Imaginary Number
Imaginary number
An imaginary number is the square root of a negative real number. (The square root of a number is a second number that, when multiplied by itself, equals the first number.) As an example, √−25 is an imaginary number.
The problem with imaginary numbers arises because the square (the result of a number multiplied by itself) of any real number is always a positive number. For example, the square of 5 is 25. But the square of −5 (−5 × −5) is also 25. What does it mean, then, to say that the square of some number is −25. In other words, what is the answer to the problem √−25 = ?
As early as the sixteenth century, mathematicians were puzzled by this question. Italian mathematician Girolamo Cardano (1501–1576) is generally regarded as the first person to have studied imaginary numbers. Eventually, a custom developed for using the lowercase letter i to represent the square root of a negative number. Thus √−1 = i, and √−25 = √25 × √−1 = 5i.
Complex numbers
Imaginary numbers were largely a stepchild in mathematics until the nineteenth century. Then, they were incorporated into another mathematical concept known as complex numbers. A complex number is a number that consists of a real part and an imaginary part. For example, the number 5 + 3i is a complex number because it contains a real number (5) and an imaginary number (3i ). One reason complex numbers are important is that they can be manipulated in ways so as to eliminate the imaginary part.
[See also Complex numbers ]
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imaginary number
imaginary number In mathematics, the square root (√) of a negative quantity. The simplest, √−1, is usually represented by i. The numbers are so called because when first discovered they were widely regarded as meaningless. But they are necessary for the solution of many quadratic equations, the roots of which can be expressed only as complex numbers, which are composed of a real part and an imaginary part.
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imaginary number
imaginary number: see number.
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