Complex Numbers

Definition

Complex numbers are the numbers that are expressed in the form of a+ib where, a, b are real numbers and  ‘i’ is an imaginary number called “iota”. The value of i = (√-1). For example, 2+3i is a complex number, where 2 is a real number (Re) and 3i is an imaginary number (Im). 

Combination of both the real number and imaginary number is a complex number.

The main application of these numbers is to represent periodic motions such as water waves, alternating current, light waves, etc., which rely on sine or cosine waves, etc. ⁵

Who

Most scientists and engineers, as well as every mathematician, should have an understanding of complex numbers. Physicists and electrical engineers may routinely work with complex numbers. You may not be aware of complex numbers, but the real-world applications of complex numbers are visible to anyone who studies or analyzes things that oscillate or resemble waves. Complex numbers are also used in finance and economics. ⁷

What

When I Solve A Practical Mathematics Problem And I Arrive At A Complex Number Solution, What Does It Mean Practically?

Complex number answers are sometimes actual solutions. The meaning of the complex solution depends on what problem you are solving. If you are solving a wave problem, a complex number may be describing oscillations and decay at the same time. If you are analyzing angles, there is even such thing as a complex angle! This happens when the planes of equal phase and equal amplitude are different.

I will try to generalize and say that complex numbers are usually describing two things at once. This makes sense since a complex number as a real and an imaginary part. Are there two aspects to the thing you are solving? Like waves have amplitude and phase, but sometimes we just think amplitude and forget about phase. ¹

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How Are Imaginary Numbers Useful?

They are useful for anything that oscillates.

That means that they are used when describing sound or radio waves, the movement of buildings and bridges in the wind, the behavior of earthquakes, and anything else where feedback can occur.

The reason why they are useful is because of Euler’s identity e^πi = −1, or rather, because of its more general form, called Euler’s formula: e^θi = cosθ + isinθ. This means that you can work with periodic functions as if they were exponential functions. I don’t know how much trigonometry you have done, but it’s difficult to do. If you express trigonometry as exponential functions instead then suddenly trigonometry becomes algebra. The math is much easier, which makes it much easier to analyze these types of systems and then to apply that analysis to designing new things.

Imaginary numbers are at the heart of all frequency analysis, including Fourier and Laplace transforms, as well as all electronic communication and signal processing. It would be possible to do all of it without imaginary numbers, but it would be much harder to do and as a result we wouldn’t have accomplished nearly as much as we have done.

I work with radio waves, and I use imaginary numbers all the time to solve real-world problems, because imaginary numbers describe things in the real world.

Calling them “imaginary” was a mistake, not because the name makes any difference in how useful they are, but because that leads people who don’t understand them to believe that they aren’t really numbers and can’t be useful. ²

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How Are Imaginary Numbers Useful?

Really short answer : rotations.

When you are working with rotations you should be working with elements that are not real numbers and whose square is -1. To appreciate this, perhaps you should also consider the dual numbers and the hyperbolic numbers.

The dual numbers contain an element epsilon that is not a real number and that squares to zero. They don’t produce rotations under multiplication, but translations.

The Hyperbolic numbers contain an element that is not a real number and that squares to one. They produce reflections under multiplication.

So three special numbers (lets say i, j and epsilon), that are not members of the Real numbers and turn multiplication into rotation, translation and reflection.

Whenever a problem involves our universe, the most adequate, concise and useful mathematical model will involve these three elements and their multiples.

For example .. complex numbers have one element that squares to -1 and so are ideal for rotations in the plane. Quaternions have three elements that square to -1 and so are ideal for rotations in three dimensions. Dual-Quaternions have three elements that square to -1 and three that square to 0 and can as such handle translations and rotations in three dimensions. All of these algebras are eloquently captured by the Clifford Algebras (that have slowly been getting the recognition they deserve under the name geometric algebra). ³

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How Are Imaginary Numbers Useful?

First, allow me to rename them during the remainder of this answer to lateral numbers, in accordance to the naming convention as was recommended by Gauss. I have a special reason for using this naming convention. It will later become apparent why I’ve done this.

