Maths Foundations for Electrical Science

1

Like & Unlike Signs

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Before we can tackle any electrical calculation, we need to be confident with positive and negative numbers. Most numbers you see are positive — the number 5 actually means +5. A negative number is shown with a minus sign, like −5.

Adding & Subtracting: The Golden Rule

Like signs → ADD | Unlike signs → SUBTRACT

Worked Examples from the Textbook

3 + (+7) = 3 + 7 = 10 (like signs \to add) 3 + (-7) = 3 - 7 = -4 (unlike signs \to subtract)

Multiplying & Dividing

Like signs → positive result | Unlike signs → negative result

+4 \times +5 = +20 (like) -4 \times +5 = -20 (unlike) -4 \times -5 = +20 (like — two negatives make positive!)

Extra Worked Examples

Example 1: A temperature drops from +8°C by 12 degrees. What is the new temperature?

1

Write it as: +8 + (−12)

2

Unlike signs → subtract: 8 − 12 = −4°C

Example 2: −6 × −3 = ?

1

Both negative = like signs → positive: 6 × 3 = +18

Example 3: −24 ÷ +6 = ?

1

Unlike signs → negative: 24 ÷ 6 = −4

⚡ Quick Memory Aid

Think of it like people: two friends (same sign) get along (+), but a friend and an enemy (different signs) don't (−).

2

BODMAS (Order of Operations)

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When a calculation has more than one operation, you must follow a strict order called BODMAS (sometimes BIDMAS).

Textbook Example 1

6 + 4 × 2 = ?

1

Multiply first: 4 × 2 = 8

2

Then add: 6 + 8 = 14 (NOT 20!)

Textbook Example 2

2 × 4 + 5 × 4(10 − 8)²

1

Brackets: (10 − 8) = 2

2

Indices: 2² = 4, and 4 × 4 = 16

3

Multiply: 2 × 4 = 8 and 5 × 16 = 80

4

Add: 8 + 80 = 88

Textbook Example 3

20 × 4 − 10 + 2

1

Multiply: 20 × 4 = 80

2

Left to right: 80 − 10 = 70, then 70 + 2 = 72

⚠ Common Mistake

When only + and − remain, work left to right!

Extra Worked Example

2 × (3 + 5)² ÷ 4 − 10

1

Brackets: (3+5) = 8

2

Indices: 8² = 64

3

Multiply: 2 × 64 = 128

4

Divide: 128 ÷ 4 = 32

5

Subtract: 32 − 10 = 22

3

Percentages

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A percentage is a fraction out of 100. To convert: divide the top by the bottom, then multiply by 100.

Textbook Example — Exam Score

42 correct out of 48:

42 \div 48 = 0.875 0.875 \times 100 = 87.5%

⚡ Electrical Application — Volt Drop

BS 7671: max volt drop is 3% for lighting and 5% for power on 230 V.

Lighting: (3 \div 100) \times 230 = 6.9 V Power: (5 \div 100) \times 230 = 11.5 V

Extra Example 1 — 3-Phase

What is 5% of 400 V?

0.05 \times 400 = 20 V

Extra Example 2 — Price Increase

Job quoted £800, add 20%:

800 \times 1.20 = £960

4

Powers of 10 & Scientific Notation

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Electrical engineering uses very large and very small numbers. Powers of 10 let us write them neatly.

10 = 10¹ | 100 = 10² | 1,000 = 10³ | 1,000,000 = 10⁶

Multiplying: Combine zeros. Dividing: Cancel zeros.

3000 = 3 × 10³ and 3,000,000 = 3 × 10⁶

Extra Example — Small Numbers

Express 0.0025 A in scientific notation:

1

Move decimal right 3 places: 2.5

2

Result: 2.5 × 10−³ A = 2.5 mA

5

SI Units & Prefixes

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Key Electrical SI Units

Quantity	Unit	Symbol
Current	Ampere	A
Voltage	Volt	V
Resistance	Ohm	Ω
Power	Watt	W
Energy	Joule	J
Frequency	Hertz	Hz
Magnetic flux	Weber	Wb
Flux density	Tesla	T
Charge	Coulomb	C

SI Prefixes

Prefix	Symbol	Factor	Example
Mega	M	×10⁶	MΩ (insulation test)
Kilo	k	×10³	kW, kV
milli	m	×10−³	mA (RCD trip)
micro	μ	×10−⁶	μF (capacitor)
nano	n	×10−⁹	nF

Extra Example — Converting Between Prefixes

Convert 4700 Ω to kilohms:

Since 1 kΩ = 1000 Ω, ask: how many thousands in 4700?

