Level 2 Engineering Maths Decimals

🔢 Decimals — Level 2 Engineering Maths

Every measurement in engineering is a decimal. Tolerances, voltages, pressures, thread pitches — they all need precise decimal understanding. Get this right and everything else becomes easier.

Level 2 Core Skill Place value Rounding & significant figures Standard form Engineering precision

📋 Decimals — Quick Reference

1
Digits left of the point = whole numbers
2
Digits right of the point = tenths, hundredths...
3
Line up decimal points when adding/subtracting
🧠

Key Words — What Do They Mean?

Decimal PointSeparates whole
from part
Tenths1st digit after
the point
PrecisionHow exact a
value is
What is a decimal?Numbers between whole numbers

A decimal number has a decimal point that separates the whole number part (left) from the fractional part (right).

4 7 . 3 8 6
↑↑ ↑↑↑
Whole number . Fractional part
💡 The decimal point is just a marker. Digits to the LEFT are whole units; digits to the RIGHT are parts of 1 (tenths, hundredths, thousandths…)
Why decimals matter in engineering
  1. Precision — a shaft of 24.75 mm is very different from 24.8 mm in precision engineering
  2. Measurements — voltmeters, micrometers and pressure gauges all display decimals
  3. Calculations — every formula in engineering produces decimal answers
  4. Standards — ISO tolerances, material densities and thread pitches are all decimals
📍Place Valueeach digit's worth
🔵Roundingd.p. & nearest
➕Operations+ − × ÷
🔟Powers of 10shift the digits
🎯Sig. Figuresaccuracy
🔬Standard Formvery big/small
⚙️Engineeringreal uses
📍

Place Value

Every digit in a decimal number has a specific value depending on its position. Understanding this is the foundation of everything else.

📋 Place Value — Quick Reference

1
Each column is 10 times smaller moving right
2
Tenths, hundredths, thousandths — in that order after the point
🧠

Key Words — What Do They Mean?

Place ValueWhat a digit
is worth
ColumnPosition of
a digit
Place Value TableThe number 4 7 3 . 5 8 6 shown in every column
THOUSANDS HUNDREDS TENS UNITS . TENTHS HUNDREDTHS THOUSANDTHS — — 4 7 . 3 8 6 4 × 10 = 40 7 × 1 = 7 3 × 1/10 = 0.3 8 × 1/100 = 0.08 6 × 1/1000 = 0.006 40 + 7 + 0.3 + 0.08 + 0.006 = 47.386
💡 Each column is worth 10 times the column to its right. So 4 in the tens column = 40, but 4 in the tenths column = 0.4 — totally different!

🔍 Place Value Explorer — type any decimal number

👆 Now try it yourself — type any decimal number in the box above and watch each digit light up in its column.
Fractional partWhat each decimal place means
1st decimal place = tenths = /10
2nd decimal place = hundredths = /100
3rd decimal place = thousandths= /1000
📐 Engineering example: A micrometer reads 12.475 mm
12 = whole mm   4 = 4/10 mm   7 = 7/100 mm   5 = 5/1000 mm
Common MistakeConfusing tenths and hundredths
❌ "0.3 and 0.30 are different" — WRONG
0.3 = 3 tenths = 0.30 = 30 hundredths — identical!
❌ "0.7 is bigger than 0.70" — WRONG
Trailing zeros after a decimal do NOT change the value
✅ But: 0.7 ≠ 0.07
0.7 = 7 tenths = 0.700
0.07 = 7 hundredths — ten times smaller!
🔵

Rounding Decimals

Rounding gives a simpler approximate value. In engineering you'll round to decimal places (d.p.) and significant figures (s.f.).

📋 Rounding — Quick Reference

1
Look at the next digit after the rounding point
2
5 or more → round up. Below 5 → round down
🧠

Key Words — What Do They Mean?

d.p.Decimal
places
ApproximateA close but not
exact value
The RuleLook at the digit AFTER where you're rounding. 0–4 → round down. 5–9 → round up.
VisualRounding 3.46 to 1 decimal place — which way does it round?
3.4 round DOWN to this 3.45 ↑ midpoint (digit = 5 → round UP) 3.5 round UP to this ✅ 3.46 6 ≥ 5 → rounds UP
✅ 3.46 rounded to 1 d.p. = 3.5 because the digit after the first decimal place is 6 (≥5), so we round up.

