Divide and solve equations

Make a complex denominator real

Check that the denominator is nonzero before manipulating the fraction. Multiply the whole fraction by the conjugate over itself, then collect the real and imaginary parts.

The surd analogy multiplies $1/(\sqrt a+\sqrt b)$ by $(\sqrt a-\sqrt b)/(\sqrt a-\sqrt b)$ to obtain $(\sqrt a-\sqrt b)/(a-b)$, for real $a,b\ge0$ and $a\ne b$.

Taking $a=5,b=1$ gives the source split $\sqrt5/4-1/4$.

The source says “integer” denominator; generally it is real, and it is an integer only when $a-b$ is an integer.

The case $a=b>0$ needs a direct simplification instead.

For division by $w=x+iy\ne0$, multiply numerator and denominator by $w^*=x-iy$.

Since $ww^*=x^2+y^2>0$, the denominator becomes real and nonzero.

This is what the notes call realising the denominator.

Example 7 then splits its fraction to identify the real part and the imaginary coefficient separately.

Solve simultaneous equations by substitution

Use the same substitution or elimination principles as for real equations.

Example 8 first writes $z=(1-i)w$, substitutes into the second equation and realises the remaining denominator; its instruction forbids a graphing calculator.

When a conjugate remains, as in Example 9, set $z=a+bi$ and $z^*=a-bi$ with real $a,b$, then compare components.

A ratio such as $w/z$ requires $z\ne0$.