If we examine lateral numbers algebraically, a pattern emerges.

i⁰ = 1

i¹ = i

i² = −1

i³ = −i

i⁴ = (i²)² = (−1)² = 1

i⁵ = i ⋅ i⁴ = i

i⁶ = i² ⋅ i⁴ = (−1)(1) = −1

i⁷ = i² ⋅ i⁵ = −1i = −i

i⁸ = i⁴ ⋅ i⁴ = 1⋅1 = 1

When we raise lateral numbers to higher powers, the answers do not get higher and higher in value like other numbers do. Instead, a pattern emerges after every 4th multiplication. This pattern never ceases.

All other numbers, besides laterals, have a place on what currently is called the ‘Real number line’.
I qualify the naming of the Real Numbers, because even their conceptualisation has come into question by some very incisive modern mathematicians. That is a very ‘volatile’ subject for conventional mathematicians and would take us off on a different tangent, so I’ll leave that idea for a different question.

If we look for laterals on any conventional Real number line, we will never ‘locate’ them. They are found there, but we need to look at numbers differently in order to ‘see’ them.

Lateral numbers solve one problem in particular: to find a number, which when multiplied by itself, yields [another] negative number.
Lateral numbers ‘unify’ the number line with the algebraic pattern shown above.

2 is positive and, when multiplied by itself, yields a positive number. It maintains direction on the number line.

When one of the numbers (leaving squaring briefly) being multiplied is negative, the multiplication yields a negative number. The direction ‘flips’ 180° into the opposite direction.

Multiplying -2 by -2 brings us back to the positive direction, because of the change resulting in multiplying by a negative number, which always flips our direction on the number line.

So, it appears as if there’s no way of landing on a negative number, right? We need a number that only rotates 90°, instead of the 180° when using negative numbers. This is where lateral numbers come into play.

If we place another lateral axis perpendicular to our ‘Real’ number line, we obtain the desired fit of geometry with our algebra.

When we multiply our ‘Real’ number 1 by i, we get i algebraically, which geometrically corresponds to a 90° rotation from 1 to i.

Now, multiplying by i again results in i squared, which is -1. This additional 90° rotation equals the customary 180° rotation when multiplying by -1 (above).

[We may even look at this point as if we were viewing it down a perpendicular axis of the origin itself (moving in towards the origin from our vantage point, through the origin, and then out the back of our screen).]

[If we allow this interpretation, we can identify the ‘spin’ of a point around the axis of its own origin! The amount of spin is determined by how much the point moves laterally in terms of i.]

[We may even determine in which direction the rotation is made. I’ll add how this is done to this answer soon. UPDATE: As promised, here is the first answer as to how to determine direction of rotation – Carey G. Butler’s answer to Why do imaginary numbers work? more coming…]

Each time we increase our rotation by multiplying by a factor of i, we increase our rotation another 90°, as seen here:

and,

The cycle repeats itself on every 4th power of i.

We could even add additional lateral numbers to any arbitrary point. This is what I do in my knowledge representations of holons. For example a point at say 5 may be expressed as any number of laterals i, j, k,… simply by adding or subtracting some amount of i, j, k,…:

5 + i + j + k + …

Or even better as:

[5, i, j, k ,…]

Seeing numbers in this fashion makes a point n-dimensional. ⁴

[Please read the comments to this post for more information.]

Why

Where do we use imaginary numbers : ⁶

• Imaginary numbers are very useful in various mathematical proofs.
• Imaginary numbers are used to represent waves.
• Imaginary numbers show up in equations that don’t touch the x axis.
• Imaginary numbers are very useful in advanced calculus.
• Combining AC currents is very difficult as they may not match properly on the waves. 
• Using imaginary currents helps in making the calculations easy.

See Theoretical Knowledge Vs Practical Application.

How

Real World Example: Oscillating Springs

This article^# examines how complex numbers of the form a + bi are used to describe the motion of an oscillating spring with damping.