🔵 Rounding Tool — see exactly which digit to look at

👆 Now try it yourself — there are 6 options in the drop-down:

1 decimal place 2 decimal places 3 decimal places 4 decimal places nearest 10 nearest 100

Use the Round to ▼ drop-down to switch between them, and type any number in the Number box. Watch the digits change colour each time!

Rounding to decimal placesStep by step
  1. Identify which decimal place you're rounding to
  2. Look at the digit immediately after that place
  3. If it's 0, 1, 2, 3 or 4 → leave the last digit as it is (round down)
  4. If it's 5, 6, 7, 8 or 9 → add 1 to the last digit (round up)
  5. Drop all digits after your chosen place
Round 6.3847 to 2 d.p.
2 d.p.: look at the 3rd decimal place → 4
4 < 5: round down — keep the 8
Answer: 6.38 ✅
Watch out!Rounding up a 9

What if rounding up gives you a 10? You carry over — just like normal addition.

Round 7.996 to 2 d.p.
2 d.p.: look at 3rd → 6 ≥ 5 → round up
9 + 1: = 10 → write 0, carry 1
9 + 1: = 10 → write 0, carry 1
7 + 1: = 8
Answer: 8.00 ✅
⚠️ Write 8.00 (not just 8) to show you rounded to 2 d.p. — the zeros are significant here!

➕ Decimal Operations Calculator — see the working

👆 Try your own numbers — change the values in the boxes, pick an operation from the drop-down, then hit Calculate → to see the full working.
➕

Adding, Subtracting, Multiplying & Dividing Decimals

The operations are the same as whole numbers — the key is keeping the decimal point in the right place.

📋 Decimal Operations — Quick Reference

1
Add/Subtract: line up the decimal points
2
Multiply: ignore points, multiply, then count d.p. back in
3
Divide: make the divisor a whole number first
🧠

Key Words — What Do They Mean?

DivisorThe number
you divide by
AlignLine numbers
up straight
Adding & SubtractingLine up the decimal points!

Write numbers so all decimal points are vertically aligned. Add zeros to fill empty spaces if needed.

Enter two decimals — watch them align in columns:

👆 Try it yourself — here are some to have a go at:

➕ Addition
5.4 + 2.73
➕ Addition
14.6 + 0.075
➖ Subtraction
9.5 − 3.27
➖ Subtraction
20.1 − 4.856

Type the numbers into the two boxes above, then use the + / − ▼ drop-down to switch operation. Notice how the decimal point column stays red and everything lines up!

✅ The decimal point column is always red — line them all up and the answer's point goes in the same column.
MultiplyingIgnore the point, then place it back
  1. Count total decimal places in both numbers
  2. Multiply as if they were whole numbers (ignore the point)
  3. Put the decimal point back — same number of places from the right

Enter two decimals — see every step:

×
DividingMake the divisor a whole number first
  1. If the divisor has decimals, multiply both numbers by 10/100/1000 to make it a whole number
  2. Then divide normally

Enter any division — see how to make the divisor whole:

÷
⚙️ Engineering CalculatorThread pitch / CNC feed rate

How many revolutions to travel a given distance?

🔟 Digit Shift — watch the digits move left or right

↓ After:
👆 Try it yourself — enter any decimal, then use the drop-down arrow to choose how many places to multiply or divide by a power of 10. Notice the digits shift along the columns!
🔟

Multiplying & Dividing by Powers of 10

The fastest and most-used decimal skill. Every time you multiply by 10, 100 or 1000, the digits shift LEFT. Divide — they shift RIGHT.

📋 Powers of 10 — Quick Reference

1
× 10, 100, 1000... → digits move LEFT
2
÷ 10, 100, 1000... → digits move RIGHT
🧠

Key Words — What Do They Mean?

