Equating Real & Imaginary Parts (Edexcel International A Level (IAL) Further Maths: Further Pure 1): Revision Note

Equating Real & Imaginary Parts

How do I equate real and imaginary parts?

• If two complex expressions are equal, then

  • their real parts are equal

  • and their imaginary parts are equal

• If $a+b\mathrm{i}=8-10\mathrm{i}$ where $a$ and $b$ are real

  • then $a=8$ (equating real parts)

  • and $b=-10$ (equating imaginary parts)

How do I solve equations using real and imaginary parts?

• Introduce $z=x+y\mathrm{i}$ for expressions in $z$

• For example, solve $z+2z*=12-\mathrm{i}$

  • Let $z=x+y\mathrm{i}$

  • Substitute in

    • $\left(x+y\mathrm{i}\right)+2\left(x-y\mathrm{i}\right)=12-i$

  • Expand and collect terms

    • $3x-y\mathrm{i}=12-\mathrm{i}$

  • Equate real and imaginary parts

    • $3x=12$ gives $x=4$

    • $-y=-1$ gives $y=1$

  • Substitute back into $z$

    • $z=4+\mathrm{i}$

How do I use real and imaginary parts with modulus signs?

• If $z=x+y\mathrm{i}$ then $\left|z\right|=\sqrt{x^2+y^2}$

• Helpful modulus rules are:

  • $\left|z_1{z}_2\right|=\left|z_1\right|\left|z_2\right|$

  • $\left|\frac{z_1}{z_2}\right|=\frac{\left|z_1\right|}{\left|z_2\right|}$

  • $\left|z\right|=|z*|$

  • $zz*=z*z=|z{|}^2$

    • Proved using $z=x+y\mathrm{i}$ and $z*=x-y\mathrm{i}$

• For example, find $p$ if $z_1=p+p\mathrm{i}$, $z_2=2\mathrm{i}$ and $\left|z_1{z}_2\right|=6\sqrt{2}$

  • Use $\left|z_1{z}_2\right|=\left|z_1\right|\left|z_2\right|$

    • $|p+p\mathrm{i}|\left|2\mathrm{i}\right|=6\sqrt{2}$

  • Use $\left|z\right|=\sqrt{x^2+y^2}$

    • $\sqrt{p^2+p^2}\times 2=6\sqrt{2}$ so $2\sqrt{2p^2}=6\sqrt{2}$

  • Square both sides and simplify

    • $p^2=9$

    • $p=\pm 3$