^# “Complex Numbers in the Real World. a+bi example explained in depth with pictures”. 2022. mathwarehouse.com. https://www.mathwarehouse.com/algebra/complex-number/real-world-example-scillating-springs-explained.php.

Many of the References and Additional Reading websites, and Videos will assist you with understanding and using complex numbers.

As some professors say: “It is intuitively obvious to even the most casual observer.“

References

¹ Rumpf, Tipper. “When I Solve A Practical Mathematics Problem And I Arrive At A Complex Number Solution, What Does It Mean Practically?” 2022. Quora. https://qr.ae/pvPKiB.

² Widdison, Eric. “How Are Imaginary Numbers Useful?” 2022. Quora. https://qr.ae/pvPKJn.

³ Mute, Enki. “How Are Imaginary Numbers Useful?” 2022. Quora. https://qr.ae/pvPKcH.

⁴ Butler, Carey G. “How Are Imaginary Numbers Useful?” 2022. Quora. https://qr.ae/pvPKJ5.

⁵ “Complex Numbers (Definition, Formulas, Examples)”. 2022. BYJUS. https://byjus.com/maths/complex-numbers/.

⁶ “What Are Imaginary Numbers? – GeeksForGeeks”. 2021. GeeksForGeeks. https://www.geeksforgeeks.org/what-are-imaginary-numbers/?ref=gcse.

⁷ Weedmark, David. “Jobs That Use Exponents”. 2022. Work – chron.com. https://work.chron.com/jobs-use-exponents-24061.html.

Additional Reading

“1.1: Complex Numbers”. 2021. Mathematics LibreTexts. https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/Introduction_to_Groups_and_Geometries_(Lyons)/01%3A_Preliminaries/1.01%3A_Complex_Numbers.

“15.4: Complex Numbers”. 2020. Mathematics LibreTexts. https://math.libretexts.org/Bookshelves/Applied_Mathematics/Math_in_Society_(Lippman)/15%3A_Fractals/15.04%3A_Complex_Numbers.

“16.4.1: Complex Numbers”. 2021. Mathematics LibreTexts. https://math.libretexts.org/Bookshelves/Applied_Mathematics/Developmental_Math_(NROC)/16%3A_Radical_Expressions_and_Quadratic_Equations/16.04%3A_Complex_Numbers/16.4.01%3A_Complex_Numbers.

“3.1: Complex Numbers”. 2015. Mathematics LibreTexts. https://math.libretexts.org/Bookshelves/Precalculus/Precalculus_(OpenStax)/03%3A_Polynomial_and_Rational_Functions/3.01%3A_Complex_Numbers.

“3.4: Complex Numbers”. 2019. Mathematics LibreTexts. https://math.libretexts.org/Bookshelves/Analysis/Book%3A_Mathematical_Analysis_(Zakon)/03%3A_Vector_Spaces_and_Metric_Spaces/3.04%3A_Complex_Numbers.

“6.1: Complex Numbers”. 2018. Mathematics LibreTexts. https://math.libretexts.org/Bookshelves/Linear_Algebra/A_First_Course_in_Linear_Algebra_(Kuttler)/06%3A_Complex_Numbers/6.01%3A_Complex_Numbers.

“8.2: Complex Numbers”. 2021. Mathematics LibreTexts. https://math.libretexts.org/Bookshelves/Differential_Equations/Introduction_to_Partial_Differential_Equations_(Herman)/08%3A_Complex_Representations_of_Functions/8.02%3A_Complex_Numbers.

Berry, Brett. “Complex Numbers Explained”. 2018. Medium. https://medium.com/i-math/complex-numbers-explained-fad9a6793019.

Berry, Brett. “The Reality Of Imaginary Numbers”. 2018. Medium. https://medium.com/i-math/imaginary-numbers-explained-e5aa63bdb7ae.

“Complex Number – Definition, Formula, Properties, Examples”. 2022. CUEMATH. https://www.cuemath.com/numbers/complex-numbers/.

“Complex Number – Encyclopedia Of Mathematics”. 2022. encyclopediaofmath.org. https://encyclopediaofmath.org/wiki/Complex_number.

