The Imaginary Unit i

Algebra 2 · Axiom Academy

Inventing one new number, i , that lets us step off the real number line. For centuries, mathematicians hit a wall: the equation x^2 = -1 has no real solution . Squaring any real number gives a result that is positive or zero — never negative — so there is nowhere on the real number line for a solution to land. Once we have i , we can rewrite the square root of any negative number. For a positive number a , split off the -1 and pull it out as an i : Factor out the largest perfect square first, then pull out the i . What happens as we compute i^1, i^2, i^3, i^4, and beyond? Each time we multiply by i , we take a quarter turn — so four multiplications carry us all the way around and back to where we started. Because the powers repeat every four steps, we get i^5 = i , i^6 = -1 , i^7 = -i , i^8 = 1 , and so on. To find any power of i , we only need the remainder when the exponent is divided by 4 . Combine a real number with an imaginary one and you get a complex number . Every complex number can be written in standard form as a real part a plus an imaginary part b : Just as we graph real numbers on a line, we graph complex numbers on a plane — the complex plane (or Argand diagram). The horizontal axis represents the real part. The vertical axis represents the imaginary part. The complex number a + bi is plotted at the point . You've met a brand-new number, learned to tame square roots of negatives, and seen where complex numbers live. Scroll up to revisit any step.

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