Maths · 1 Number

1.8 Standard form

Before you start

Place value with decimals: tenths, hundredths, thousandths, and how a zero holds an empty place.
Multiplying and dividing by 10, 100 and 1000 in your head.
Index notation and the index laws from Indices I: $10^{3} = 10 \times 10 \times 10$ , $10^{a} \times 10^{b} = 10^{a + b}$ and $10^{a} \div 10^{b} = 10^{a - b}$ .
Negative integers on a number line, so that $- 2$ is greater than $- 5$ , and subtracting a negative: $2 - (- 5) = 7$ .
Rounding a decimal to 3 significant figures, e.g. $2.53125 \to 2.53$ .

Syllabus objectives covered (3)

Use the standard form A × 10ⁿ where n is a positive or negative integer and 1 ⩽ A < 10.
Convert numbers into and out of standard form.
Calculate with values in standard form.

Part 1 of 8

Powers of ten, including negative powers

$10^{3}$ means $10 \times 10 \times 10 = 1000$ . For a positive power, the power tells you how many zeros follow the 1: $10^{6} = 1 000 000$ .
Now count down. Each time the power goes down by 1, you divide by 10. $10^{2} = 100$ , $10^{1} = 10$ , and one more step down gives $10^{0} = 1$ .
Keep going below zero, and keep dividing by 10. $10^{- 1} = \frac{1}{10} = 0.1$ , $10^{- 2} = \frac{1}{100} = 0.01$ , $10^{- 3} = \frac{1}{1000} = 0.001$ .
$10^{- n} = \frac{1}{10^{n}}$
So a negative power of ten is not a negative number. $10^{- 3}$ is $0.001$ , a small positive number. The minus sign in the power means 'divide by $10^{3}$ ', not 'below zero'.
Multiplying by $10^{4}$ makes a number $10 000$ times bigger: $3.2 \times 10^{4} = 32 000$ . Multiplying by $10^{- 4}$ is the same as dividing by $10 000$ : $3.2 \times 10^{- 4} = 0.000 32$ . Everything in standard form rests on these two facts.

Part 2 of 8

What counts as standard form

A number in standard form is written as $A \times 10^{n}$ , with two rules. $A$ is at least 1 and less than 10. $n$ is an integer: positive, negative or zero.
$A \times 10^{n}, 1 ⩽ A < 10, n an integer$
Why use it? $0.000 000 58$ is hard to read and easy to miscopy. $5.8 \times 10^{- 7}$ is the same number, and it shows the size at once: the power tells you how big, and $A$ gives the digits.
The rule $1 ⩽ A < 10$ means $A$ has exactly one non-zero digit before its decimal point. That makes the standard form of a number unique. $52 000$ is $5.2 \times 10^{4}$ , and only that. $52 \times 10^{3}$ and $0.52 \times 10^{5}$ have the same value, but they are not standard form.
Test both rules every time. $3.07 \times 10^{5}$ : $A = 3.07$ is between 1 and 10 and $5$ is an integer, so yes. $1 \times 10^{- 6}$ : yes, because $A = 1$ is allowed. $10 \times 10^{4}$ : no, because $A$ must be less than 10. $0.8 \times 10^{3}$ : no, because $A$ is less than 1.
$n = 0$ is allowed, because 0 is an integer. $4.5 \times 10^{0} = 4.5 \times 1 = 4.5$ . Every number from 1 up to (but not including) 10 has $n = 0$ .

Part 3 of 8

Large numbers into standard form and back

To write $52 000 000$ in standard form, first find $A$ . Put a decimal point after the first digit to get $5.2$ . The zeros at the end only held places, so you drop them.
Then find $n$ . Count how many places the decimal point moves. In $52 000 000$ the point is at the end. It moves 7 places to the left to sit after the 5. So $n = 7$ .
$52 000 000 = 5.2 \times 10^{7}$
Check it another way: for a number of 1 or more, $n$ is the number of digits before the decimal point, minus 1. $52 000 000$ has 8 digits, so $n = 7$ . Counting the zeros does not work. There are 6 zeros, but the 2 also moved past the point.
The number does not have to be whole. $386.4$ has 3 digits before the point, so $386.4 = 3.864 \times 10^{2}$ .
To go back to an ordinary number, multiply out. $3.07 \times 10^{5}$ means move the decimal point 5 places to the right and fill the empty places with zeros: $3.07 \to 307 000$ .
Sense-check the direction. A positive power gives a number of 10 or more. If your ordinary number comes out smaller than 10, you moved the point the wrong way.

