Arithmetic, Units & Significant Figures

Accurate arithmetic is the backbone of engineering calculations. Learn order of operations, fractions, decimals, percentages and percentage change, ratios and proportions, powers and roots, approximation and rounding, significant figures, scientific and engineering notation, SI prefixes and unit conversion.

Most real engineering mistakes are not advanced mathematics errors — they are unit slips, misplaced decimal points and wrong percentages. The Mars Climate Orbiter was lost in 1999 because one team used pound-force seconds and another newton-seconds.

An analogy

Units are like the language a number speaks: 5 means nothing until you know whether it's 5 mm, 5 m or 5 km — and you must translate before combining.

Fractions, decimals and percentages

3/8 = 0.375 = 37.5 %. To add fractions, use a common denominator. Percent of a percent multiplies: a 10 % rise then 10 % fall gives 0.99 of the original, not 1.

Powers, roots and notation

Scientific notation 4.7 × 10³; engineering notation uses powers that are multiples of 3 to match SI prefixes (4.7 kΩ). Roots are fractional powers: √x = x^(1/2).

Significant figures and rounding

Report results to the precision of the least precise input: 12.3 × 4.567 = 56.2 (3 s.f.). For sums, keep the least number of decimal places. Round only at the end of multi-step calculations.

Unit conversion discipline

Write units on every number, convert everything to SI first, and cancel units like algebra. Dimensional checks catch most formula errors (see Dimensional Analysis module).

Formulas

• Order of operations: Brackets → Exponents → × ÷ (left to right) → + − (left to right)
• Percentage change: Δ% = (new − old)/old × 100
• Proportion: a/b = c/d ⇒ ad = bc
• Unit conversion: value × (target unit / source unit) — multiply by factors equal to 1

Worked example: Percentage change

Given: Pump flow rises from 45 L/min to 54 L/min

Find: Percentage increase.

• Δ = 9 L/min
• 9/45 × 100 = 20 %

Answer: +20 %

Worked example: Unit conversion

Given: Speed 72 km/h

Find: m/s.

• 72 km/h × 1000 m/km × 1 h/3600 s
• = 20 m/s

Answer: 20 m/s

Common mistakes

• Wrong: Adding percentages of different bases. — Right: Convert each change to multipliers and multiply.
• Wrong: Rounding intermediate results. — Right: Keep extra digits; round only the final answer.

Applications

• Datasheets: Reading µA, nF, MPa and ppm correctly.
• Cost & efficiency: Percentages and ratios in estimates.
• Lab reports: Significant figures and units.

Key takeaways

• BODMAS/PEMDAS order.
• Percent changes multiply.
• Significant figures follow the weakest input.
• Convert to SI and carry units.

Going further

• Interval arithmetic: Propagating ranges through calculations.
• Guard digits: Extra precision during computation.
• Unit-aware software: Libraries that enforce units.