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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Imaginary Number
Imaginary Number
The number is the basis of any imaginary number, which, in general, is any real number times i. For example, 5i is an imaginary number and is equivalent to –1 ÷ 5. The real numbers are those numbers that can be expressed as terminating, repeating, or nonrepeating decimals; they include positive and negative numbers. The product of two negative real numbers is always positive. Thus, there is no real number that equals–1 when multiplied by itself—that is, no real number satisfies the equation x^{2} =–1 in the real number system. The imaginary number i was invented to provide a solution to this equation, and every imaginary number represents the solution to a similar equation (e.g., 5i is a solution to the equation x^{2} =–25).
In addition to providing solutions for algebraic equations, the imaginary numbers, when combined with the real numbers, form the complex numbers. Each complex number is the sum of a real number and an imaginary number, such as (6 + 9i). The complex numbers are very useful in mathematical analysis, the study of electricity and magnetism, the physics of quantum mechanics, and in the practical field of electrical engineering. In terms of the complex numbers, the imaginary numbers are equivalent to those complex numbers for which the real part is zero.
See also Square root.
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Imaginary Number
Imaginary number
The number i = √ –1 is the basis of any imaginary number, which, in general, is any real number times i. For example, 5i is an imaginary number and is equivalent to –1 ÷ 5. The real numbers are those numbers that can be expressed as terminating, repeating, or nonrepeating decimals; they include positive and negative numbers. The product of two negative real numbers is always positive. Thus, there is no real number that equals –1 when multiplied by itself—that is, no real number satisfies the equation x2 = –1 in the real number system. The imaginary number i was invented to provide a solution to this equation, and every imaginary number represents the solution to a similar equation (e.g., 5i is a solution to the equation x2 = –25).
In addition to providing solutions for algebraic equations, the imaginary numbers, when combined with the real numbers, form the complex numbers . Each complex number is the sum of a real number and an imaginary number, such as (6 + 9i). The complex numbers are very useful in mathematical analysis, the study of electricity and magnetism , the physics of quantum mechanics , and in the practical field of electrical engineering . In terms of the complex numbers, the imaginary numbers are equivalent to those complex numbers for which the real part is zero .
See also Square root.
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