Part 4 of 8

Small numbers into standard form and back

For a number less than 1, find $A$ in the same way: put the point after the first non-zero digit. In $0.000 63$ the first non-zero digit is 6, so $A = 6.3$ .
The point moves 4 places to the right, from after the leading 0 to after the 6. That made the number bigger, so the power must make it small again. The power is negative: $n = - 4$ .
$0.000 63 = 6.3 \times 10^{- 4}$
Quick check: the first non-zero digit is in the 4th decimal place, so $n = - 4$ .
To go back, a negative power means divide. $2.5 \times 10^{- 3}$ means divide by $1000$ : move the point 3 places to the left and fill the empty places with zeros. $2.5 \to 0.0025$ .
Sense-check the direction. A negative power gives a number between 0 and 1. If you get a number bigger than 1 from a negative power, you moved the point the wrong way.

Part 5 of 8

Correcting A, and comparing sizes

A calculation can give an answer such as $34 \times 10^{5}$ . The value is correct, but it is not standard form, because $34$ is not less than 10.
To correct it, write $A$ itself in standard form, then combine the powers of ten with the index law.
$34 \times 10^{5} = 3.4 \times 10^{1} \times 10^{5} = 3.4 \times 10^{6}$
The same works when $A$ is too small. $0.72 = 7.2 \times 10^{- 1}$ , so:
$0.72 \times 10^{- 3} = 7.2 \times 10^{- 1} \times 10^{- 3} = 7.2 \times 10^{- 4}$
The shortcut, and why it works: if you make $A$ ten times smaller, make the power one bigger, so the value does not change. If you make $A$ ten times bigger, make the power one smaller. With negative powers, one bigger means $- 3 \to - 2$ : $45 \times 10^{- 3} = 4.5 \times 10^{- 2}$ .
To compare two numbers in standard form, look at the powers first. The bigger power gives the bigger number, whatever $A$ is: $1.2 \times 10^{6}$ is bigger than $9.8 \times 10^{5}$ . Compare the values of $A$ only when the powers are equal.
With negative powers, $- 2$ is bigger than $- 5$ . So $3 \times 10^{- 2}$ ( $= 0.03$ ) is bigger than $8 \times 10^{- 5}$ ( $= 0.000 08$ ).
If a list mixes ordinary numbers and standard form, put them all into one form first. Then write your answer with the numbers as they were given in the question, not your converted versions.

Part 6 of 8

Multiplying and dividing in standard form

To multiply, use the fact that you can multiply in any order. Group the $A$ parts together and the powers of ten together.
$(3 \times 10^{4}) \times (2 \times 10^{5}) = (3 \times 2) \times (10^{4} \times 10^{5}) = 6 \times 10^{9}$
The powers add because $10^{4} \times 10^{5}$ is four tens times five tens, which is nine tens multiplied together: $10^{4 + 5} = 10^{9}$ .
Then check the answer is still standard form. $(4 \times 10^{3}) \times (5 \times 10^{6}) = 20 \times 10^{9}$ . Here $20$ is too big, so finish with $2 \times 10^{10}$ .
To divide, divide the $A$ parts and subtract the powers.
$(8 \times 10^{7}) \div (2 \times 10^{3}) = (8 \div 2) \times 10^{7 - 3} = 4 \times 10^{4}$
Take care when the power you subtract is negative. $10^{2} \div 10^{- 5} = 10^{2 - (- 5)} = 10^{7}$ . So $(3 \times 10^{2}) \div (6 \times 10^{- 5}) = 0.5 \times 10^{7}$ . Here $0.5$ is too small, so the answer is $5 \times 10^{6}$ .
To square, square both parts. The power doubles, because $10^{5} \times 10^{5} = 10^{10}$ .
${(6 \times 10^{5})}^{2} = 36 \times 10^{10} = 3.6 \times 10^{11}$

