NZ NCEA Level 3 Mathematics and Statistics

## The imaginary unit No real number squares to a negative, so the equation $x^2 = -1$ has no real solution. The **imaginary unit** $i$ is defined to be a number with $$ i^2 = -1 $$ which lets every negative number have a square root: $\sqrt{-16} = 4i$ and $\sqrt{-3} = \sqrt{3}\,i$. Take the factor $i$ out before doing anything else with a root of a negative number, because the usual rule $\sqrt{a}\sqrt{b} = \sqrt{ab}$ is only safe when at least one of $a$ and $b$ is positive. Powers of $i$ repeat in a cycle of four: $$ i^1 = i, \qquad i^2 = -1, \qquad i^3 = -i, \qquad i^4 = 1 $$ so $i^{n}$ depends only on the remainder of $n$ divided by 4. For example $i^{27} = i^{24}\cdot i^{3} = -i$. ## Rectangular form A **complex number** has the form $$ z = x + iy $$ where $x$ and $y$ are real. The number $x$ is the **real part**, written $\operatorname{Re}(z)$, and $y$ is the **imaginary part**, written $\operatorname{Im}(z)$. The imaginary part is the real number $y$, so it does not include the $i$. For $z = 5 - 2i$, $\operatorname{Re}(z) = 5$ and $\operatorname{Im}(z) = -2$. Two complex numbers are equal exactly when their real parts are equal and their imaginary parts are equal. This is what lets one complex equation be split into two real equations, and it is used constantly when solving for unknown constants. A number with $y = 0$ is real, so every real number is also a complex number. A number with $x = 0$ and $y \neq 0$ is **purely imaginary**. ### Worked example: equating…