1.1.2 Unit prefixes, conversions, significant figures and standard form

Unit prefixes

Definition

Unit prefix

A unit prefix is a symbol placed before a unit to show that the unit has been multiplied by a power of ten, such as kilo for one thousand or milli for one thousandth.

Giga, symbol $G\text{G}$ , means $×109\times 10^{9}$ , so $W1\ \text{GW}=1\times10^{9}\ \text{W}$ .
Mega, symbol $M\text{M}$ , means $×106\times 10^{6}$ , so $J1\ \text{MJ}=1\times10^{6}\ \text{J}$ .
Kilo, symbol $k\text{k}$ , means $×103\times 10^{3}$ , so $m1\ \text{km}=1\times10^{3}\ \text{m}$ .
Centi, symbol $c\text{c}$ , means $×10−2\times 10^{-2}$ , so $m1\ \text{cm}=1\times10^{-2}\ \text{m}$ .
Milli, symbol $m\text{m}$ , means $×10−3\times 10^{-3}$ , so $A1\ \text{mA}=1\times10^{-3}\ \text{A}$ .
Micro, symbol $μ\mu$ , means $×10−6\times 10^{-6}$ , so $C1\ \mu\text{C}=1\times10^{-6}\ \text{C}$ .
Nano, symbol $n\text{n}$ , means $×10−9\times 10^{-9}$ , so $m1\ \text{nm}=1\times10^{-9}\ \text{m}$ .
To remove a prefix, multiply the numerical value by the prefix factor, and to introduce a prefix, divide by that factor.

Example

A current of $mA850\ \text{mA}$ is converted using milli $10^{-3}$ .
$A850\ \text{mA}=850\times10^{-3}\ \text{A}=0.850\ \text{A}$ .
A power of $W2.4\times10^{6}\ \text{W}$ is converted using mega $10^{6}$ .
$MW2.4\times10^{6}\ \text{W}=2.4\ \text{MW}$ .

Converting units

Definition

Unit conversion

A unit conversion changes the unit used to express a quantity without changing the physical quantity itself.

Convert values to the units required by the equation before substitution, because a correct equation with inconsistent units gives an incorrect number.
$s1\ \text{minute}=60\ \text{s}$ and $s1\ \text{hour}=3600\ \text{s}$ .
Convert hours to seconds by multiplying by $3600$ , and convert seconds to hours by dividing by $3600$ .
$mm1\ \text{m}=100\ \text{cm}=1000\ \text{mm}$ .
$g1\ \text{kg}=1000\ \text{g}$ , so grams are converted to kilograms by dividing by $1000$ .
Converting to a smaller unit gives a larger numerical value, while converting to a larger unit gives a smaller numerical value.

Converting compound units

A compound unit contains more than one unit, so every part of the unit must be converted.
For speed, $m/s1\ \text{km/h}=\dfrac{1000\ \text{m}}{3600\ \text{s}}=\dfrac{1}{3.6}\ \text{m/s}$ .
Convert kilometres per hour to metres per second by dividing by $3.6$ , and convert metres per second to kilometres per hour by multiplying by $3.6$ .
For squared and cubed units, the conversion factor must also be squared or cubed, so $m21\ \text{cm}^{2}=(10^{-2}\ \text{m})^{2}=10^{-4}\ \text{m}^{2}$ .
Likewise, $m31\ \text{cm}^{3}=(10^{-2}\ \text{m})^{3}=10^{-6}\ \text{m}^{3}$ .

Example

A journey time of $h1.75\ \text{h}$ is converted by multiplying by $3600$ .
$st=1.75\times3600=6300\ \text{s}$ .
A speed of $km/h126\ \text{km/h}$ is converted by dividing by $3.6$ .
$m/sv=126\div3.6=35.0\ \text{m/s}$ .

Standard form

Definition

Standard form

Standard form writes a number as A multiplied by ten to the power n, where A is between 1 and 10 and n is an integer.

