Converting units, significant figures and standard form

What you'll learn

How to convert between common GCSE Physics units, including hours to seconds.
How to handle compound units such as metres per second (m/s).
How to round answers using significant figures.
How to write very large and very small numbers in standard form.

Why these skills matter

Physics calculations are not just about getting a number. The unit tells you what the number means, and the way you round or write the number tells the examiner how precise your answer is.

A correct method can lose marks if the units are wrong, or if a value is rounded badly too early.

Definition

Physical quantity and unit

A physical quantity is something measurable, such as distance, time, mass or current. A unit is the agreed scale used to measure it, such as metres, seconds, kilograms or amperes.

SI units: the “default language” of physics

SI units are the international standard units used in science. GCSE equations usually work best when values are in SI units.

Common SI units include:

distance in metres (m)
mass in kilograms (kg)
time in seconds (s)
current in amperes (A)
temperature in kelvin (K) or degrees Celsius (°C), depending on the topic

Key Idea

Convert before you calculate

If a question gives values in kilometres, minutes, hours, grams or centimetres, check whether you need to convert them into SI units before using an equation.

Metric prefixes

A prefix is a short part added to the start of a unit symbol to show a multiplier.

The most useful ones for this topic are:

kilo (k) means 1000 times the base unit, so $\text{ km} = 1000 \text{ m}$
centi (c) means one hundredth of the base unit, so $\text{ m} = 100 \text{ cm}$
milli (m) means one thousandth of the base unit, so $\text{ m} = 1000 \text{ mm}$

Mass is slightly unusual: the SI unit is the kilogram (kg), but questions often give masses in grams (g). Use $\text{ kg} = 1000 \text{ g}$ .

This conversion sheet summarises the common relationships you should know.

Unit conversion cheat sheet showing common length, mass and time conversions

Bigger unit to smaller unit

When changing to a smaller unit, the number gets bigger.

For example, kilometres are bigger than metres, so a distance in metres has a larger number.

Smaller unit to bigger unit

When changing to a bigger unit, the number gets smaller.

For example, grams are smaller than kilograms, so a mass in kilograms has a smaller number.

Example

Converting common metric units

Convert 3.2 km to metres and 450 g to kilograms.

Kilometres are larger than metres, so changing from kilometres to metres means multiplying by 1000: $\text{ km} = 3.2 \times 1000 \text{ m}$ .
Calculate the length conversion: $3.2 \times 1000 = 3200$ , so the distance is 3200 m.
Grams are smaller than kilograms, so changing from grams to kilograms means dividing by 1000: $\text{ g} = 450 \div 1000 \text{ kg} = 0.450 \text{ kg}$ .

Converting time units

Time conversions come up a lot in Physics, especially in speed, power, energy and electricity calculations.

You need these:

$\text{ min} = 60 \text{ s}$
$\text{ h} = 60 \text{ min}$
$\text{ h} = 3600 \text{ s}$

That last one matters: one hour is not 100 seconds or 1000 seconds. It is 60 minutes, and each minute is 60 seconds.

Example

Converting hours and minutes to seconds

Convert 2 h 15 min into seconds.

Convert the hours into seconds using $\text{ h} = 3600 \text{ s}$ , so $\text{ h} = 2 \times 3600 \text{ s} = 7200 \text{ s}$ .
Convert the minutes into seconds using $\text{ min} = 60 \text{ s}$ , so $\text{ min} = 15 \times 60 \text{ s} = 900 \text{ s}$ .
Add the two parts: $\text{ s} + 900 \text{ s} = 8100 \text{ s}$ .

Common Mistake

Decimal hours are not minutes

1.5 h means 1 and a half hours, which is 1 h 30 min. It does not mean 1 h 50 min.

Converting rates and compound units

A rate tells you how much of one quantity happens for each unit of another quantity. Speed is a rate because it tells you distance travelled per unit time.

For example, metres per second (m/s) means metres travelled in each second.

To convert a compound unit like kilometres per hour into metres per second, convert both parts:

kilometres to metres
hours to seconds

Example

Converting a speed

Convert 36 km/h into m/s.

Convert the distance part: $\text{ km} = 36000 \text{ m}$ .
Convert the time part: $\text{ h} = 3600 \text{ s}$ .
Rewrite the rate and divide: $\text{ km/h} = \frac{36000 \text{ m}}{3600 \text{ s}} = 10 \text{ m/s}$ .