Power of 1010, 100, 1000
and so on
ShiftDigits move
columns
Digit Shift DiagramWatch the digits move — 3.47 × 10, × 100, ÷ 10
100s 10s 1s . 0.1s 0.01s 0.001s 0.0001s 3.47 × 10 = 34.7 (digits shift ONE place LEFT) 3 . 4 7 ← shift left 1 3.47 × 100 = 347 (digits shift TWO places LEFT) 3 4 7 . ←← shift left 2 3.47 ÷ 10 = 0.347 (digits shift ONE place RIGHT) 0 . 3 4 7 → shift right 1
🔑 Multiply → shift LEFT (number gets bigger). Divide → shift RIGHT (number gets smaller). Count the zeros in 10, 100, 1000 — that's how many places to shift!
Quick ReferenceMultiply by powers of 10
× 10 → shift 1 place LEFT (÷ 0.1)
× 100 → shift 2 places LEFT (÷ 0.01)
× 1000 → shift 3 places LEFT (÷ 0.001)
Examples:
0.056 × 10 = 0.56
0.056 × 100 = 5.6
0.056 × 1000 = 56
Quick ReferenceDivide by powers of 10
÷ 10 → shift 1 place RIGHT (× 0.1)
÷ 100 → shift 2 places RIGHT(× 0.01)
÷ 1000 → shift 3 places RIGHT(× 0.001)
Examples:
4500 ÷ 10 = 450
4500 ÷ 100 = 45
4500 ÷ 1000 = 4.5

🎯 Significant Figures Tool — highlights which digits count

👆 Have a go — type any number into the box and use the drop-down to change how many significant figures you want. Great for checking your rounding!
🎯

Significant Figures

Significant figures (s.f.) express how precise a measurement is — more meaningful than decimal places for very large or very small numbers.

📋 Significant Figures — Quick Reference

1
Count from the first non-zero digit
2
Zeros between other digits do count
🧠

Key Words — What Do They Mean?

s.f.Significant
figures
Non-zeroAny digit
1-9
RulesStart counting sig figs from the FIRST non-zero digit. Then round normally.
Which digits are significant?The rules with examples
✅ Non-zero digits are always significant
4.37 → 3 s.f.   125 → 3 s.f.
✅ Zeros BETWEEN non-zero digits are significant
4.07 → 3 s.f.   1005 → 4 s.f.
✅ Trailing zeros AFTER decimal point are significant
3.40 → 3 s.f.   12.00 → 4 s.f.
⚠️ Leading zeros are NOT significant
0.0047 → 2 s.f. (the 4 and 7 are sig figs)
⚠️ Trailing zeros in whole numbers — ambiguous without context
3400 → could be 2, 3 or 4 s.f.
📌 In engineering, always state the number of s.f. used, or write in standard form to avoid ambiguity.
Round 0.006473 to 3 s.f.
1st s.f.: 6 (first non-zero digit)
2nd s.f.: 4
3rd s.f.: 7 — look at next digit: 3 < 5 → round down
Answer: 0.00647 ✅
⚙️ Round 34,872 N to 3 s.f.
1st s.f.: 3
2nd s.f.: 4
3rd s.f.: 8 — look at next: 7 ≥ 5 → round up
Answer: 34,900 N ✅

🔬 Standard Form Converter

👆 Type any large or small number above and watch it convert to standard form instantly.
 × 10
👆 Enter a value in the a × 10ⁿ boxes and see it written as an ordinary number.
🔬

Standard Form (Scientific Notation)

A way to write very large or very small numbers neatly. Used constantly in engineering for forces, voltages, wavelengths and material properties.

📋 Standard Form — Quick Reference

1
Written as a × 10ⁿ, where 1 ≤ a < 10
2
Big numbers → positive n. Small numbers → negative n
🧠

Key Words — What Do They Mean?

