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Imaginary and Complex Numbers

Issue No 31, 12 June 2023

By: Anthony O. Ives

A previous article [1] introduced the concept that it was not possible to take the square root of a negative number. However, the solution to this problem is quite simple you just use an imaginary number 'i' to represent the square root of a negative number and treat it like you would any unknown number. When imaginary numbers are combined with other numbers (known as real numbers) that are not imaginary numbers such as the type of ordinary numbers most people are familar with they are called complex numbers. Imaginary and complex numbers have important applications and a lot of technology would not be possible without them. For helicopters imaginary and complex numbers are needed for control and stability analysis as well as aircraft control systems, but beyond helicopters they are used for analysing alternating current, vibrations, etc.

Usually in engineering especially electrical and electronical engineering 'i' is not used to represent imaginary numbers to avoid confusion with current as 'i' is usually used to represent current, so typically in engineering 'j' is used to represent imaginary numbers. So therefore:

\[ j = \sqrt{-1} \]

You can simplfy the square root of a negative number by taking the square root of the number in the same way as you would if it was positive, see example below.

\[ \sqrt{-16} = \sqrt{16} \sqrt{-1} \]

\[= \sqrt{16} j = 4j \]

The arithmetic operations with imaginary numbers are more or less the same as they with other numbers:

\[16j + 2j = 18j\]

\[6j - 4j = 2j\]

Multplying two imaginary numbers gives you a real number however, multiplying three imaginary numbers gives you an imaginery number the examples below will hopefully demostrate why this is the case.

\[j^2 = j \times j \]

\[= \sqrt{-1} \times \sqrt{-1} \]

\[= -1\]

\[ 16j \times 4j \]

\[= 16 \sqrt{-1} \times 4 \sqrt{-1}\]

\[= -64\]

\[j^3 = j \times j \times j \]

\[= \sqrt{-1} \times \sqrt{-1} \times \sqrt{-1} \]

\[= -1j \]

\[ 2j \times 4j \times 7j \]

\[= 2 \sqrt{-1} \times 4 \sqrt{-1} \times 7 \sqrt{-1} \]

\[= -56j \]

Therefore an imaginary number to the power of an even number will give a real number, but an imaginary number to the power of an odd number will give a imaginary number as below:

\[j^2 = -1\]

\[j^3 = -1j\]

\[j^4 = 1\]

\[j^5 = 1j\]

etc....

Note all also that the sign will change as was discussed in a previous article [1], see references [2] and [3] for an explanation of indices. As already discussed if you combine an imaginary number with a real number you get a complex number as in the examples below:

\[34 + \sqrt{-16} = 34 + 4j\]

\[-3 + \sqrt{-4} = -3 + 4j\]

\[10 - \sqrt{-49} = 10 - 7j\]

A later article will discuss quadratic equations and ploynomials. Quadratic equations and ploynomials can in certain circumstances have complex numbers as their roots. Complex numbers often come as conjugate pairs particularly as roots of polynomial equations. Conjugate pair of complex numbers is a pair of complex numbers when multpied together give a real number. Doing arithmetic on complex numbers is similar to adding any numbers together with unknown numbers such as the example below:

\[(10-7j) + (34+4j)\]

\[ = 44-3j\]

\[(10-7j) - (34+4j) \]

\[ = -24-13j\]

However, multiplying complex numbers together is a little bit more complex:

\[(10-7j)(3+4j)\]

\[ = (10)(3+4j)+(-7j)(3+4j) \]

\[ = (30+40j)+(-21j-28j^2) \]

\[ = (30+40j)+(-21j+28j) \]

\[ = 9+68j \]

The multiplication method for complex number is the same as used for quadratic and ploynomial equations, in future articles on quadratic and ploynomial equations we will discuss it in more detail but a conjugate pair of complex numbers multiplied together give a real number:

\[(10+7j)(10-7j)\]

\[ = (10)(10-7j)+(7j)(10-7j) \]

\[ = 20 -70j + 70j -49j^2 \]

\[ = 20 + 49 = 69 \]

So a conjugate pair of complex numbers is two almost identical pair of complex numbers with the exception the sign of the imaginary number is different. Conjugate pairs of complex numbers will be discussed in more detail in future articles on quadratic and ploynomial equations.

A future article will also discuss vectors but complex numbers also use the same arithmetic operations of vectors. Complex numbers can also be represented in polar notation just like vectors with similar conversion between cartesian and polar coordinates as below:

\[z = x + yj \]

\[R = \sqrt{(x^2 + y^2)} \]

\[ \theta = atan \frac{x}{y} \]

\[z = x + yj \]

\[ = R(cos \theta + j sin \theta)\]

\[ = Re^{j\theta} \]

Complex numbers is big topic in mathematics which can related to hyperbolic trignometery [4] as well as other advanced topics such as complex mapping [5].

Please leave a comment on my facebook page or via email and let me know if you found this blog article useful and if you would like to see more on this topic. Most of my blog articles are on:

Mathematics
Helicopters
VTOL UAVs (RC Helicopters)
Sailing and Sailboat Design

If there is one or more of these topics that you are specifically interested in please also let me know in your comments this will help me to write blog articles that are more helpful.

References:

[1] http://www.eiteog.com/EiteogBLOG/No28EiteogBlogNumbers.html

[2] http://www.eiteog.com/EiteogBLOG/No8EiteogBlogLiftCL.html

[3] http://www.eiteog.com/EiteogBLOG/No9EiteogBlogLogs.html

[4] Engineering Mathematics, K. A. Stroud, Fourth Edition, 1995, Macmillan

[5] Further Engineering Mathematics, K. A. Stroud, Fourth Edition, 1996, Palgrave Macmillan

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