Imaginary Numbers

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Imaginary Numbers

Like real numbers, imaginary numbers are a subset of the set of all. Imaginary numbers are typically represented as either the constant i (in mathematics) or j (in engineering and the sciences), where i² = -1 or sqrt(-1) = i. Like the real numbers, imaginary numbers have relative magnitude and can be plotted along a number line, with 3i > i. Imaginary numbers may also be negative, with 3i > i > -i > -3i. Any imaginary number ki can be written as the complex number 0 + ki.

Properties of Imaginary Numbers

Imaginary numbers follow the same rules of addition and subtraction available to real numbers. For example, 3 + 2 = 5 and 4 - 6 = -2 in the real number plane. Similarly, 3i + 2i = 5i and 4i - 6i = -2i in the imaginary number plane. Note, however, that 3 + 3i ( 6, as 3 = 3 * sqrt(1) and 3i = 3 * sqrt(-1), which are not equal.

Scalar multiples of imaginary numbers also follow the conventions of real numbers. For example, 2 x 3 = 6. Similarly, 2 x 3i = 6i. Note, however, that if the scalar factor is imaginary itself, such as 2i, the product 2i x 2i = 4i² = -4. If an imaginary scalar number is multiplied against an imaginary number, then, the product will be a real number; the product of any two imaginary numbers is real.

Division within imaginary numbers also follows the rules of real numbers, but the quotient will also be a real number. For example, 4i / 2i = 2 and 6i / 2i = 3. If an imaginary number is divided by a real number (in effect, multiplying by the reciprocal of the division), then the number will be imaginary. For example 20i / 4 = 20i x 1/4 = 5i.

Just as i can be multiplied by itself to form a square, so can i be raised to any power. A pattern quickly develops:

• i¹ = i
• i² = -1
• i³ = i x i² = -i
• i⁴ = i² x i² = 1
• i⁵ = i x i⁴ = i
• i⁶ = i² x i⁴ = -1...

It is also possible to raise i to a power that is not a positive integer, such as -1 or 1/2. Interestingly, i ^ -1 = 1/i = 1/i x i/i = i/i² = -i, which provides the interesting result that the reciprocal of i is equal to its opposite, a useful relationship in engineering. Further, it is straightforward to show that sqrt(i) = (1 + i)/sqrt(2), indicating that i itself is a square.

One of the first known explorations of imaginary numbers occurred in Heron's Stereometica. In his work, Heron developed the expression sqrt(81 - 144), for which no understanding existed at the time. Later, mathematicians seeking solutions to quadratic equations encountered equations for which they contended no solution existed, such as x² + 1 = 0 and x² + 2x² + 2 = 0. The equations do, however, have solutions as valid as the solutions to the equation x² - 1 = 0 (that is, x = 1 and x = -1. In the former case, the solutions are x = i and x = -i; in the latter case, the solutions are x = -1 + i and x = -1 - i. This understanding of imaginary numbers had led to many mathematical and scientific advances; imaginary numbers are used widely in science and engineering applications across the world today.