COMPLEX NUMBERS · HL

Complex Numbers 1 — Rules, Add and Subtract

Surds, i, and adding and subtracting complex numbers.

Section 1 of 6

Surds

Rule

1.$\sqrt{ab} = \sqrt{a}\,\sqrt{b}$

Know your perfect squares: $\sqrt{4},\ \sqrt{9},\ \sqrt{16},\ \sqrt{25},\ \sqrt{36},\ \sqrt{100}$

Simplify

(i) $\sqrt{8} = \sqrt{4(2)} = \sqrt{4}\,\sqrt{2} = 2\sqrt{2}$

(ii) $\sqrt{12} = \sqrt{6(2)} = \sqrt{6}\,\sqrt{2}$ no good

$\sqrt{12} = \sqrt{4(3)} = \sqrt{4}\,\sqrt{3} = 2\sqrt{3}$

(iii) $\sqrt{72} = \sqrt{36(2)} = \sqrt{36}\,\sqrt{2} = 6\sqrt{2}$

(iv) $\sqrt{75} = \sqrt{15(5)} = \sqrt{15}\,\sqrt{5}$ no good

$\sqrt{75} = \sqrt{25(3)} = \sqrt{25}\,\sqrt{3} = 5\sqrt{3}$

(v) $\sqrt{27} = \sqrt{9}\,\sqrt{3} = 3\sqrt{3}$

(vi) $\sqrt{50} = \sqrt{25}\,\sqrt{2} = 5\sqrt{2}$

(vii) $\sqrt{45} = \sqrt{9}\,\sqrt{5} = 3\sqrt{5}$

Section 2 of 6

$i = \sqrt{-1}$

Definition

1.$i = \sqrt{-1}$

2.$i = $ iota

Simplify

(i) $\sqrt{4} = 2$

(ii) $\sqrt{-9} = \sqrt{9}\,\sqrt{-1} = 3i$

(iii) $\sqrt{-16} = \sqrt{16}\,\sqrt{-1} = 4i$

(iv) $\sqrt{-100} = \sqrt{100}\,\sqrt{-1} = 10i$

Section 3 of 6

Quadratic Equations

$ax^2 + bx + c = 0$

$x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Solve $x^2 - 2x - 5 = 0$ and leave answer in surd form

$a = 1, \quad b = -2, \quad c = -5$

$x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

$x = \dfrac{+2 \pm \sqrt{(-2)^2 - 4(1)(-5)}}{2(1)}$

$x = 1 \pm \sqrt{6}$

$x = -1.4 \quad$ or $\quad 3.4$

Solve $x^2 - 2x + 5 = 0$

$a = 1, \quad b = -2, \quad c = 5$

$\dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

$\dfrac{2 \pm \sqrt{(-2)^2 - 4(5)}}{2(1)} \qquad 4 - 20$

$\dfrac{2 \pm \sqrt{-16}}{2} \qquad \sqrt{-16} = \sqrt{16}\,\sqrt{-1} = 4i$

$\dfrac{2 \pm 4i}{2}$

$1 \pm 2i$

Section 4 of 6

$z = x + yi$

Every complex number $z$ is written as

1.$z = x + yi$

2.$x = $ Real part $= \operatorname{Re}(z)$

3.$y = $ Imaginary part $= \operatorname{Im}(z)$

Section 5 of 6

Add and Subtract

$z = 2 + 3i, \quad w = 4 + 7i$. Find:

(i) $z + w$

$2 + 3i + 4 + 7i$

$6 + 10i$

(ii) $2z + 3w$

$2(2 + 3i) + 3(4 + 7i)$

$4 + 6i + 12 + 21i$

$16 + 27i$

Method

1.Add real to real and imaginary to imaginary.

2.Multiply everything inside by constant outside.

$z_1 = 5 - 3i, \quad z_2 = -2 - 4i$. Find:

(i) $z_1 + z_2$

$5 - 3i - 2 - 4i$

$= 3 - 7i$

(ii) $z_1 - z_2$

Put a $1$ in front of $z_2$.

$5 - 3i - 1(-2 - 4i)$

$5 - 3i + 2 + 4i$

$= 7 + i$

(iii) $2z_1 + 3z_2$

$2(5 - 3i) + 3(-2 - 4i)$

$10 - 6i - 6 - 12i$

$4 - 18i$

$w_1 = 5 - 2i, \quad w_2 = 3 - 5i$. Find:

(i) $w_1 + w_2$

$5 - 2i + 3 - 5i = 8 - 7i$

(ii) $4w_1 - 3w_2$

$4(5 - 2i) - 3(3 - 5i)$

$20 - 8i - 9 + 15i$

$11 + 7i$

Section 6 of 6

Recap

1.$\sqrt{ab} = \sqrt{a}\,\sqrt{b}$

2.$\sqrt{-1} = i$

3.Complex number $z = x + yi$, $x = $ real, $y = $ imaginary

Simplify

(i) $\sqrt{-81} = \sqrt{81}\,\sqrt{-1} = 9i$

(ii) $\sqrt{-100} = \sqrt{100}\,\sqrt{-1} = 10i$

(iii) $\sqrt{-4} = \sqrt{4}\,\sqrt{-1} = 2i$

$z = 2 + 5i, \quad w = 4 + 8i$. Find:

(i) $z + w$

$2 + 5i + 4 + 8i$

$6 + 13i$

(ii) $3z + 5w$

$3(2 + 5i) + 5(4 + 8i)$

$6 + 15i + 20 + 40i$

$26 + 55i$

(iii) $4z - 2w$

$4(2 + 5i) - 2(4 + 8i)$

$8 + 20i - 8 - 16i$

$= 0 + 4i$

$z_1 = 3 - 5i, \quad z_2 = 4 - 3i$. Find:

(i) $z_1 + z_2$

$3 - 5i + 4 - 3i = 7 - 8i$

(ii) $z_1 - z_2$

Put in $-1$.

$3 - 5i - 1(4 - 3i)$

$3 - 5i - 4 + 3i = -1 - 2i$

$z = 2 - i, \quad w = 3 - 5i$. Find:

(i) $2z + 5w$

$2(2 - i) + 5(3 - 5i)$

$4 - 2i + 15 - 25i$

$19 - 27i$

(ii) $3w - z$

$3(3 - 5i) - 1(2 - i)$

$9 - 15i - 2 + i$

$7 - 14i$

SUM

The lot in one box

Rules and Add/Subtract toolkit

1.$\sqrt{ab} = \sqrt{a}\,\sqrt{b}$

2.$i = \sqrt{-1}$

3.$z = x + yi$: $x = \operatorname{Re}(z)$, $y = \operatorname{Im}(z)$

4.Add real to real and imaginary to imaginary.

5.Multiply everything inside by the constant outside.

End of lesson

Complex Numbers 1 — Rules, Add and Subtract · HL · Mathslive.ie