Move the decimal point until exactly one non-zero digit remains before it, then use the number of places moved as the magnitude of the power.
A large number has a positive power, such as $000=6.5×1066\,500\,000=6.5\times10^{6}$ .
A number between zero and one has a negative power, such as $45=4.5×10−70.000\,000\,45=4.5\times10^{-7}$ .
When multiplying standard-form values, multiply the leading numbers and add the powers, then rewrite the result so the leading number lies between $1$ and $10$ .
When dividing standard-form values, divide the leading numbers and subtract the powers, then normalise the result.

Example

$(3.0×108)(2.0×10−6)=(3.0×2.0)×108−6(3.0\times10^{8})(2.0\times10^{-6})=(3.0\times2.0)\times10^{8-6}$ .
The result is $6.0×1026.0\times10^{2}$ .
$8.4×1072.0×103=8.42.0×107−3\dfrac{8.4\times10^{7}}{2.0\times10^{3}}=\dfrac{8.4}{2.0}\times10^{7-3}$ .
The result is $4.2×1044.2\times10^{4}$ .

Significant figures

Definition

Significant figures

Significant figures are the digits in a number that show its measured precision, starting from the first non-zero digit.

All non-zero digits are significant, and zeros between significant digits are also significant.
Zeros before the first non-zero digit are not significant, so $0.00456$ has three significant figures.
Trailing zeros after a decimal point are significant, so $2.50$ has three significant figures.
To round to a stated number of significant figures, keep the required digits and inspect the next digit.
If the next digit is $5$ or more, increase the last retained digit by one, and if it is below $5$ , leave the last retained digit unchanged.
Keep unrounded calculator values during working and round only the final answer.

Example

The value $0.007864$ rounded to three significant figures keeps $7$ , $8$ and $6$ .
The next digit is $4$ , so the rounded value is $0.00786$ .
The value $74658\,746$ rounded to three significant figures keeps $5$ , $8$ and $7$ .
The next digit is $4$ , so the rounded value is $70058\,700$ , or $5.87×1045.87\times10^{4}$ .

Example

A train moves at $km/h126\ \text{km/h}$ for $min1\ \text{h}\ 45\ \text{min}$ .
Convert the speed: $m/s126\div3.6=35.0\ \text{m/s}$ .
Convert the time: $s(1\times3600)+(45\times60)=6300\ \text{s}$ .
Use $d = v t$ to obtain $md=35.0\times6300=220\,500\ \text{m}$ .
Write the result in standard form and round to three significant figures: $md=2.21\times10^{5}\ \text{m}$ .

Common Mistake

Do not confuse lower-case $m\text{m}$ for milli with upper-case $M\text{M}$ for mega.
Do not treat significant figures as decimal places, because $0.00456$ to two significant figures is $0.0046$ .
Do not convert $cm2\text{cm}^{2}$ or $cm3\text{cm}^{3}$ using only the linear factor.
Do not round intermediate calculator values, because early rounding can change the final answer.

Exam technique

Write each conversion on a separate line before substitution so that the method remains clear if later arithmetic is incorrect.
Check the requested output unit before calculating, because the final conversion may carry a separate mark.
Give the answer to the stated number of significant figures, or use the precision of the supplied data when the question asks for an appropriate value.
Include the final unit and use the expected size of the answer as a conversion check.

Self review

What powers of ten correspond to giga, mega, kilo, centi, milli, micro and nano?
How are kilometres per hour converted to metres per second?
What conditions must $A$ and $n$ satisfy in standard form?
Which zeros count as significant figures?
Why should rounding be left until the final line?

1.1.2 Unit prefixes, conversions, significant figures and standard form

Unit prefixes

Definition

Unit prefix

A unit prefix is a symbol placed before a unit to show that the unit has been multiplied by a power of ten, such as kilo for one thousand or milli for one thousandth.

