The first mention of complex numbers is around 50 BC when finding a square root of a negative number. However, it was in 1545 that Cardano found these numbers formally while trying to obtain the roots of an equation. Complex numbers were standardized in 1777 when Euler used *i* to represent the square root of negative one.

Here, we will look at the detailed history of complex numbers. Furthermore, we will also look at some of the reasons why these numbers are important.

##### ALGEBRA

**Relevant for**…

Learning about the history of complex numbers and their importance.

##### ALGEBRA

**Relevant for**…

Learning about the history of complex numbers and their importance.

## History of complex numbers

The first known mention of people trying to use imaginary numbers is in the first century. Around 50 BC, Heron of Alexandria studied the volume of an impossible section of a pyramid. What made this impossible is that he had to calculate the result of $latex \sqrt{81-114}$.

However, he considered this to be impossible and gave up. After this, no one tried to manipulate imaginary numbers for long. Once negative numbers were “made up,” mathematicians tried to find a number that would result in a negative number when squared. Since no one found the answer, they gave up again.

In the 1500s, speculation about the square roots of negative numbers returned. When the formulas for solving third and fourth degree equations were discovered, mathematicians realized that some work with square roots of negative numbers will occasionally be required. Finally in 1545, the first major work with imaginary numbers occurred.

In 1545, Girolamo Cardano wrote a book called *Ars Magna*. Cardano solved the equation $latex x(10-x)=40$, finding that the answer is 5 and $latex \pm\sqrt{-15}$. Although he found this to be the answer to the equation, he was not satisfied with the idea of imaginary numbers. He said that working with them would be useless and referred to working with them as “mental torture.” For some time, most people agreed with him.

In 1637, René Descartes came up with the standard form of complex numbers, which is $latex a+bi$. However, he also did not like the idea of complex numbers. He also came up with the term “imaginary,” even though he meant it was negative. Issac Newton agreed with René Descartes, and Albert Girad even called them “impossible solutions.” Although they did not like the idea of imaginary numbers, other mathematicians continued to study them.

Rafael Bombelli was very interested in complex numbers. He helped introduce them, but since he didn’t really know what to do with these numbers, hardly anyone believed him. Bombelli understood that *i* multiplied by *i* should be equal to -1. He also understood that *i* multiplied by –*i*, must be equal to 1.

However, many people did not believe this either. Furthermore, Bombelli also had the idea that imaginary numbers could be used to get the real answers. This is now known as conjugation.

For a long time, people began to believe more and more in complex numbers and decided to accept and understand them. One of the ways they wanted them to be accepted was by plotting them on a graph. In this case, the *x*-axis would represent the real numbers and the *y*-axis would represent the imaginary numbers. The first person to consider this type of graph was John Wallis. In 1685, he said that a complex number is simply a number on a plane.

In 1777, Euler came up with the symbol *i* to represent the expression $latex \sqrt{-1}$, which made it easier to understand. In 1806, Jean Robert Argand wrote about how to draw complex numbers on a plane, and now that plane is called the Argand diagram. In 1831, Carl Friedich Gauss made Argand’s idea popular.

Furthermore, he also took the notation $latex a + bi$ from Descartes and called them complex numbers. In 1833, Willian Rowan Hamilton expressed complex numbers as pairs of real numbers (as $latex 3+5i$ which is expressed as (3, 5). This made complex numbers less confusing.

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## Importance of complex numbers

Complex numbers are of great importance in many areas as they can be applied in real life situations, which includes engineering fields, especially electricity. Obviously due to their origin, complex numbers are also used to find solutions to quadratic equations. When we have equations that do not cross the *x*-axis, their solutions can be represented by complex numbers.

Perhaps the most important application of complex numbers is in electricity, especially in alternating current (AC) electronics. This type of current produces a sine wave because it changes between positive and negative.

Performing calculations with alternating current would be extremely difficult without the use of complex numbers because sine waves do not fit together properly. However, with the help of complex numbers, the calculation of alternating currents becomes simpler.

Signal processing is another important application of complex numbers. Signal processing is used in mobile phone technology, wireless technology, radar, and other signal transmission applications. In short, complex numbers particularly used to simplify process calculations involving sine or cosine waves.

## See also

Interested in learning more about imaginary and complex numbers? Take a look at these pages:

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