Complex Numbers - Arithmetic

Addition is performed componentwise:

• (𝑎 + 𝑏𝑖) + (𝑐 + 𝑑𝑖) = (𝑎+𝑐) + (𝑏+𝑑)𝑖

Multiplication is performed using distributivity and 𝑖² = -1:

• (𝑎 + 𝑏𝑖)(𝑐 + 𝑑𝑖) = 𝑎𝑐 + 𝑎𝑑𝑖 + 𝑏𝑐𝑖 + 𝑏𝑑𝑖²
• (𝑎 + 𝑏𝑖)(𝑐 + 𝑑𝑖) = (𝑎𝑐 - 𝑏𝑑) + (𝑎𝑑 + 𝑏𝑐)𝑖

Complex Conjugation replaces 𝑖 with -𝑖, and is denoted with a bar:

• $\overline{a+bi}=a-bi$

One checks that for any 2 complex numbers 𝑧, 𝑤, we have:

• $\overline{z+w}=\overline{z}+\overline{w}$
• $\overline{zw}=\overline{z}\cdot \overline{w}$

Also, (𝑎 + 𝑏𝑖)(𝑎 - 𝑏𝑖) = 𝑎² + 𝑏², so 𝑧𝑧̅ is a non-negative real number for any complex number 𝑧

The absolute value of a complex number 𝑧 is the real number$|z|=\sqrt{z\overline{z}}:$

• $|a+bi|=\sqrt{a^2+b^2}$

One checks that |𝑧𝑤| = |𝑧|·|𝑤|

Division by a nonzero real number proceeds componentwise:

• $\frac{a+bi}{c}=\frac{a}{c}+\frac{b}{c}i$

Division by a nonzero complex number requires multiplying the numerator and denominator by the complex conjugate of the denominator:

• $\frac{z}{w}=\frac{z\overline{w}}{w\overline{w}}=\frac{z\overline{w}}{|w{|}^2}$

For example:

• $\frac{1+i}{1-i}=\frac{(1+i{)}^2}{1^2+(-1{)}^2}=\frac{1+2i+i^2}{2}=i$

The real and imaginary parts of a complex number are:

• 𝑅𝑒(𝑎 + 𝑏𝑖) = 𝑎
• 𝐼𝑚(𝑎 + 𝑏𝑖) = 𝑏