Giga, symbol $G\text{G}$ , means $×109\times 10^{9}$ , so $W1\ \text{GW}=1\times10^{9}\ \text{W}$ .
Mega, symbol $M\text{M}$ , means $×106\times 10^{6}$ , so $J1\ \text{MJ}=1\times10^{6}\ \text{J}$ .
Kilo, symbol $k\text{k}$ , means $×103\times 10^{3}$ , so $m1\ \text{km}=1\times10^{3}\ \text{m}$ .
Centi, symbol $c\text{c}$ , means $×10−2\times 10^{-2}$ , so $m1\ \text{cm}=1\times10^{-2}\ \text{m}$ .
Milli, symbol $m\text{m}$ , means $×10−3\times 10^{-3}$ , so $A1\ \text{mA}=1\times10^{-3}\ \text{A}$ .
Micro, symbol $μ\mu$ , means $×10−6\times 10^{-6}$ , so $C1\ \mu\text{C}=1\times10^{-6}\ \text{C}$ .
Nano, symbol $n\text{n}$ , means $×10−9\times 10^{-9}$ , so $m1\ \text{nm}=1\times10^{-9}\ \text{m}$ .
To remove a prefix, multiply the numerical value by the prefix factor, and to introduce a prefix, divide by that factor.

Example

A current of $mA850\ \text{mA}$ is converted using milli $10^{-3}$ .
$A850\ \text{mA}=850\times10^{-3}\ \text{A}=0.850\ \text{A}$ .
A power of $W2.4\times10^{6}\ \text{W}$ is converted using mega $10^{6}$ .
$MW2.4\times10^{6}\ \text{W}=2.4\ \text{MW}$ .

Converting units

Definition

Unit conversion

A unit conversion changes the unit used to express a quantity without changing the physical quantity itself.

Convert values to the units required by the equation before substitution, because a correct equation with inconsistent units gives an incorrect number.
$s1\ \text{minute}=60\ \text{s}$ and $s1\ \text{hour}=3600\ \text{s}$ .
Convert hours to seconds by multiplying by $3600$ , and convert seconds to hours by dividing by $3600$ .
$mm1\ \text{m}=100\ \text{cm}=1000\ \text{mm}$ .
$g1\ \text{kg}=1000\ \text{g}$ , so grams are converted to kilograms by dividing by $1000$ .
Converting to a smaller unit gives a larger numerical value, while converting to a larger unit gives a smaller numerical value.

Converting compound units

A compound unit contains more than one unit, so every part of the unit must be converted.
For speed, $m/s1\ \text{km/h}=\dfrac{1000\ \text{m}}{3600\ \text{s}}=\dfrac{1}{3.6}\ \text{m/s}$ .
Convert kilometres per hour to metres per second by dividing by $3.6$ , and convert metres per second to kilometres per hour by multiplying by $3.6$ .
For squared and cubed units, the conversion factor must also be squared or cubed, so $m21\ \text{cm}^{2}=(10^{-2}\ \text{m})^{2}=10^{-4}\ \text{m}^{2}$ .
Likewise, $m31\ \text{cm}^{3}=(10^{-2}\ \text{m})^{3}=10^{-6}\ \text{m}^{3}$ .

Example

A journey time of $h1.75\ \text{h}$ is converted by multiplying by $3600$ .
$st=1.75\times3600=6300\ \text{s}$ .
A speed of $km/h126\ \text{km/h}$ is converted by dividing by $3.6$ .
$m/sv=126\div3.6=35.0\ \text{m/s}$ .

Standard form

Definition

Standard form

Standard form writes a number as A multiplied by ten to the power n, where A is between 1 and 10 and n is an integer.

Move the decimal point until exactly one non-zero digit remains before it, then use the number of places moved as the magnitude of the power.
A large number has a positive power, such as $000=6.5×1066\,500\,000=6.5\times10^{6}$ .
A number between zero and one has a negative power, such as $45=4.5×10−70.000\,000\,45=4.5\times10^{-7}$ .
When multiplying standard-form values, multiply the leading numbers and add the powers, then rewrite the result so the leading number lies between $1$ and $10$ .
When dividing standard-form values, divide the leading numbers and subtract the powers, then normalise the result.

Example

$(3.0×108)(2.0×10−6)=(3.0×2.0)×108−6(3.0\times10^{8})(2.0\times10^{-6})=(3.0\times2.0)\times10^{8-6}$ .
The result is $6.0×1026.0\times10^{2}$ .
$8.4×1072.0×103=8.42.0×107−3\dfrac{8.4\times10^{7}}{2.0\times10^{3}}=\dfrac{8.4}{2.0}\times10^{7-3}$ .
The result is $4.2×1044.2\times10^{4}$ .