Part 7 of 8

Adding and subtracting in standard form

You cannot add the powers when you add numbers. $10^{4}$ and $10^{5}$ are different sizes, in the same way that tens and hundreds are, so the $A$ parts do not line up.
The safe method: change both numbers to ordinary numbers, add them, then convert the answer back.
$4.2 \times 10^{5} + 3.1 \times 10^{4} = 420 000 + 31 000 = 451 000 = 4.51 \times 10^{5}$
The other method: rewrite one number so that both have the same power, then add the $A$ parts. $3.1 \times 10^{4} = 0.31 \times 10^{5}$ , so the sum is $(4.2 + 0.31) \times 10^{5} = 4.51 \times 10^{5}$ . This is quicker when the ordinary numbers would be very long.
Subtraction works in the same way. Ordinary numbers line up the place values for you.
$6 \times 10^{- 3} - 4 \times 10^{- 4} = 0.006 - 0.0004 = 0.0056 = 5.6 \times 10^{- 3}$

Part 8 of 8

Standard form on your calculator

Enter a standard-form number with the calculator's power-of-ten key. Depending on the model, it is labelled $\times 10^{x}$ , EXP or EE. For $3.2 \times 10^{5}$ , type $3.2$ , press that key, then type $5$ . For a negative power, type the negative sign before the power (on many calculators this is the $(-)$ key, not the subtract key).
If you type a calculation with $\times$ , 10 and a power key instead, put brackets round each number. Without brackets, $8.1 \times 10^{7} \div 3.2 \times 10^{- 4}$ is worked out from left to right as $(8.1 \times 10^{7} \div 3.2) \times 10^{- 4} = 2531.25$ . The correct value is $2.53125 \times 10^{11}$ . Brackets round each number are always safe, whichever way you type it.
$\frac{8.1 \times 10^{7}}{3.2 \times 10^{- 4}} = (8.1 \times 10^{7}) \div (3.2 \times 10^{- 4})$
The display may show a large or small answer in a short form such as 2.53125E11. 'E11' means $\times 10^{11}$ . On the paper, write it as $2.53125 \times 10^{11}$ . The E form is calculator notation, not standard form.
To round, round $A$ and keep the power. To 3 significant figures, $2.53125 \times 10^{11}$ is $2.53 \times 10^{11}$ .
The awkward case: rounding can push $A$ up to 10. $9.996 \times 10^{4}$ to 3 significant figures is $10.0 \times 10^{4}$ , which is not standard form. Write it as $1.00 \times 10^{5}$ .
If a question does not ask for standard form, a correct ordinary number is also an answer. When it says 'Give your answer in standard form', the form is part of what is asked: write $A \times 10^{n}$ with $1 ⩽ A < 10$ .

Worked examples

Example 1. (a) Write $4 070 000$ in standard form. (b) Write $0.000 030 9$ in standard form. (c) Write $6.04 \times 10^{- 3}$ as an ordinary number. (d) Write $52.8 \times 10^{- 6}$ in standard form.

Show the working

(a) Put the point after the first digit: $A = 4.07$ . The zero between the 4 and the 7 stays, because it holds a place. Drop only the zeros at the end.
$4 070 000$ has 7 digits, so $n = 7 - 1 = 6$ .
$4 070 000 = 4.07 \times 10^{6}$
(b) The first non-zero digit is the 3, so $A = 3.09$ . The 3 is in the 5th decimal place, so $n = - 5$ .
$0.000 030 9 = 3.09 \times 10^{- 5}$
Check by multiplying out: $3.09 \div 100 000 = 0.000 030 9$ . It matches.
(c) A negative power means divide. Move the point 3 places to the left and fill with zeros: $6.04 \to 0.006 04$ .
(d) $52.8$ is not less than 10, so write it as $5.28 \times 10^{1}$ first. Then add the powers: $1 + (- 6) = - 5$ .
$52.8 \times 10^{- 6} = 5.28 \times 10^{1} \times 10^{- 6} = 5.28 \times 10^{- 5}$

