Working with Imaginary Numbers (i)

  Imaginary Numbers are simply real numbers multiplied by the imaginary unit i which is representative of  √−1. The reason we use i as the imaginary unit is that we simply cannot multiply any number by itself to get a negative square, neither rationals, irrationals, negatives, nor zero will work, so we imagine a unit, i, that equals the square root of a negative (i = √−1). Now when we square i, we can get a negative number that is a square (i² = -1).

  Now that we understand the basic concept of i,  we can move on to its uses and other interesting properties. i's main use is to simplify a negative root, by basically pulling out the negative. So, the square root of -4 would be 2i! Find the root as though it was not negative then multiply it by the unit i. So simple. The unit i can be multiplied by integers, fractions, decimals, radicals, and can even be negative. We mainly use it in the quadratic equation, because quite often, we take square roots of negatives, thus we get an imaginary root.

  Now, look at this property of i. You could say it cycles between 4 numbers when you multiply i by i or take the powers of i. Observe:

• i × i = -1
• -1 × i = -i
• -i × i = 1
• then 1 × i = i again

Look at it as powers:

• i¹ = √−1or i
• i² = -1 
• i³ = -√−1or -i
• i⁴ = 1
• i⁵ =  √−1or i again

-Zech