# equality of complex numbers

The equality relation “=” among the is determined as consequence of the definition of the complex numbers^{} as elements of the quotient ring $\mathbb{R}/({X}^{2}+1)$, which enables the of the complex numbers as the ordered pairs $(a,b)$ of real numbers and also as the sums $a+ib$ where ${i}^{2}=-1$.

${a}_{1}+i{b}_{1}={a}_{2}+i{b}_{2}\mathit{\hspace{1em}}\u27fa\mathit{\hspace{1em}}{a}_{1}={a}_{2}\wedge {b}_{1}={b}_{2}$ | (1) |

This condition may as well be derived by using the field properties of $\u2102$ and the properties of the real numbers:

${a}_{1}+i{b}_{1}={a}_{2}+i{b}_{2}$ | $\u27f9{a}_{2}-{a}_{1}=-i({b}_{2}-{b}_{1})$ | ||

$\u27f9{({a}_{2}-{a}_{1})}^{2}=-{({b}_{2}-{b}_{1})}^{2}$ | |||

$\u27f9{({a}_{2}-{a}_{1})}^{2}+{({b}_{2}-{b}_{1})}^{2}=0$ | |||

$\u27f9{a}_{2}-{a}_{1}=0,{b}_{2}-{b}_{1}=0$ | |||

$\u27f9{a}_{1}={a}_{2},{b}_{1}={b}_{2}$ |

The implication in the reverse direction is apparent.

If $a+ib\ne 0$, then at least one of the real numbers $a$ and $b$ differs from 0. We can set

$a=r\mathrm{cos}\phi ,b=r\mathrm{sin}\phi ,$ | (2) |

where $r$ is a uniquely determined positive number and $\phi $ is an angle which is uniquely determined up to an integer multiple of $2\pi $. In fact, the equations (2) yield

$${a}^{2}+{b}^{2}={r}^{2}({\mathrm{cos}}^{2}\phi +{\mathrm{sin}}^{2}\phi )={r}^{2},$$ |

whence

$r=\sqrt{{a}^{2}+{b}^{2}}.$ | (3) |

Thus (2) implies

$\mathrm{cos}\phi ={\displaystyle \frac{a}{\sqrt{{a}^{2}+{b}^{2}}}},\mathrm{sin}\phi ={\displaystyle \frac{b}{\sqrt{{a}^{2}+{b}^{2}}}}.$ | (4) |

The equations (4) are , since the sum of the squares of their is 1. So these equations determine the angle $\phi $ up to a multiple of $2\pi $. We can write the

Theorem. Every complex number may be represented in the polar form

$$r(\mathrm{cos}\phi +i\mathrm{sin}\phi ),$$ |

where $r$ is the modulus and $\phi $ the argument of the number. Two complex numbers are equal if and only if they have equal moduli and, if the numbers do not vanish, their arguments differ by a multiple of $2\pi $.

Title | equality of complex numbers |
---|---|

Canonical name | EqualityOfComplexNumbers |

Date of creation | 2015-05-09 17:11:34 |

Last modified on | 2015-05-09 17:11:34 |

Owner | pahio (2872) |

Last modified by | pahio (2872) |

Numerical id | 7 |

Author | pahio (2872) |

Entry type | Topic |

Classification | msc 30-00 |

Related topic | ModulusOfAComplexNumber |

Related topic | ArgumentOfProductAndQuotient |

Related topic | ComplexLogarithm |