Euler's Formula and the Exponential Form
e to the i theta traces the unit circle, tying the exponential function to sine and cosine. In exponential form, multiplying complex numbers means adding angles, and De Moivre's theorem falls out for free.
What a learner can do afterwards
- Write a complex number in exponential form and convert back
- Multiply and divide complex numbers by adding and subtracting arguments
- Use De Moivre's theorem to compute a small integer power of a complex number
1 · Read
Euler packs direction into an exponential: e to the iT equals cos T plus i sin T, with the angle T in radians. As T grows, the value traces the unit circle, so its modulus is always 1. Landmarks fall out free: e to the i pi over 2 is i, and e to the i pi is minus one.
Multiplying splits into two easy jobs: multiply the moduli, add the arguments. So 2e to the i pi over 6 times 3e to the i pi over 3 is 6e to the i pi over 2, which is 6i. Division mirrors it: 2e to the i pi over 4 over 4e to the i pi over 4 is 0.5, since the arguments cancel and 2 over 4 is 0.5.
De Moivre extends the trick to powers: raise the modulus to n and multiply the angle by n. Squaring cos 30 plus i sin 30 gives cos 60 plus i sin 60, and cubing cos 20 plus i sin 20 gives cos 60 plus i sin 60. (1 + i) to the 4th works the same way: modulus root 2 to the 4th is 4, angle 4 times 45 is 180 degrees, so the answer is minus 4.
To write a + bi in exponential form, find the modulus r and the argument T in radians, as on the Argand plane, then write r e to the iT. For 1 + i, r is root 2 and T is pi over 4, so 1 + i = root 2 e to the i pi over 4. To go back, use a = r cos T and b = r sin T. A conjugate keeps r and negates T. Add and subtract in a + bi form; multiply, divide and take powers in exponential form.
Exponential form turns multiplication into adding angles, and De Moivre turns powers into multiplying them.
Readings 'Size and direction', 'Two routes to the same answer' and 'Multiplying as turning' are original to this dataset. All 10 extra questions are original to this dataset.
2 · Watch
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18 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.