“Complex Number — From Wolfram MathWorld”. 2022. mathworld.wolfram.com. https://mathworld.wolfram.com/ComplexNumber.html.

“Complex Numbers | Algebra 2 | Math | Khan Academy”. 2022. Khan Academy. https://www.khanacademy.org/math/algebra2/x2ec2f6f830c9fb89:complex#x2ec2f6f830c9fb89:complex-num.

“Complex Numbers | Brilliant Math & Science Wiki”. 2022. brilliant.org. https://brilliant.org/wiki/complex-numbers/.

“Complex Number Multiplication”. 2022. mathsisfun.com. https://www.mathsisfun.com/algebra/complex-number-multiply.html.

“Imaginary Number – Wikipedia”. 2022. en.wikipedia.org. https://en.wikipedia.org/wiki/Imaginary_number.

Originally coined in the 17th century by René Descartes as a derogatory term and regarded as fictitious or useless, the concept gained wide acceptance following the work of Leonhard Euler (in the 18th century) and Augustin-Louis Cauchy and Carl Friedrich Gauss (in the early 19th century).

“Imaginary Numbers”. 2022. mathsisfun.com. https://www.mathsisfun.com/numbers/imaginary-numbers.html.

⭐ McNulty, Keith. “The Magic of Demoivre’s Theorem”. 2023. Medium. https://keith-mcnulty.medium.com/the-magic-of-demoivres-theorem-893e94911fdd.

DeMoivre’s Theorem ( (cos θ + i sin θ)ⁿ = cos nθ + i sin nθ ) is a simple theorem relating complex numbers to trigonometry. It was proposed and proved for all positive integers n by Abraham DeMoivre (1667–1754), a French mathematician who was a religious exile in England from a young age and a friend of such greats as Isaac Newton and Edmond Halley.

Padavic-Callaghan, Karmela. “How Imaginary Numbers Describe The Fundamental Shape Of Nature | Aeon Essays”. 2022. Aeon. https://aeon.co/essays/how-imaginary-numbers-describe-the-fundamental-shape-of-nature.

Quantum theory has historically challenged many seemingly ‘common sense’ assumptions about nature. It has, for example, changed the way physicists think about an experimenter’s ability to measure something with certainty, or the claim that objects can be affected only by other objects in their immediate surroundings. When quantum theory was first formulated, it scandalised many luminaries of science at the time, including Einstein who contributed to its foundations himself. Working with quantum ideas and poking quantum systems has always, by default, come with the possibility of uncovering something unexpected at best, and bizarre at worst. Now quantum physics has revealed that we’ve misunderstood imaginary numbers all along. They may have, for a time, seemed to be just a mental device inhabiting the minds of physicists and mathematicians, but since the real world that we inhabit is indeed quantum, it’s no surprise that imaginary numbers can be found, quite clearly, within it.

Rizk, M. A. “The Complex Numbers In Real Life – Science Use”. 2019. Science Use. https://www.scienceuse.com/the-complex-numbers-in-real-life/.

In this article, I will show the utility of complex numbers, and how physicists describe physical phenomena using this kind of numbers.

Videos

Errors: [1] On the first sketch of a complex plane, there is a “2i” written instead of “-2i”. [2] At the end, in writing the angle sum identity, the last term should be sin(beta) instead of sin(alpha). [3] During Q9, the terms in parentheses should include an i, (1/2 + sqrt(3)/2 i)

This algebra 2 video tutorial explains how to perform operations using complex numbers such as simplifying radicals, adding and subtracting complex numbers, simplifying it in standard form, graphing complex numbers and calculating the absolute value of complex numbers. This video contains plenty of examples and practice problems. It’s useful for students taking algebra 2, precalculus or college algebra.

This precalculus video tutorial provides a basic introduction into imaginary numbers. it explains how to simplify imaginary numbers as well as adding, subtracting, multiplying and dividing imaginary numbers.

Images

 

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⭐ I suggest that you read the entire reference. Other references can be read in their entirety but I leave that up to you.