🔢 Decimals — T Level / Level 3 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.

T Level Core 4.1 Place value Rounding & significant figures Standard form Engineering precision

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

Precision — a shaft of 24.75 mm is very different from 24.8 mm in precision engineering
Measurements — voltmeters, micrometers and pressure gauges all display decimals
Calculations — every formula in T Level engineering produces decimal answers
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 TableThe number 4 7 3 . 5 8 6 shown in every column

💡 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

Enter a 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.).

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?

✅ 3.46 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

Number

Round to

👆 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

Identify which decimal place you're rounding to
Look at the digit immediately after that place
If it's 0, 1, 2, 3 or 4 → leave the last digit as it is (round down)
If it's 5, 6, 7, 8 or 9 → add 1 to the last digit (round up)
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

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

First number

Operation

Second number

👆 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.

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

Count total decimal places in both numbers
Multiply as if they were whole numbers (ignore the point)
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

If the divisor has decimals, multiply both numbers by 10/100/1000 to make it a whole number
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?

Total distance (mm)

Feed per rev (mm)

🔟 Digit Shift — watch the digits move left or right

Number

Operation

↓ 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.

Digit Shift DiagramWatch the digits move — 3.47 × 10, × 100, ÷ 10

🔑 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

Number

Round to

👆 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.

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

Ordinary number → Standard Form

👆 Type any large or small number above and watch it convert to standard form instantly.

Standard Form → Ordinary number

A

× 10

n

👆 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.

FormatA × 10ⁿ where 1 ≤ A < 10 and n is a whole number (positive or negative)

AnatomyBreaking down 3.47 × 10⁴

→ Standard FormConverting to standard form

Move the decimal point until you have a number between 1 and 10
Count how many places you moved — this is the power of 10
If you moved LEFT → power is positive. RIGHT → power is negative

Type any ordinary number:

← Ordinary NumberConverting back from standard form

Positive power → move decimal point RIGHT (number gets bigger)
Negative power → move decimal point LEFT (number gets smaller)
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.

🔩Manufacturing: Tolerances on drawings: 25.00±0.05 mm. CNC to 0.001 mm.
⚡Electrical: Resistor values: 4.7 kΩ. Component ratings to 2 d.p.
🌡️Daily Life: Body temp 36.8°C, fuel economy 42.3 mpg, recipes.
🏗️Construction: Concrete quantities and mix ratios to 2 decimal places.

⚙️

Decimals in Engineering

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

🔩 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

🔧 T Level Exam Style — 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 — T Level / Level 3 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 — T Level 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 = T Level challenge (18 marks)