Tip

Quick speed conversion

To convert from km/h to m/s, divide by 3.6. To convert from m/s to km/h, multiply by 3.6. This works because $\text{ m/s} = 3.6 \text{ km/h}$ .

Significant figures

Definition

Significant figures

Significant figures (s.f.) are the digits in a value that show its size and precision. Precision means how finely a measurement is given.

The first significant figure is the first non-zero digit.

Rules to remember:

Non-zero digits are significant.
Zeros between non-zero digits are significant.
Zeros at the start are not significant.
Zeros at the end after a decimal point are significant.

For example, 3.0 has two significant figures. The zero matters because it shows the value was measured or rounded to the nearest tenth.

Example

Rounding to significant figures

Round 0.004872 to 2 significant figures, and 15,960 to 3 significant figures.

For 0.004872, ignore the zeros at the start. The first two significant digits are 4 and 8.
The next digit is 7, so round 48 up to 49. The answer is 0.0049.
For 15,960, the first three significant digits are 1, 5 and 9. The next digit is 6, so the value rounds to 16,000.
To show clearly that 16,000 is to 3 significant figures, write it as $1.60 \times 10^4$ .

Common Mistake

Rounding too early

Do not round every line of working. Keep extra digits in your calculator, then round the final answer to the number of significant figures requested.

Standard form

Standard form is used for very large or very small numbers. It is common in topics like space, atoms, waves and electricity.

Definition

Standard form

A number in standard form is written as $\times 10^n$ , where $\le A < 10$ and $n$ is an integer, meaning a whole number that can be negative, zero or positive.

The number before the power of ten must be at least 1 but less than 10.

Examples:

$\times 10^4$
$\times 10^{-4}$
$\times 10^6$ if written to 2 significant figures

A positive power means the original number is large. A negative power means the original number is small, between 0 and 1.

Example

Writing values in standard form

Write 48,000 and 0.000305 in standard form.

For 48,000, move the decimal point 4 places left to make 4.8.
Because the original number was large, use a positive power: $\times 10^4$ .
For 0.000305, move the decimal point 4 places right to make 3.05.
Because the original number was small, use a negative power: $\times 10^{-4}$ .

Common Mistake

Negative powers are not negative numbers

$4.0 \times 10^{-3}$ is still positive. It means 4.0 divided by 1000, which is 0.0040.

Tip

Calculator shortcut

Use the EXP, EE or ×10^x key for standard form. A calculator display such as 2.5E-3 means $2.5 \times 10^{-3}$ .

Putting it together in a physics calculation

A typical GCSE calculation may require all three skills: convert the units, calculate, then round appropriately.

Example

Calculating speed with converted units

A cyclist travels 2.4 km in 3.0 min. Calculate the cyclist’s speed in m/s, giving your answer to 2 significant figures.

Convert the distance to metres: $\text{ km} = 2.4 \times 1000 \text{ m} = 2400 \text{ m}$ .
Convert the time to seconds: $\text{ min} = 3.0 \times 60 \text{ s} = 180 \text{ s}$ .
Use speed equals distance divided by time: $\frac{d}{t} = \frac{2400 \text{ m}}{180 \text{ s}} = 13.3 \text{ m/s}$ .
Round to 2 significant figures because the data were given to 2 significant figures: 13 m/s.

Exam technique

In the exam

Check the units before substituting into an equation, especially minutes, hours, centimetres, kilometres and grams.
Keep full calculator values during working, then round only the final answer unless the question tells you otherwise.
Use standard form for very large or very small answers, and make sure the first number is at least 1 but less than 10.

Self review

Check yourself

Convert 0.45 h into seconds.
Round 0.006738 to 2 significant figures.
Write 92,000,000 in standard form.

Converting units, significant figures and standard form

What you'll learn

How to convert between common GCSE Physics units, including hours to seconds.
How to handle compound units such as metres per second (m/s).
How to round answers using significant figures.
How to write very large and very small numbers in standard form.

Why these skills matter

Physics calculations are not just about getting a number. The unit tells you what the number means, and the way you round or write the number tells the examiner how precise your answer is.

A correct method can lose marks if the units are wrong, or if a value is rounded badly too early.

Definition

Physical quantity and unit

A physical quantity is something measurable, such as distance, time, mass or current. A unit is the agreed scale used to measure it, such as metres, seconds, kilograms or amperes.

