@|xy|)(sigma(x,y))@i.

Proof:

Case 1. x ³ 0 and y ³ 0. Then sigma(x,y) = 0 and |xy| = xy. The conclusion follows.

Case 2. x < 0 and y ³ 0. Then sigma(x,y) = 1. The conclusion follows.

Case 3. x ³ 0 and y < 0. Interchange x and y in Case 2. The conclusion follows.

Case 4. x < 0 and y < 0. Then sigma(x,y) = 2, (1/2)@x = ((1/2)@ (-x))i, and

(1/2)@y = ((1/2)@(-y))i, and so (1/2)@x(1/2)@y

= (((1/2)@(-x))i)(((1/2)@(-y))i) = ((1/2)@((-x)(-y)))(2@i) = ((1/2)@|xy|)(sigma(x,y))@i.

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Example 4. (1/2)@(-2)(1/2)@(-8) = ((1/2)((-(-2))(-(-8))))(sigma(-2,-8))@i

= ((1/2)@(2´8)(2@i) = ((1/2)@(2´8))(-1) = ((1/2)@16)(-1) = 4(-1) = -4.

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