Answer: (a) $4.07 \times 10^{6}$ (b) $3.09 \times 10^{- 5}$ (c) $0.006 04$ (d) $5.28 \times 10^{- 5}$

Example 2. Without a calculator, work out the following. Give each answer in standard form. (a) $(6 \times 10^{5}) \times (4 \times 10^{- 2})$ (b) $(6 \times 10^{5}) \div (4 \times 10^{- 2})$

Show the working

(a) Multiply the $A$ parts: $6 \times 4 = 24$ . Add the powers: $5 + (- 2) = 3$ .
$(6 \times 10^{5}) \times (4 \times 10^{- 2}) = 24 \times 10^{3}$
$24$ is not less than 10, so write $24 = 2.4 \times 10^{1}$ and add 1 to the power.
$24 \times 10^{3} = 2.4 \times 10^{4}$
Check with ordinary numbers: $600 000 \times 0.04 = 24 000 = 2.4 \times 10^{4}$ .
(b) Divide the $A$ parts: $6 \div 4 = 1.5$ . Subtract the powers: $5 - (- 2) = 7$ . Subtracting a negative is the step to slow down on.
$(6 \times 10^{5}) \div (4 \times 10^{- 2}) = 1.5 \times 10^{7}$
$1.5$ is between 1 and 10, so this is already standard form. Check: $600 000 \div 0.04 = 15 000 000$ .

Answer: (a) $2.4 \times 10^{4}$ (b) $1.5 \times 10^{7}$

Example 3. Without a calculator, work out $3.6 \times 10^{4} + 8.5 \times 10^{3}$ . Give your answer in standard form.

Show the working

The powers are different, so you cannot add the $A$ parts as they are. Change both to ordinary numbers.
$3.6 \times 10^{4} = 36 000$ and $8.5 \times 10^{3} = 8500$ . Add them.
$36 000 + 8500 = 44 500$
Convert back: $44 500$ has 5 digits, so $n = 4$ , and $A = 4.45$ .
$44 500 = 4.45 \times 10^{4}$
Check with the other method: $8.5 \times 10^{3} = 0.85 \times 10^{4}$ , and $3.6 + 0.85 = 4.45$ . Same answer.

Answer: $4.45 \times 10^{4}$

Example 4. One grain of a fine powder has a mass of $2.4 \times 10^{- 5}$ grams. A bag holds $1.5$ kg of the powder. (a) Calculate the number of grains in the bag. Give your answer in standard form. (b) The powder is shared equally between $7$ jars. Calculate the number of grains in each jar. Give your answer in standard form, correct to 3 significant figures.

Show the working

(a) The units must match first. The grain is in grams and the bag is in kilograms, so change the bag to grams: $1.5$ kg $= 1500$ g $= 1.5 \times 10^{3}$ g.
The number of grains is the total mass divided by the mass of one grain. Type it with brackets round each number.
$\frac{1.5 \times 10^{3}}{2.4 \times 10^{- 5}} = 62 500 000$
Convert to standard form: $62 500 000$ has 8 digits, so $n = 7$ .
$62 500 000 = 6.25 \times 10^{7}$
Sense check: $1.5 \div 2.4$ is a bit more than $0.5$ , and $10^{3} \div 10^{- 5} = 10^{8}$ . So the answer is a bit more than $0.5 \times 10^{8} = 5 \times 10^{7}$ , which fits.
(b) Divide your unrounded answer to (a) by 7.
$\frac{6.25 \times 10^{7}}{7} = 8 928 571.42 \dots = 8.928 571 \dots \times 10^{6}$
Round $A$ to 3 significant figures and keep the power: $8.93 \times 10^{6}$ .

Answer: (a) $6.25 \times 10^{7}$ grains (b) $8.93 \times 10^{6}$ grains

Where people go wrong

Mistake: Counting the zeros to find the power, e.g. writing $3 400 000 = 3.4 \times 10^{5}$ .

Instead: Count the places the decimal point moves, or use the number of digits minus 1. $3 400 000$ has 7 digits, so it is $3.4 \times 10^{6}$ .

