Imaginary numbers and Cartesian form | CAIE 9709 Pure Mathematics 3 Notes

Chapters

Pure Mathematics 3

Mathematics9709

The one new rule is i squared equals minus one

Definition

What is meant by imaginary unit?

Model answer: The number i is defined by $i^2=-1$ .

For $b>0$ , the two solutions of $x^2=-b$ are $x=\pm i\sqrt b$ .
Powers repeat every four: $i,\,-1,\,-i,\,1$ .
To reduce a large exponent, divide it by 4. The remainder tells you where you are in the cycle.

Worked example

Reduce a power of i

i^{37}=i^{4(9)+1}

=(i^4)^9i=1^9i

Answer

i^{37}=i

Cartesian form stores two real numbers

Definition

What is meant by complex number?

Model answer: A number expressible as $x+iy$ , where x and y are real.

z = 3 + 2i is the point (3, 2)

ReIm

3 + 2i

horizontal = Re(z) = 3

vertical = Im(z) = 2

A complex number in Cartesian form is a point with coordinates (Re z, Im z).

z=x+iy

$\operatorname{Re}z=x$ and $\operatorname{Im}z=y$ .
The imaginary part is y, not iy.
Two complex numbers are equal only when both real parts match and both imaginary parts match.

Equate real parts and imaginary parts

Worked example

One complex equation gives two real equations

Given $(2a-1)+(b+3)i=5-2i$ , compare the two parts.

2a-1=5

b+3=-2

Answer

a=3

b=-5

Your turnIndependent foundation check [3 marks]

Simplify $i^{27}+\sqrt{-48}$ into Cartesian form.

Show worked answer

i^{27}=i^3=-i

\sqrt{-48}=i\sqrt{48}

=i\sqrt{16\times3}=4i\sqrt3

Answer

(4\sqrt3-1)i

Common mistake

Convert each negative square root to i times a positive square root before multiplying. The real-root rule

\sqrt a\sqrt b=\sqrt{ab}

is unsafe when both a and b are negative.

(Pure Mathematics 2 & 3 Coursebook, Ch. 11.1–11.2)

The one new rule is i squared equals minus one

Definition

What is meant by imaginary unit?

Model answer: The number i is defined by $i^2=-1$ .

For $b>0$ , the two solutions of $x^2=-b$ are $x=\pm i\sqrt b$ .
Powers repeat every four: $i,\,-1,\,-i,\,1$ .
To reduce a large exponent, divide it by 4. The remainder tells you where you are in the cycle.

Worked example

Reduce a power of i

i^{37}=i^{4(9)+1}

=(i^4)^9i=1^9i

Answer

i^{37}=i

Cartesian form stores two real numbers

Definition

What is meant by complex number?

Model answer: A number expressible as $x+iy$ , where x and y are real.

z = 3 + 2i is the point (3, 2)

ReIm

3 + 2i

horizontal = Re(z) = 3

vertical = Im(z) = 2

A complex number in Cartesian form is a point with coordinates (Re z, Im z).

z=x+iy

$\operatorname{Re}z=x$ and $\operatorname{Im}z=y$ .
The imaginary part is y, not iy.
Two complex numbers are equal only when both real parts match and both imaginary parts match.

Equate real parts and imaginary parts

Worked example

One complex equation gives two real equations

Given $(2a-1)+(b+3)i=5-2i$ , compare the two parts.

2a-1=5

b+3=-2

Answer

a=3

b=-5

Your turnIndependent foundation check [3 marks]

Simplify $i^{27}+\sqrt{-48}$ into Cartesian form.

Show worked answer

i^{27}=i^3=-i

\sqrt{-48}=i\sqrt{48}

=i\sqrt{16\times3}=4i\sqrt3

Answer

(4\sqrt3-1)i

Common mistake

Convert each negative square root to i times a positive square root before multiplying. The real-root rule

\sqrt a\sqrt b=\sqrt{ab}

is unsafe when both a and b are negative.

(Pure Mathematics 2 & 3 Coursebook, Ch. 11.1–11.2)