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Unit 3: Complex Numbers

The number that squares to negative one, what it does to the number line, and why multiplying by it is really a rotation.

8 topics · 32 questions

Each topic is a path, not a menu: work down it in order and every question builds on the one above. Click any of them to watch the answer draw itself.

The Imaginary Unit i

  1. How it worksHow does defining i as the square root of negative one let you simplify any negative square root?
  2. Worked exampleFind the simplified form of the square root of negative 50 in terms of i.
  3. Common mistakeWhy doesn't a negative number under a square root mean the equation has no answer at all?
Your turn1 of 2
Simplify the square root of negative 72, writing your answer in terms of i.

Powers of i

  1. How it worksHow does multiplying by i repeatedly rotate you through the same four values on a loop?
  2. Worked exampleFind the value of i37.
  3. Common mistakeWhy isn't i4 just left as another power of i instead of simplifying to 1?
  4. Going furtherWhat happens when you need a negative power of i, like i to the power negative 3?
Your turn1 of 2
Find the value of i4·18 + 1, where 18 is a positive integer.

Adding and Subtracting Complex Numbers

  1. How it worksHow do you combine the real parts and imaginary parts of two complex numbers separately?
  2. Worked exampleFind the sum of (5 - 3i) and (-2 + 7i).
  3. Common mistakeWhy can't 5 and 3i be combined into a single term the way 5x and 3x can?
Your turn1 of 2
Find the sum of (7 - 6i) and (8 + 3i).

Multiplying Complex Numbers

  1. How it worksHow does i2 turning into negative one collapse a complex product down to a single complex number?
  2. Worked exampleFind the product of (3 + 2i) and (1 - 4i).
  3. Common mistakeWhy isn't the product of two complex numbers found by just multiplying the real parts and the imaginary parts?
  4. Going furtherWhat happens to the product of two complex numbers when one of them is purely real?
Your turn1 of 2
Find the product of (7 + 6i) and (8 - 3i).

Complex Conjugates and Division

  1. How it worksHow does multiplying the top and bottom of a fraction by the conjugate clear i out of the denominator?
  2. Worked exampleFind the quotient of (4 + i) divided by (2 - 3i), written in a + bi form.
  3. Common mistakeWhy doesn't dividing complex numbers work by just dividing the real parts and imaginary parts separately?
  4. Ties togetherHow does clearing i from a denominator with a conjugate mirror clearing a square root from a denominator with its conjugate?
Your turn1 of 2
Find the quotient of (7 + 6i) divided by (8 - 3i), written in a + bi form.

The Complex Plane and the Modulus

  1. How it worksHow does the Pythagorean theorem give you the distance of a complex number from the origin?
  2. Worked exampleFind the modulus of the complex number 3 - 4i.
  3. Common mistakeWhy isn't the modulus of a complex number just the sum of its real and imaginary parts?
  4. Going furtherWhat happens to a complex number's position in the plane when you take its conjugate?
  5. Ties togetherHow does the modulus of a complex number generalize the absolute value of an ordinary real number?
Your turn1 of 2
Find the modulus of the complex number 14 - 13i.

Multiplying by i as a Rotation

  1. How it worksHow does multiplying 3 + 4i by i swap and flip its coordinates into a 90 degree rotation?
  2. Worked exampleFind where 2 + 5i lands after multiplying it by i, and plot both points.
  3. Common mistakeWhy isn't multiplying by i just another way of scaling a complex number bigger or smaller?
  4. Common mistakeWhy doesn't multiplying by i three times return a point to where it started, the way multiplying by i four times does?
  5. Going furtherWhat rotation do you get from multiplying by i twice in a row, and does that match i squared?
Your turn1 of 2
A student says multiplying 7 + 6i by i just makes the number 'bigger', like scaling it. What's the actual effect?

Complex Solutions of Quadratic Equations

  1. How it worksHow does a negative discriminant turn the quadratic formula's two real roots into a complex conjugate pair?
  2. Worked exampleSolve x2 + 2x + 5 = 0 using the quadratic formula.
  3. Common mistakeWhy do complex roots of a real quadratic always come in a conjugate pair instead of appearing alone?
  4. Ties togetherHow does the discriminant's sign predict whether a quadratic's roots will be complex before you even solve it?
Your turn1 of 2
Solve x2 + 5x + 16 = 0 using the quadratic formula. Write the solution that uses the PLUS sign, in a + bi form.