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Introduction to Complex Numbers

Hint 1; Complex numbers are not more complicated than any other numbers, just different.

Complex number = real number + imaginary number

Hint 2; imaginary numbers are not more ethereal than real numbers, just different.

Imaginary numbers were invented to solve problems involving square roots of negative numbers.

Hint 3; we lied to you when we told you that you cannot take the square root of a negative number. We just waited until now to tell you how to do it.

  • i = √(-1)

The letter i is used to signify the square root of -1. Any number multiplied by i is an imaginary number.

Thus the square root of any negative number equals the square root of its positive value multiplied by i. Yes, it really is that easy. Examples

  • √(-25) = 5i
  • √(-7) =  i√(7)
  • √(-12) = i√(12) = 2i√(3)

The powers of i follow a repeating pattern that is illustrated below.

  • i0 = 1 because any number (even imaginary ones) raised to the power of zero equals 1
  • i1 = i because any number raised to the power of 1 equals that number
  • i2 = -1 because just like [√(3)]2 = 3, so does [√(-1)]2 = -1
  • i3 = -i because i3 = (i)(i2) = (i)(-1) = -i

Ok, that was harder but go over it a couple times to convince yourself and the next part is easy.

The value of in can be found by dividing n by 4 and matching the remainder with the power of i in the pattern above.

For example;

  • i53 = i because the remainder of 53 divided by 4 is 1 and i1 = i
  • i18 = -1 because the remainder of 18 divided by 4 is 2 and i2 = -1
  • i31 = -i because the remainder of 31 divided by 4 is 3 and i3 = -i
  • i48 = 1 because the remainder of 50 divided by 4 is 0 and i0 = 1

The conjugate of a complex number is another complex number except the imaginary part have opposite signs.

  • 6 + 3i has a complex conjugate of 6 - 3i
  • 6 - 3i has a complex conjugate of 6 + 3i
  • -6 + 3i has a complex conjugate of -6 - 3i
  • -6 - 3i has a complex conjugate of -6 + 3i

The conjugate is important because the product of a complex number and its conjugate is always a real number because the sum of the inner products is always zero. For example

(4 - 3i)(4 + 3i) = 16 - 12i + 12i - 9i^2) = 16 - (9)(-1) = 16 + 9 = 25

Hint 4: If you have gotten to here then you have a basic introduction to complex numbers.

Carry on!

Steven G.

Comments

Instead of defining i as sqrt(-1), it's a bit more instructive to use the definition i2 = -1. This looks equivalent, but it shows that imaginary numbers come in conjugate pairs, since i2 = -1 implies i = ±sqrt(-1) or sqrt(-1) = ±i.  Since imaginary numbers show up in pairs, complex numbers also exist in conjugate pairs. So, using that definition makes the complex conjugate seem natural. It's oftentimes introduced as merely a mysterious definition.

$40p/h

Steven G.

Mathematics Tutor; Algebra, Pre-Calc, Statistics

200+ hours
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