SI units: the “default language” of physics

SI units are the international standard units used in science. GCSE equations usually work best when values are in SI units.

Common SI units include:

distance in metres (m)
mass in kilograms (kg)
time in seconds (s)
current in amperes (A)
temperature in kelvin (K) or degrees Celsius (°C), depending on the topic

Key Idea

Convert before you calculate

If a question gives values in kilometres, minutes, hours, grams or centimetres, check whether you need to convert them into SI units before using an equation.

Metric prefixes

A prefix is a short part added to the start of a unit symbol to show a multiplier.

The most useful ones for this topic are:

kilo (k) means 1000 times the base unit, so $\text{ km} = 1000 \text{ m}$
centi (c) means one hundredth of the base unit, so $\text{ m} = 100 \text{ cm}$
milli (m) means one thousandth of the base unit, so $\text{ m} = 1000 \text{ mm}$

Mass is slightly unusual: the SI unit is the kilogram (kg), but questions often give masses in grams (g). Use $\text{ kg} = 1000 \text{ g}$ .

This conversion sheet summarises the common relationships you should know.

Unit conversion cheat sheet showing common length, mass and time conversions

Bigger unit to smaller unit

When changing to a smaller unit, the number gets bigger.

For example, kilometres are bigger than metres, so a distance in metres has a larger number.

Smaller unit to bigger unit

When changing to a bigger unit, the number gets smaller.

For example, grams are smaller than kilograms, so a mass in kilograms has a smaller number.

Example

Converting common metric units

Convert 3.2 km to metres and 450 g to kilograms.

Kilometres are larger than metres, so changing from kilometres to metres means multiplying by 1000: $\text{ km} = 3.2 \times 1000 \text{ m}$ .
Calculate the length conversion: $3.2 \times 1000 = 3200$ , so the distance is 3200 m.
Grams are smaller than kilograms, so changing from grams to kilograms means dividing by 1000: $\text{ g} = 450 \div 1000 \text{ kg} = 0.450 \text{ kg}$ .

Converting time units

Time conversions come up a lot in Physics, especially in speed, power, energy and electricity calculations.

You need these:

$\text{ min} = 60 \text{ s}$
$\text{ h} = 60 \text{ min}$
$\text{ h} = 3600 \text{ s}$

That last one matters: one hour is not 100 seconds or 1000 seconds. It is 60 minutes, and each minute is 60 seconds.

Example

Converting hours and minutes to seconds

Convert 2 h 15 min into seconds.

Convert the hours into seconds using $\text{ h} = 3600 \text{ s}$ , so $\text{ h} = 2 \times 3600 \text{ s} = 7200 \text{ s}$ .
Convert the minutes into seconds using $\text{ min} = 60 \text{ s}$ , so $\text{ min} = 15 \times 60 \text{ s} = 900 \text{ s}$ .
Add the two parts: $\text{ s} + 900 \text{ s} = 8100 \text{ s}$ .

Common Mistake

Decimal hours are not minutes

1.5 h means 1 and a half hours, which is 1 h 30 min. It does not mean 1 h 50 min.

Converting rates and compound units

A rate tells you how much of one quantity happens for each unit of another quantity. Speed is a rate because it tells you distance travelled per unit time.

For example, metres per second (m/s) means metres travelled in each second.

To convert a compound unit like kilometres per hour into metres per second, convert both parts:

kilometres to metres
hours to seconds

Example

Converting a speed

Convert 36 km/h into m/s.

Convert the distance part: $\text{ km} = 36000 \text{ m}$ .
Convert the time part: $\text{ h} = 3600 \text{ s}$ .
Rewrite the rate and divide: $\text{ km/h} = \frac{36000 \text{ m}}{3600 \text{ s}} = 10 \text{ m/s}$ .

Tip

Quick speed conversion

To convert from km/h to m/s, divide by 3.6. To convert from m/s to km/h, multiply by 3.6. This works because $\text{ m/s} = 3.6 \text{ km/h}$ .

Significant figures

Definition

Significant figures

Significant figures (s.f.) are the digits in a value that show its size and precision. Precision means how finely a measurement is given.

The first significant figure is the first non-zero digit.

Rules to remember:

Non-zero digits are significant.
Zeros between non-zero digits are significant.
Zeros at the start are not significant.
Zeros at the end after a decimal point are significant.

For example, 3.0 has two significant figures. The zero matters because it shows the value was measured or rounded to the nearest tenth.

