Starting from the 16th-century, mathematicians faced the special numbers' necessity, also known nowadays as complex numbers. A complex number is a number of the form a+bi, where a,b — real numbers, and i — imaginary unit is a solution of the equation: i2=-1. Show It's interesting to trace the evolution of the mathematician opinions on complex number problems. Here are some quotes from ancient works on this topic:
It is used in several ways in complex number defining. We'll show three of them Algebraic formwhere a and b - real numbers, i - imaginary unit, so that i2=-1. a - corresponds to real part, b - imaginary part. Polar form, Exponential form (Euler's form)is a simplified version of the polar form derived from Euler's formula. Complex numberCalculation precision Digits after the decimal point: 2 Argument principal value (rad) Argument principal value (degrees) Complex plane The file is very large. Browser slowdown may occur during loading and creation. Complex number argument is a multivalued function , for integer k. Principal value of the argument is a single value in the open period (-π..π]. All elementary arithmetic operations are defined for complex number: Complex number elementary operationsCalculation precision Digits after the decimal point: 2 The file is very large. Browser slowdown may occur during loading and creation. Complex number additionOne complex number can be added to another in the same way as polynomials: Complex number multiplicationUsing complex number definition i*i=-1, we can easily explain complex number multiplication
formula: Complex number divisionTo derive complex number division formula we multiply both numerator and denominator by the complex number conjugate (to eliminate imaginary unit in denominator): Complex number exponentiationUsing Euler's form it is simple: n-th degree rootFrom De Moivre's formula, n nth roots of z (the power of 1/n) are given by: What is the absolute value of the complex number − 4 − √ 2i?Summary: The absolute value of the complex number -4 - √2i is √18.
Why is the absolute value of a complex number?The absolute value of a complex number , a+bi (also called the modulus ) is defined as the distance between the origin (0,0) and the point (a,b) in the complex plane.
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