4700 \div 1000 = 4.7 So 4700 Ω = 4.7 kΩ

Convert 0.03 A to milliamps:

Since 1 A = 1000 mA, multiply the amp value by 1000 to get the milliamp value:

0.03 \times 1000 = 30 So 0.03 A = 30 mA (the RCD trip threshold!)

6

Fractions

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A fraction is a part of a whole. Top = numerator, bottom = denominator.

Adding Fractions

Find a common denominator.

Textbook Example

¼ + ½ → common denominator 4: 1/4 + 2/4 = 3/4

Multiplying Fractions

1/4 \times 4/8 = 4/32 = 1/8

Dividing Fractions

Keep, Change, Flip: flip the second fraction, then multiply.

Textbook Example

1/4 ÷ 3/7 → 1/4 × 7/3 = 7/12

Extra Example

2/3 + 1/5: common denominator 15 → 10/15 + 3/15 = 13/15

7

Algebra Basics

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Letters stand for numbers. No sign between letters = multiply. So F = BIL means F = B × I × L.

Textbook Example

2PR − 1PR where P=4, R=2:

(2\times4\times2) - (1\times4\times2) = 16 - 8 = 8

Extra — Ohm's Law

V = IR. If I=13 A, R=2.5 Ω: V = 13 × 2.5 = 32.5 V

Extra — Power

P = I²R. If I=10 A, R=5 Ω: P = 100 × 5 = 500 W

8

Laws of Indices

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An index tells you how many times to multiply a number by itself. 2³ = 2×2×2 = 8.

Multiplying (same base): ADD indices | Dividing (same base): SUBTRACT indices

2⁹ \times 2³ = 2¹² | 2⁹ \div 2³ = 2⁶

Factorising with HCF

Textbook Example

Factorise 27a + 18: HCF is 9 → 9(3a + 2)

Extra Example

10⁵ × 10³ ÷ 10⁴ = 10⁸ ÷ 10⁴ = 10⁴ = 10,000

9

Pythagoras' Theorem

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In a right-angled triangle: c² = a² + b². Used in electrical work for impedance, power factor, and phasor diagrams.

Textbook Example — 3:4:5

c² = 3² + 4² = 9 + 16 = 25 c = \sqrt25 = 5

Extra — Finding a Shorter Side

Hypotenuse = 13, one side = 5:

a² = 13² - 5² = 169 - 25 = 144 a = \sqrt144 = 12

⚡ Impedance Triangle

Z = \sqrt(R² + X²) Z = \sqrt(30² + 40²) = \sqrt2500 = 50 Ω

10

Trigonometry — SOH CAH TOA

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Pythagoras finds sides; trigonometry finds angles. Essential for power factor calculations.

Textbook — Aircraft Angle

Opposite=300, Adjacent=400. Use TOA:

tan θ = 300/400 = 0.75 θ = tan⁻¹(0.75) = 36.87°

Extra — Power Factor

True Power=3kW, Apparent=4kVA. Use CAH:

cos θ = 3000/4000 = 0.75 θ = cos⁻¹(0.75) = 41.4°, PF = 0.75

11

Types of Triangles

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All angles in any triangle add up to 180°

Textbook Example

Angles of 60° and 40°: 180° − 100° = 80°

Extra Example

Isosceles with top angle 110°: (180−110)÷2 = 35° each

12

Transposition of Formula

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Rearrange formulas using the opposite rule: + ↔ −, × ↔ ÷, x² ↔ √

⚡ Three Steps

1: Write the letter you want =
2: Bring over what's already on the other side
3: Move the rest using the opposite rule

Textbook — Ohm's Law

V = IR, find I: → I = V ÷ R

Textbook — Pythagoras

a² + b² = c, find b: → b² = c − a² → b = √(c − a²)

Extra — Power

P = I²R, find R: → R = P ÷ I²

If P=500W, I=10A: R = 500÷100 = 5 Ω

Extra — Resistivity

R = ρL ÷ A, find A: → A = ρL ÷ R

13

Statistics — Mean & Median

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Mean = Sum of all values ÷ Number of values
Median = Middle value when ordered (if even count, average the two middle values)

Textbook Example — Insulation Resistance

Values: 2, 1.6, 2, 1, 0.5, 100, 300, 5, 10 MΩ

Mean = 422.1 \div 9 = 46.9 MΩ

Ordered: 0.5, 1, 1.6, 2, 2, 5, 10, 100, 300 → Median = 2 MΩ

The median is better here because outliers (100, 300) skew the mean.

Extra — Loop Impedance

Zs readings: 0.82, 0.95, 0.78, 0.91, 0.84 Ω

Mean = 4.30 \div 5 = 0.86 Ω

Ordered: 0.78, 0.82, 0.84, 0.91, 0.95 → Median = 0.84 Ω

⚡ When to Use Which

Mean when values are similar. Median when there are extreme outliers.