My Coding > Mathematics > Complex numbers > How to divide complex numbers

How to divide complex numbers

In this article, you will read how to divide complex numbers. To make complex number division, you need to do the following steps:

  1. Multiply the numerator and denominator of the fraction by the complex conjugate of the denominator. The complex conjugate of a complex number a + bi is a - bi.
  2. Simplify the numerator and denominator using the distributive property of multiplication.
  3. Use the fact that \(i^2 = -1\) to simplify any occurrences of \(i^2\).
  4. If necessary, simplify the result by combining like terms.

Example: \( \frac{1}{i} = ? \)

We can simplify this expression by multiplying both the numerator and denominator by \(-i\), which gives:

\[\frac{1}{i} = \frac{1}{i}\cdot\frac{i}{i} = \frac{i}{i^2} = \frac{i}{-1} = -i\]

Therefore, the reciprocal of \(i\) is equal to \(i\).

Example: \( \frac{1+i}{1-i} = ? \)

To solve \((1+i)/(1-i)\), we can first multiply the numerator and denominator by the complex conjugate of the denominator:

\[\begin{align*}\frac{1+i}{1-i} = \frac{(1+i)(1+i)}{(1-i)(1+i)} = \\ = \frac{1+2i+i^2}{1-i^2} = \frac{2i}{2} = i\end{align*}\]

Therefore, \((1+i)/(1-i) = i\).

Example: \( \frac{1}{a + bi} = ? \)

The reciprocal of a complex number \(z=a+bi\) can be found by multiplying both the numerator and denominator of \(\frac{1}{a+bi}\) by the complex conjugate of the denominator, \(a-bi\). This gives:

\[\frac{1}{a+bi} = \frac{1}{a+bi} \cdot \frac{a-bi}{a-bi} = \frac{a}{a^2 + b^2} - \frac{b}{a^2 + b^2}i\]

Thus, the reciprocal of a complex number is equal to the complex conjugate of the denominator divided by the sum of the squares of the real and imaginary parts of the original complex number. Applying this formula to the specific case of \(1/(a+bi)\), we get:

\[\frac{1}{a+bi} = \frac{a - bi}{a^2 + b^2} \]


Published: 2023-05-08 08:47:24
Updated: 2023-05-08 08:58:22

Last 10 artitles


9 popular artitles
© 2020 MyCoding.uk -My blog about coding and further learning. This blog was writen with pure Perl and front-end output was performed with TemplateToolkit.