Every idea in this chapter — from the mysterious i to the Argand plane — is something you can see move. Drag the points below, watch the algebra update live, and build the picture in your head that the textbook can only describe in words.
For every real number x, x² ≥ 0. So x² + 1 = 0 has no solution among the reals — the parabola y = x² + 1 never touches the x-axis. Mathematicians defined a new number, i = √−1, with i² = −1, just so equations like this could have an answer.
A complex number z = a + ib is just an ordered pair (a, b). Plot a on the horizontal real axis and b on the vertical imaginary axis, and every complex number becomes a single point — drag it around.
Tip: you can also drag the point directly on the plane. Points on the real axis look like a + i0; points on the imaginary axis look like 0 + ib.
z₁ + z₂ = (a+c) + i(b+d): add real parts, add imaginary parts. Geometrically it's the same "tip-to-tail" rule you use for vectors — the parallelogram rule. Drag either point.
z₁z₂ = (ac − bd) + i(ad + bc). Notice multiplying by i rotates a point 90° about the origin — watch what happens to z as you change z₂ toward pure i.
Every power of i lands on one of just four points: 1, i, −1, −i, then repeats forever. Slide n and watch iⁿ spin around the unit circle in steps of 90°.
The modulus |z| = √(a²+b²) is the straight-line distance from the origin to z. The conjugate z̄ = a − ib is z's mirror image flipped across the real axis. Drag z and watch its shadow.
Notice z · z̄ = |z|² always — this is exactly the trick used to divide complex numbers (multiply top and bottom by the conjugate).
For a positive real number a, √−a is defined as √a · i. But the familiar rule √a × √b = √(ab) breaks down the moment both a and b are negative — try it below.