Statistics Definitions > Imaginary Numbers
- What is an Imaginary Number?
- The More Technical Definition.
- A Geometrical Interpretation of Imaginary Numbers.
- What are Imaginary Numbers Used For in Real Life?
- The History of Imaginary Numbers.
- Complex Numbers vs. Imaginary Numbers.
An imaginary number is any number, that when squared, results in a negative number.
Sounds easy, right? The problem comes when you try and figure out which numbers are imaginary. Let’s take a few random numbers and square them:
- 10 = 102 = 100
- .99 = .992 = 39801
- -4 = -42
- pi2 = .986960440109.
OK, none of those seem to be working. In fact, it’s almost impossible to “guess” an imaginary number. It would be like me asking you to guess what the word for “bicycle” is in Swahili. Or what the fifty-seventh digit in pi is. You’ll need to learn what they are, much in the same way that you learned what a “variable” means. Let’s take “x” for example. You probably know that “x” stands for “a variable.” Well, “i” stands for a very particular type of variable…an imaginary number. Unlike the variable x, which could be anything on the planet, i is equal to the square root of -1:
i = √-1
Now before you say “hang on…where did that come from? How do we know that i – √-1?”
If you’re just learning about imaginary numbers, at this point it’s really important to take a deep breath, and accept that i = √-1. Remember learning about pi? The hardest thing was remembering it was equal to about 3.14. Then you got to use it to find circumferences and diameters of circles. And then, sometime down the road, you might have learned the history of pi and how it was derived. The imaginary number i is just like that. To reiterate what I said at the top of this section: all you need to know for most elementary math/algebra/statistics classes is that i = √-1.*
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You might see imaginary numbers defined this way: i2 = -1 (which is just i = √ -1 rewritten). You’ll be using the definition to try and solve equations. And if you’ve worked enough algebra equations you’ll know the first thing you do to an exponent is to get rid of it. How to get rid of it? By taking the square root of both sides. If you want to use the definition of i2, instead of √ 1, go ahead. For me, it makes equations more complicated. But if it works for you, there’s no reason at all why you can’t use it.
That said, there are some occasions when you’ll want to know some variations on i2 so that you can plug them into equations. Some you are likely to come across (and most of these are based on rules you probably already know, such as any number raised to zero is just one):
- i0 = 1
- i1 = i
- i2 = -1
- i3 = i2 * i = -1 * i = -i
- i4 = i3 * i = -i * i = (-1)(i2) = -1 * -1 = 1
- i5 = i4 * i = 1 * i = i
- i6 = i5 * i = i * i = i2 = -1
- i7 = i6 * i = -1 * i = -i
- i8 = i7 * i = -i * i = i2 = -1
Take a look at that fourth term for a moment and notice it’s the same as i0. Do you see a pattern starting? The pattern continues (1, i, -1, i,) so that very high powers of i become relatively easy to work out. Odd powers like i20 will be equal to -1 or 1. Even powers like i57 are going to be equal to i or -i.
“i” raised to a negative exponent are seen less often, but they do exist (have a chuckle at that paradox!).
- i-1 = -i
- i-2 = -1
- i-3 = -i
A final note on the technical stuff: for basic algebra, the definition that i = √-1 works. But in more advanced classes (a complex numbers class, for example), you should know that –i is also a valid solution.
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A complex number (a + bi) is just the rotation of a regular number. With a negative number, you count backwards from the origin (zero) on the number line. With an imaginary number, you rotate around the origin, like in the image above. The + and – signs in a negative number tell you which direction to go: left or right on the number line. In the same way, i tells you where to go on a Cartesian plane (well, it looks like a Cartesian plane, just that the y-axis is labeled i instead). Take the basic equation for i:
X2 = – 1
You can rewrite this as:
x * x = -1
Which is the same as saying:
What number, times itself is -1?
If you’re trying to think of a real number, like 2 or 3, remember that we’re working in a different real. If you have x * x = 4, you could use the number line to deduce that 2 * 2 = 4 by going right on the line 2 spaces and then 2 spaces. With imaginary numbers, you want to rotate, not multiply. If you rotate a number “x” 90 degrees and then 90 degrees again (which is x* x in the imaginary realm), you get -x.
If you find that hard to wrap your head around, let it sink in for a while. Consider this: not so long ago, people couldn’t comprehend negative numbers. In 1798, British mathematician said they “… darken the very whole doctrines of the equations and make dark of the things which are in their nature excessively obvious and simple.” Nowadays, every grade school student knows that a negative number is the opposite of a positive number.
Right now, in the everyday world, imaginary numbers aren’t used. But imagine some day down the road, they might become part of our everyday language, much like the number zero has become commonplace (not so long ago, the number zero didn’t exist, but that’s a story for another article). If you become a mathematician, engineer or physicist, imaginary numbers become very important.
Imaginary numbers are mainly used in mathematical modeling. They can affect values in models where the state of a model at a particular moment in time is affected by the state of a model at an earlier time. You’re most likely to use imaginary numbers in fields like quantum mechanics and engineering where differential equations are used (differential equations are part of calculus). For example, they can be used to monitor the phase and amplitude of an audio signal or electrical currents. You’ll also come across these numbers in computer science, where some programming languages (like C#) use imaginary numbers in their routines. Imaginary numbers are very rarely used in statistics; you’ll come across them in advanced topics like Fourier Analysis.
Heron of Alexandria (CE 100) is thought to be the first person proposing that the square root of a number (√63) could be a solution to a problem.
Niccolò Fontana (Tartaglia), Gerolamo Cardano and Lodovico Ferrari developed a formula in the early 16th century for finding the roots of cubic equations. Their work was published in the 1545 book Ars Magna. The formula included the roots of -1, which they realized didn’t exist. At the time, the numbers were called these non-existent numbers “numeri ficti.” Although they appeared in the equations, they ended up canceling out, so there was no need to figure out what they actually were. In 1572, Rafael Bombelli explained what the numeri ficti were and what they could be used for.
Rene Descartes came up with the phrase “imaginary numbers,” in the 17th century; mentioned in La Geometrie, it was meant to be a derogatory term. In the 18th century, the Swiss mathematician Leonhard Euler came up with the notation i as being equal to the square root of -1. Carl Friedrich Gauss popularized the use of imaginary numbers in the 19th century.
Complex numbers = Imaginary Numbers + Real Numbers
For example, 8 + 4i, -6 + πi and √3 + i/9 are all complex numbers.
Imaginary numbers and complex numbers are often confused, but they aren’t the same thing. Take the following definition:
“The term “imaginary number” now means simply a complex number with a real part equal to 0, that is, a number of the form bi.”
Some people read the first part of the sentence (bolded) to read that the two are equivalent. They aren’t: read the second part of the sentence carefully. What it is saying is:
- Take a random complex number like 7 + 4i.
- Make the “real” part of the equation zero. 0 + 4i.
That equals 4i, which is imaginary. In other words, you can get an imaginary number from a complex number. But they are not the same thing.
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Hargittai, Istvan (Ed.). (1994). Fivefold Symmetry. World Publishing.
Maseres, Francis. (1758). Dissertation on the Use of the Negative Sign in Algebra. Retrieved December 31, 2015
The Story of Mathematics. (2010).16th Century Mathematics — Tartaglia, Cardano & Ferrari. Retrieved December 31, 2015.
Waldemar Dos Passos. (2011). Numerical Methods, Algorithms, and Tools in C#. CRC Press.
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