Significant figures

Definition

Significant figures

Significant figures are the digits in a number that show its measured precision, starting from the first non-zero digit.

All non-zero digits are significant, and zeros between significant digits are also significant.
Zeros before the first non-zero digit are not significant, so $0.00456$ has three significant figures.
Trailing zeros after a decimal point are significant, so $2.50$ has three significant figures.
To round to a stated number of significant figures, keep the required digits and inspect the next digit.
If the next digit is $5$ or more, increase the last retained digit by one, and if it is below $5$ , leave the last retained digit unchanged.
Keep unrounded calculator values during working and round only the final answer.

Example

The value $0.007864$ rounded to three significant figures keeps $7$ , $8$ and $6$ .
The next digit is $4$ , so the rounded value is $0.00786$ .
The value $74658\,746$ rounded to three significant figures keeps $5$ , $8$ and $7$ .
The next digit is $4$ , so the rounded value is $70058\,700$ , or $5.87×1045.87\times10^{4}$ .

Example

A train moves at $km/h126\ \text{km/h}$ for $min1\ \text{h}\ 45\ \text{min}$ .
Convert the speed: $m/s126\div3.6=35.0\ \text{m/s}$ .
Convert the time: $s(1\times3600)+(45\times60)=6300\ \text{s}$ .
Use $d = v t$ to obtain $md=35.0\times6300=220\,500\ \text{m}$ .
Write the result in standard form and round to three significant figures: $md=2.21\times10^{5}\ \text{m}$ .

Common Mistake

Do not confuse lower-case $m\text{m}$ for milli with upper-case $M\text{M}$ for mega.
Do not treat significant figures as decimal places, because $0.00456$ to two significant figures is $0.0046$ .
Do not convert $cm2\text{cm}^{2}$ or $cm3\text{cm}^{3}$ using only the linear factor.
Do not round intermediate calculator values, because early rounding can change the final answer.

Exam technique

Write each conversion on a separate line before substitution so that the method remains clear if later arithmetic is incorrect.
Check the requested output unit before calculating, because the final conversion may carry a separate mark.
Give the answer to the stated number of significant figures, or use the precision of the supplied data when the question asks for an appropriate value.
Include the final unit and use the expected size of the answer as a conversion check.

Self review

What powers of ten correspond to giga, mega, kilo, centi, milli, micro and nano?
How are kilometres per hour converted to metres per second?
What conditions must $A$ and $n$ satisfy in standard form?
Which zeros count as significant figures?
Why should rounding be left until the final line?

Converting units, significant figures and standard form

1.1.2 Unit prefixes, conversions, significant figures and standard form

Unit prefixes

Converting units

Converting compound units

Standard form

Significant figures

Recap questions

Review 1.1.2 Converting units, significant figures and standard form by teaching Genie

Recap your knowledge with an interactive lesson

Put it into practice with exam-style questions

Remember key concepts with flashcards

Converting units, significant figures and standard form

1.1.2 Unit prefixes, conversions, significant figures and standard form

Unit prefixes

Converting units

Converting compound units

Standard form

Significant figures

Recap questions

Review 1.1.2 Converting units, significant figures and standard form by teaching Genie

Recap your knowledge with an interactive lesson

Put it into practice with exam-style questions

Remember key concepts with flashcards

Converting units, significant figures and standard form

1.1.2 Unit prefixes, conversions, significant figures and standard form

Unit prefixes

Converting units

Converting compound units

Standard form

Significant figures

Recap questions

Review 1.1.2 Converting units, significant figures and standard form by teaching Genie

Recap your knowledge with an interactive lesson

Put it into practice with exam-style questions

Remember key concepts with flashcards

1.1.2 Converting units, significant figures and standard form Revision Guide

Converting units, significant figures and standard form

1.1.2 Unit prefixes, conversions, significant figures and standard form

Unit prefixes

Converting units

Converting compound units

Standard form

Significant figures

Recap questions

Review 1.1.2 Converting units, significant figures and standard form by teaching Genie

Recap your knowledge with an interactive lesson

Put it into practice with exam-style questions

Remember key concepts with flashcards

1.1.2 Converting units, significant figures and standard form Revision Guide