ACT Math: Imaginary and Complex Numbers
You might be surprised that not all numbers are real–some are imaginary. No, imaginary numbers aren’t as interesting as you might imagine them to be. They’re merely numbers invented by mathematicians to signify the even roots of negative numbers. Yup, just when you thought the test-writers packed in enough math material for a standardized test, they incorporated a whole set of numbers that doesn’t correspond to anything in reality.
Imaginary numbers, represented by the letter i, represent the even roots of negative numbers. The square root of -1, for example, is i. If you never took Algebra 2, or you slept through the portion on imaginary numbers, you might still think that the square root of any negative number is mathematically impossible, or undefined (like 1/0). Well, in the world of real numbers, it is. That’s why a bunch of bored mathematicians invented imaginary numbers.
Let’s look at a couple of examples of how imaginary numbers work. As long as you remember that the definition of i is √(-1), you should be fine.
√(-16)= √ (16) * √ (-1) = 4i
See? It’s that simple. Just tack on that little i to the roots of negative numbers.
A complex number is what we call the sum of a real number and an imaginary number. Think of it as a marriage of the real and imaginary, a tasty cocktail of Morpheus’s proffered red and blue pills. Complex numbers are written in the form a+bi, where a and b are real numbers; for example, 6+7i, is a complex number.
The Powers of i
To work with complex numbers, you must remember the pattern of the powers of i. Luckily, the pattern works in cycles of four:
i ^1= i
It’s much easier to simply remember the pattern than to work out the powers as products of √(-1).
By knowing the pattern, you can easily figure out a much larger exponent, say i^99. To figure this out, think of the closest multiple of 4 that’s less than the exponent; in this case, it’s 96. So, i^99 is the same thing as i^(96+3), which means that the corresponding exponent in the pattern is 3 (96=exponent of 4, 97=exponent of 1, 98=exponent of 2 , 99=exponent of 3, 100=exponent of 4, and so on). With an exponent of 3, i^99 must be -i.
Operations on complex numbers is virtually identical to simplifying or expanding real numbers with variables; the only difference is that you must remember to apply the exponent rule whenever necessary.
Example: Expand (2x+i)(4x+3i)
First, we just use a basic FOIL method to expand:
8x^2 +6xi +4xi + 3i^2
8x^2 +10xi + 3i^2
Notice the i squared, and remember the pattern.
8x^2 +10xi + 3 (-1)
8x^2 +10xi -3
The most important thing to remember about imaginary numbers is the pattern of exponents. For the most part, dealing with imaginary numbers is pretty similar to dealing with polynomials (though do not mistake i for just another variable–it hates that). Just think of complex numbers as polynomials with a new set of rules to follow, and you’ll be fine.