Chapter 1 - Numbers
MM - Page 13 of 14
|Numbers - Imaginary Numbers - Contents|
Outside the set of real numbers is the set of imaginary numbers. They use a mathematical contrivance that allows for taking the square root of a negative number. The letter i (j in electrical engineering) usually represents a special imaginary number. This number is equal to the square root of negative one. It then makes it possible to solve equations such as x squared equals negative one. Before imaginary numbers there is not a solution. With them, x can equal positive i or negative i as a solution. The set of imaginary numbers consists of i times every number in the set of real numbers. These two mirrored sets, combined in the equation form z = a + bi, create the complex numbers. Here a is a constant real number and b is a constant real number times i the special imaginary number. A coordinate system with the horizontal axis the set of real numbers and the vertical axis the set of imaginary numbers are used to map complex numbers. The creation of these complex numbers makes it possible to solve certain quadratic equations.
|Numbers - Imaginary Numbers - Examples|
Imaginary numbers usage:
To solve x squared = -1,
set i = the square root of -1,
then x = plus or minus the square root of -1,
and therefore x = +i or - i.
Complex numbers examples:
z = 3 + 2i,
z = -7 + 9i,
z = 2 - 6.5i.
|Numbers - Imaginary Numbers - Mysteries|
Math Mysteries: Even though the imaginary numbers are a mathematical impossibility they are useful in solving otherwise insolvable problems and are invaluable in some fields. This may show that the impossible should never be discounted.
|Numbers - Sections - Chapters|
|1 - Natural Numbers||2 - Zero||3 - Negative Numbers|
|4 - Integers||5 -Rational Numbers||6 - Common Fractions|
|7 - Decimal Fractions||8 - Irrational Numbers||9 - Absolute Value|
|10 - Infinity||11 - Special Numbers||12 - Prime Numbers|
|13 - Imaginary Numbers||14 - Systems of Numeration|
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