Standard Forma × 10ⁿ
notation
CoefficientThe 'a' part,
1 to 9.99
FormatA × 10ⁿ where 1 ≤ A < 10 and n is a whole number (positive or negative)
AnatomyBreaking down 3.47 × 10⁴
3.47 × 10 4 = 34,700 coefficient (must be 1 ≤ A < 10) power of 10 (+ve → big number) ordinary number (shift point 4 right)
→ Standard FormConverting to standard form
  1. Move the decimal point until you have a number between 1 and 10
  2. Count how many places you moved — this is the power of 10
  3. If you moved LEFT → power is positive. RIGHT → power is negative

Type any ordinary number:

← Ordinary NumberConverting back from standard form
  1. Positive power → move decimal point RIGHT (number gets bigger)
  2. Negative power → move decimal point LEFT (number gets smaller)
  3. Fill in zeros as needed

Type coefficient and power:

× 10 (power)
⚙️ Engineering — Standard Form in Practice

Try any engineering value — or use the preset examples:

⚙️

Applications

Where this topic is used in engineering, manufacturing, maintenance and daily life.

🧠

Where You'll See Decimals

MeasuringMicrometers,
gauges
MoneyPrices, invoices,
wages
SizingShaft diameters,
tolerances
⚙️

Decimals in Engineering

Every single measurement in engineering is a decimal. Here's where each skill actually appears on the job.

📋 Engineering Decimals — Quick Reference

1
Measurements always carry units (mm, V, A, N)
2
Check your answer's precision matches the tool used
🧠

Key Words — What Do They Mean?

ToleranceAllowed range
above/below
ReadingA measured
value

🔩 Thread Pitch & Tolerances

M8 bolt: thread pitch = 1.25 mm. Shaft diameter tolerance: 25.000 ± 0.015 mm. Precision to 3 decimal places is essential.

Shaft acceptable range:
25.000 − 0.015 = 24.985 mm (min)
25.000 + 0.015 = 25.015 mm (max)
Total tolerance = 0.030 mm = 30 μm

⚡ Electrical Values

Resistor: 4.7 kΩ = 4,700 Ω = 4.7 × 10³ Ω. Capacitor: 47 μF = 0.000047 F = 4.7 × 10⁻⁵ F.

Ohm's law: V = I × R
V = 0.025 × 4700
= 117.5 V
(0.025 A = 25 mA)

🌡️ Material Properties

Density of aluminium = 2.71 g/cm³. Tensile strength of mild steel = 4.0 × 10⁸ Pa. Standard form keeps these manageable.

Mass = density × volume
= 2.71 × 45.5 cm³
= 123.305 g
≈ 123.3 g (4 s.f.)

📐 CNC & Machining

CNC programs use coordinates to 3 decimal places in mm. A move from X=12.475 to X=15.000 is 2.525 mm.

Feed rate: 0.12 mm/rev
Spindle speed: 1200 rpm
Material removal rate:
= 0.12 × 1200 = 144 mm/min
🔧 Engineering Example — Gear Module Calculation
Problem: A spur gear has 45 teeth and a pitch circle diameter of 112.5 mm. Calculate the module (m = PCD ÷ number of teeth) to 3 significant figures.
Formula: m = PCD ÷ teeth = 112.5 ÷ 45
Calculate: 112.5 ÷ 45 = 2.5
Answer: m = 2.50 (3 s.f.) ✅
🎮

Quick Fire Quiz

Test your knowledge — 10 questions, instant feedback.

⚡ Decimals Blitz

Score: 0 Streak: 0 🔥 Q 1/10
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📝

Practice Questions

15 questions in 3 tiers — Level 2 Engineering style with full working.

