Introduction to Complex Numbers (Edexcel A-Level Further Mathematics): Revision Notes

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1.1.1 Introduction to Complex Numbers

Overview

A complex number is any number that can be written in the form:

z=a+biz = a + bi

where:

  • aa is the real part (Re(z)=a)(\text{Re}(z) = a)
  • bb is the imaginary part (Im(z)=b)(\text{Im}(z) = b)
  • ii is the imaginary unit, defined by i2=−1i^2 = -1
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Example: For the complex number z=3+4iz = 3 + 4i

  • Real part = 33
  • Imaginary part = 44

Basic Operations on Complex Numbers

Addition

To add two complex numbers, add their real and imaginary parts separately.

(a+bi)+(c+di)=(a+c)+(b+d)i(a + bi) + (c + di) = (a + c) + (b + d)i
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Example:

(2+3i)+(1+4i)(2 + 3i) + (1 + 4i)
=(2+1)+(3+4)i= (2 + 1) + (3 + 4)i =3+7i= 3 + 7i

Subtraction

To subtract two complex numbers, subtract their real and imaginary parts separately.

(a+bi)−(c+di)=(a−c)+(b−d)i(a + bi) - (c + di) = (a - c) + (b - d)i
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Example:

(5+2i)−(3+6i)(5 + 2i) - (3 + 6i)
=(5−3)+(2−6)i= (5 - 3) + (2 - 6)i=2−4i= 2 - 4i

Multiplication

To multiply two complex numbers, expand using the distributive property and simplify using:

i2=−1i^2 = -1

(a+bi)(c+di)=ac+adi+bci+bdi2=(ac−bd)+(ad+bc)i(a + bi)(c + di) = ac + adi + bci + bdi^2 = (ac - bd) + (ad + bc)i
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Example:

(3+2i)(1+4i)(3 + 2i)(1 + 4i)

Step 1**:** Expand using distributive property:

=3(1)+3(4i)+2i(1)+2i(4i)= 3(1) + 3(4i) + 2i(1) + 2i(4i)

Step 2: Simplify:

=3+12i+2i+8i2=3+14i−8=−5+14i= 3 + 12i + 2i + 8i^2 = 3 + 14i - 8 = -5 + 14i
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Example:

(3+4i)(−5+6i)(3 + 4i)(-5 + 6i)

Step 1: Expand:

=3(−5)+3(6i)+4i(−5)+4i(6i)= 3(-5) + 3(6i) + 4i(-5) + 4i(6i)

Step 2: Simplify:

=−15+18i−20i+24i2= -15 + 18i - 20i + 24i^2 =−15−2i−24= -15 - 2i - 24 =−39−2i= -39 - 2i

Division

To divide two complex numbers, multiply both the numerator and denominator by the conjugate of the denominator.

Conjugate of a+bia + bi is a−bia - bi

Formula:

a+bic+di=(a+bi)(c−di)(c+di)(c−di)\frac{a + bi}{c + di} = \frac{(a + bi)(c - di)}{(c + di)(c - di)}
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Example:

2−i4+3i\frac{2 - i}{4 + 3i}

Step 1: Multiply by the conjugate 4−3i4 - 3i

=(2−i)(4−3i)(4+3i)(4−3i)= \frac{(2 - i)(4 - 3i)}{(4 + 3i)(4 - 3i)}

Step 2: Expand numerator:

=(2−i)(4−3i)= (2 - i)(4 - 3i) =8−6i−4i+3i2= 8 - 6i - 4i + 3i^2 =5−10i= 5 - 10i

Step 3: Expand denominator:

=(4+3i)(4−3i)= (4 + 3i)(4 - 3i) =16−9i2= 16 - 9i^2 =25= 25

Step 4: Simplify:

=5−10i25= \frac{5 - 10i}{25} =15−25i= \frac{1}{5} - \frac{2}{5}i
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Additional Example: Simplify

3+2i4+5i\frac{3 + 2i}{4 + 5i}

Step 1: Multiply by conjugate 4−5i4 - 5i

=(3+2i)(4−5i)(4+5i)(4−5i)= \frac{(3 + 2i)(4 - 5i)}{(4 + 5i)(4 - 5i)}

Step 2: Expand numerator:

=12−15i+8i−10i2= 12 - 15i + 8i - 10i^2 =22−7i= 22 - 7i

Step 3: Expand denominator:

=16−25i2= 16 - 25i^2 =41= 41

Step 4: Simplify:

=22−7i41= \frac{22 - 7i}{41} =2241−741i= \frac{22}{41} - \frac{7}{41}i

Worked Examples

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Example 1: Simplify (2+3i)(4−i)(2 + 3i)(4 - i)

Step 1: Expand:

=(2+3i)(4−i)= (2 + 3i)(4 - i) =2(4)+2(−i)+3i(4)+3i(−i)= 2(4) + 2(-i) + 3i(4) + 3i(-i)

Step 2: Simplify:

=8−2i+12i−3i2= 8 - 2i + 12i - 3i^2

Step 3: Combine like terms:

=8+10i+3= 8 + 10i + 3 =11+10i= 11 + 10i
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Example 2: Simplify 1+2i3+4i\frac{1 + 2i}{3 + 4i}

Step 1: Multiply by the conjugate 3−4i3 - 4i

(1+2i)(3−4i)(3+4i)(3−4i)\frac{(1 + 2i)(3 - 4i)}{(3 + 4i)(3 - 4i)}

Step 2: Expand numerator:

=(1+2i)(3−4i)= (1 + 2i)(3 - 4i) =3−4i+6i−8i2= 3 - 4i + 6i - 8i^2 =3+2i+8= 3 + 2i + 8 =11+2i= 11 + 2i

Step 3: Expand denominator:

=(3+4i)(3−4i)= (3 + 4i)(3 - 4i) =9+16=25= 9 + 16 = 25

Step 4: Simplify:

=11+2i25= \frac{11 + 2i}{25} =1125+225i= \frac{11}{25} + \frac{2}{25}i
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Example 3: Solve z2−2z+17=0z^2 - 2z + 17 = 0 by completing the square

Step 1: Rewrite:

z2−2z=−17z^2 - 2z = -17

Step 2: Complete the square:

(z−1)2−1=−17(z - 1)^2 - 1 = -17 \quad ⇒(z−1)2=−16\Rightarrow \quad (z - 1)^2 = -16

Step 3: Solve:

z−1=±4iz - 1 = \pm 4i \quad ⇒z=1±4i\Rightarrow \quad z = 1 \pm 4i

Note Summary

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Common Mistakes:

  1. Forgetting i2=−1i^2 = -1 when simplifying terms.
  2. Incorrectly applying the conjugate in division.
  3. Mixing up real and imaginary parts during addition or subtraction.
  4. Expanding incorrectly in multiplication.
  5. Skipping simplification steps leads to wrong final answers.
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Key Formulas:

  1. Addition**:** (a+bi)+(c+di)=(a+c)+(b+d)i(a+bi) + (c+di) = (a+c) + (b+d)i
  2. Subtraction**:** (a+bi)−(c+di)=(a−c)+(b−d)i(a+bi) - (c+di) = (a-c) + (b-d)i
  3. Multiplication: (a+bi)(c+di)=(ac−bd)+(ad+bc)i(a+bi)(c+di) = (ac - bd) + (ad + bc)i
  4. Division: a+bic+di=(a+bi)(c−di)c2+d2\frac{a+bi}{c+di} = \frac{(a+bi)(c-di)}{c^2+d^2}
  5. Conjugate**:** a+bi‾=a−bi\overline{a+bi} = a - bi

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