The zeros are only some of the places. The 4 also moves past the decimal point. Check: $3.4 \times 10^{5} = 340 000$ , which is ten times too small.

Mistake: Giving a small number a positive power, e.g. $0.0063 = 6.3 \times 10^{3}$ .

Instead: $0.0063 = 6.3 \times 10^{- 3}$ . A number between 0 and 1 always has a negative power in standard form.

$6.3 \times 10^{3}$ is $6300$ . Multiply your answer out to check that it gives back the number you started with.

Mistake: Leaving $A$ outside the range after a calculation, e.g. giving $20 \times 10^{9}$ or $0.5 \times 10^{7}$ as the final answer.

Instead: Correct $A$ and adjust the power: $20 \times 10^{9} = 2 \times 10^{10}$ and $0.5 \times 10^{7} = 5 \times 10^{6}$ .

The value is right, but standard form needs $1 ⩽ A < 10$ . When the question asks for standard form, the form is part of the answer.

Mistake: Adding the powers when adding numbers, e.g. $2 \times 10^{3} + 3 \times 10^{4} = 5 \times 10^{7}$ .

Instead: Change to ordinary numbers first: $2000 + 30 000 = 32 000 = 3.2 \times 10^{4}$ .

Adding the powers is the rule for multiplying powers of ten. Adding numbers needs the place values lined up, as in ordinary column addition.

Mistake: Copying the calculator display as the answer, e.g. writing 2.53E11, or turning it into ${2.53}^{11}$ .

Instead: Write $2.53 \times 10^{11}$ .

E11 is the calculator's way of showing $\times 10^{11}$ . ${2.53}^{11}$ means $2.53$ multiplied by itself 11 times, which is only about $27 000$ .

Check yourself

Which of these are written in standard form? $7.2 \times 10^{5}$ , $12 \times 10^{3}$ , $0.9 \times 10^{- 2}$ , $1 \times 10^{- 6}$ , $3.5 \times 10^{0}$
Answer
$7.2 \times 10^{5}$ , $1 \times 10^{- 6}$ and $3.5 \times 10^{0}$ . $12 \times 10^{3}$ is not, because $12$ is not less than 10. $0.9 \times 10^{- 2}$ is not, because $0.9$ is less than 1.
Write in standard form: (a) $86 500 000$ (b) $0.000 407$
Answer
(a) $8.65 \times 10^{7}$ (8 digits, so $n = 7$ ) (b) $4.07 \times 10^{- 4}$ (the 4 is in the 4th decimal place)
Write as ordinary numbers: (a) $5.08 \times 10^{- 3}$ (b) $9.1 \times 10^{4}$
Answer
(a) $0.005 08$ (b) $91 000$
Write $270 \times 10^{- 5}$ in standard form.
Answer
$270 = 2.7 \times 10^{2}$ , so $270 \times 10^{- 5} = 2.7 \times 10^{2 + (- 5)} = 2.7 \times 10^{- 3}$ . Check: both equal $0.0027$ .
Write these in order of size, smallest first: $3.1 \times 10^{- 2}$ , $2.9 \times 10^{- 1}$ , $8 \times 10^{- 3}$ , $0.05$
Answer
$8 \times 10^{- 3}$ , $3.1 \times 10^{- 2}$ , $0.05$ , $2.9 \times 10^{- 1}$ . As ordinary numbers they are $0.008$ , $0.031$ , $0.05$ , $0.29$ .
Without a calculator, work out $(5 \times 10^{4}) \times (3 \times 10^{6})$ . Give your answer in standard form.
Answer
$15 \times 10^{10} = 1.5 \times 10^{11}$
Without a calculator, work out $(2.4 \times 10^{- 3}) \div (8 \times 10^{5})$ . Give your answer in standard form.
Answer
$2.4 \div 8 = 0.3$ and $- 3 - 5 = - 8$ , giving $0.3 \times 10^{- 8} = 3 \times 10^{- 9}$ .
Without a calculator, work out $7.4 \times 10^{5} - 9 \times 10^{4}$ . Give your answer in standard form.
Answer
$740 000 - 90 000 = 650 000 = 6.5 \times 10^{5}$

Exam questions

Try each part first. Open the mark scheme after you have written your answer.