🟢 Tier 1 — Foundations
Q1
Write the value of the digit 7 in the number 3.47082.
[1]
Hint: which decimal place is the 7 in?
7 is in the hundredths column
Value = 7 hundredths = 0.07
Q2
Round 8.4762 to (a) 1 d.p.   (b) 2 d.p.   (c) 3 d.p.
[3]
(a) look at 2nd d.p. → 4 < 5 → round down: 8.5
(b) look at 3rd d.p. → 6 ≥ 5 → round up: 8.48
(c) look at 4th d.p. → 2 < 5 → round down: 8.476
Q3
Calculate 12.6 + 3.045
[2]
12.600 + 3.045 (align points, add trailing zero)
= 15.645
Q4
Calculate 0.38 × 100 and 4700 ÷ 1000.
[2]
0.38 × 100 = 38 (shift 2 left)
4700 ÷ 1000 = 4.7 (shift 3 right)
Q5
Write 45,000 in standard form.
[2]
Move decimal 4 places left: 4.5 × 10⁴
= 4.5 × 10⁴
🟡 Tier 2 — Core Skills
Q6
Calculate 2.35 × 1.4. Show working.
[3]
Total d.p. = 1+1 = 2
235 × 14 = 3290
Place 2 d.p.: 3.29
Q7
Calculate 9.6 ÷ 0.032
[3]
×1000 both: 9600 ÷ 32
9600 ÷ 32 = 300
Q8
Round 0.007652 to (a) 2 s.f.   (b) 3 s.f.
[3]
1st s.f. = 7, 2nd s.f. = 6, 3rd s.f. = 5
(a) look at 3rd s.f.=5 ≥ 5 → round up: 0.0077
(b) look at 4th s.f.=2 < 5 → round down: 0.00765
Q9
Write 6.3 × 10⁻⁴ as an ordinary number.
[2]
Negative power → move decimal LEFT 4 places
= 0.00063
Q10
A micrometer reads 12.375 mm. Round this to (a) the nearest 0.1 mm   (b) the nearest 0.01 mm.
[2]
(a) nearest 0.1mm = 1 d.p.: look at 2nd d.p.=7 ≥ 5 → 12.4 mm
(b) nearest 0.01mm = 2 d.p.: look at 3rd d.p.=5 ≥ 5 → 12.38 mm
🔴 Tier 3 — Extension Challenge
Q11
A shaft diameter is specified as 30.000 mm ± 0.025 mm. (a) State the maximum and minimum acceptable diameters. (b) A measured shaft is 30.027 mm — is it acceptable?
[4]
(a) Max = 30.000 + 0.025 = 30.025 mm
Min = 30.000 − 0.025 = 29.975 mm
(b) 30.027 > 30.025 → NOT acceptable (oversize) ✅
Q12
A capacitor is labelled 470 μF. Write this value in farads in standard form.
[3]
470 μF = 470 × 10⁻⁶ F
= 4.70 × 10² × 10⁻⁶
= 4.70 × 10⁻⁴ F ✅
Q13
The Young's modulus of copper is 1.17 × 10¹¹ Pa. Write this as an ordinary number and explain what it represents.
[3]
1.17 × 10¹¹ = move point 11 right
= 117,000,000,000 Pa = 117 GPa
It represents the stiffness of copper — how much stress is needed per unit strain.
Q14
A CNC machine moves from coordinate X=14.375 mm to X=27.050 mm. Calculate the distance moved. Give your answer to 3 d.p.
[3]
Distance = 27.050 − 14.375
= 12.675 mm ✅ (already 3 d.p.)
Q15
Calculate the resistance of a wire using R = ρL/A where ρ = 1.72 × 10⁻⁸ Ωm, L = 2.5 m, A = 3.14 × 10⁻⁶ m². Give your answer in standard form to 3 s.f.
[5]
R = (1.72 × 10⁻⁸ × 2.5) / (3.14 × 10⁻⁶)
Numerator: 1.72 × 2.5 = 4.30 → 4.30 × 10⁻⁸
R = (4.30 × 10⁻⁸) / (3.14 × 10⁻⁶)
= (4.30/3.14) × 10⁻⁸⁺⁶
= 1.369... × 10⁻²
= 1.37 × 10⁻² Ω (3 s.f.) ✅
🎯 Score guide: Q1–5 = foundations (10 marks) · Q6–10 = core skills (13 marks) · Q11–15 = extension challenge (18 marks)
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