Complex numbers: watch it solved step by step

Add, subtract, multiply and divide with i² = −1, then modulus, argument and polar form, on an Argand diagram. Level: A-level. Type any problem of this kind and the Lab works it out and animates every move, with a plain-words explanation of why.

Worked example: (2 + i) / (1 - i)

1. The problem. Treat i like a letter, but remember the one special rule: i^2 = -1.
2. Multiply top and bottom by the conjugate of the bottom. The conjugate of p + qi is p - qi. Multiplying a number by its conjugate gives a real number, which removes i from the denominator.
3. The bottom becomes real. (p+qi)(p-qi) = p^2 + q^2.
4. Multiply out the top. Expand, using i^2 = -1.
5. Divide each part by the real denominator. Split into real and imaginary parts.
6. Check. Multiply the answer by the divisor: you should get the original top number.

Answer: 1/2+3/2i

Practise complex numbers with problems the Lab marks for you.