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.
- 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.