Example

Rounding to significant figures

Round 0.004872 to 2 significant figures, and 15,960 to 3 significant figures.

For 0.004872, ignore the zeros at the start. The first two significant digits are 4 and 8.
The next digit is 7, so round 48 up to 49. The answer is 0.0049.
For 15,960, the first three significant digits are 1, 5 and 9. The next digit is 6, so the value rounds to 16,000.
To show clearly that 16,000 is to 3 significant figures, write it as $1.60 \times 10^4$ .

Common Mistake

Rounding too early

Do not round every line of working. Keep extra digits in your calculator, then round the final answer to the number of significant figures requested.

Standard form

Standard form is used for very large or very small numbers. It is common in topics like space, atoms, waves and electricity.

Definition

Standard form

A number in standard form is written as $\times 10^n$ , where $\le A < 10$ and $n$ is an integer, meaning a whole number that can be negative, zero or positive.

The number before the power of ten must be at least 1 but less than 10.

Examples:

$\times 10^4$
$\times 10^{-4}$
$\times 10^6$ if written to 2 significant figures

A positive power means the original number is large. A negative power means the original number is small, between 0 and 1.

Example

Writing values in standard form

Write 48,000 and 0.000305 in standard form.

For 48,000, move the decimal point 4 places left to make 4.8.
Because the original number was large, use a positive power: $\times 10^4$ .
For 0.000305, move the decimal point 4 places right to make 3.05.
Because the original number was small, use a negative power: $\times 10^{-4}$ .

Common Mistake

Negative powers are not negative numbers

$4.0 \times 10^{-3}$ is still positive. It means 4.0 divided by 1000, which is 0.0040.

Tip

Calculator shortcut

Use the EXP, EE or ×10^x key for standard form. A calculator display such as 2.5E-3 means $2.5 \times 10^{-3}$ .

Putting it together in a physics calculation

A typical GCSE calculation may require all three skills: convert the units, calculate, then round appropriately.

Example

Calculating speed with converted units

A cyclist travels 2.4 km in 3.0 min. Calculate the cyclist’s speed in m/s, giving your answer to 2 significant figures.

Convert the distance to metres: $\text{ km} = 2.4 \times 1000 \text{ m} = 2400 \text{ m}$ .
Convert the time to seconds: $\text{ min} = 3.0 \times 60 \text{ s} = 180 \text{ s}$ .
Use speed equals distance divided by time: $\frac{d}{t} = \frac{2400 \text{ m}}{180 \text{ s}} = 13.3 \text{ m/s}$ .
Round to 2 significant figures because the data were given to 2 significant figures: 13 m/s.

Exam technique

In the exam

Check the units before substituting into an equation, especially minutes, hours, centimetres, kilometres and grams.
Keep full calculator values during working, then round only the final answer unless the question tells you otherwise.
Use standard form for very large or very small answers, and make sure the first number is at least 1 but less than 10.

Self review

Check yourself

Convert 0.45 h into seconds.
Round 0.006738 to 2 significant figures.
Write 92,000,000 in standard form.

Converting units, significant figures and standard form

What you'll learn

Why these skills matter

SI units: the “default language” of physics

Metric prefixes

Bigger unit to smaller unit

Smaller unit to bigger unit

Converting time units

Converting rates and compound units

Significant figures

Standard form

Putting it together in a physics calculation

Recap questions

You've reached the end

Converting units, significant figures and standard form

What you'll learn

Why these skills matter

SI units: the “default language” of physics

Metric prefixes

Bigger unit to smaller unit

Smaller unit to bigger unit

Converting time units

Converting rates and compound units

Significant figures

Standard form

Putting it together in a physics calculation

Recap questions

You've reached the end

Converting units, significant figures and standard form

What you'll learn

Why these skills matter

SI units: the “default language” of physics

Metric prefixes

Bigger unit to smaller unit

Smaller unit to bigger unit

Converting time units

Converting rates and compound units

Significant figures

Standard form

Putting it together in a physics calculation

Recap questions

You've reached the end

Converting units, significant figures and standard form Revision Guide

Converting units, significant figures and standard form

What you'll learn

Why these skills matter

SI units: the “default language” of physics

Metric prefixes

Bigger unit to smaller unit

Smaller unit to bigger unit

Converting time units

Converting rates and compound units

Significant figures

Standard form

Putting it together in a physics calculation

Recap questions

You've reached the end

Converting units, significant figures and standard form Revision Guide