Question 1.

(a) Explain why $23.5 \times 10^{4}$ is not written in standard form. 1 mark No calculator

Mark scheme

B1 23.5 is not less than 10 oe, e.g. 'the number in front must be between 1 and 10'; may also give 2.35 × 10⁵

(b) Write $6 300 000$ in standard form. 1 mark No calculator

Mark scheme

B1 6.3 × 10⁶ cao

(c) Write $0.000 048$ in standard form. 1 mark No calculator

Mark scheme

B1 4.8 × 10⁻⁵ cao

(d) Write $2.06 \times 10^{- 4}$ as an ordinary number. 1 mark No calculator

Mark scheme

B1 0.000206 cao

(e) Write these numbers in order of size, starting with the smallest. $4.1 \times 10^{- 3}$ , $0.0039$ , $3.9 \times 10^{- 2}$ , $4 \times 10^{- 4}$ 2 marks No calculator

Mark scheme

M1 all four in the same form, e.g. 0.0041, 0.0039, 0.039, 0.0004, or 4.1 × 10⁻³, 3.9 × 10⁻³, 3.9 × 10⁻², 4 × 10⁻⁴
A1 4 × 10⁻⁴, 0.0039, 4.1 × 10⁻³, 3.9 × 10⁻² oe; correct order implies M1

Question 2. Do not use a calculator in this question. Give each answer in standard form.

(a) Work out $(4 \times 10^{6}) \times (7 \times 10^{- 2})$ . 2 marks No calculator

Mark scheme

M1 28 × 10⁴ seen, or 280 000 seen oe
A1 2.8 × 10⁵ cao

(b) Work out $(9 \times 10^{3}) \div (1.5 \times 10^{- 4})$ . 2 marks No calculator

Mark scheme

M1 9 ÷ 1.5 = 6 and 10³ ÷ 10⁻⁴ = 10⁷ both seen, or 60 000 000 seen
A1 6 × 10⁷ cao

(c) Work out $5.2 \times 10^{5} + 8 \times 10^{4}$ . 2 marks No calculator

Mark scheme

M1 520 000 + 80 000, or 5.2 × 10⁵ + 0.8 × 10⁵, or 52 × 10⁴ + 8 × 10⁴, or 600 000 seen
A1 6 × 10⁵ cao

Question 3. A reservoir holds $4.62 \times 10^{9}$ litres of water. Take one drop of water to have a volume of $5 \times 10^{- 5}$ litres.

(a) Calculate the number of drops of water in the reservoir. Give your answer in standard form. 2 marks

Mark scheme

M1 4.62 × 10⁹ ÷ (5 × 10⁻⁵) oe
A1 9.24 × 10¹³ cao

(b) Each day $3.8 \times 10^{7}$ litres of water are taken from the reservoir, and no water is added. Calculate the volume of water left in the reservoir after $30$ days. Give your answer in standard form. 3 marks

Mark scheme

M1 30 × 3.8 × 10⁷ [= 1.14 × 10⁹] oe
M1 dep 4.62 × 10⁹ − their 1.14 × 10⁹
A1 3.48 × 10⁹ cao

Question 4.

(a) Calculate $\frac{6.4 \times 10^{- 3} + 2.9 \times 10^{- 4}}{1.6 \times 10^{5}}$ . Give your answer in standard form, correct to 3 significant figures. 2 marks

Mark scheme

M1 numerator 6.69 × 10⁻³ oe seen, e.g. 0.00669
A1 4.18 × 10⁻⁸ [from 4.18125 × 10⁻⁸]

How this is examined

Dividing or multiplying two standard-form values in a context, on the calculator: A 2- or 3-mark Calculate after a stem that gives quantities in standard form, ending 'Give your answer in standard form'. If the units differ (grams and kilograms), converting them is part of the method. The method mark is for the right operation written with the given values in matching units, e.g. $1500 \div (2.4 \times 10^{- 5})$ : total divided by the amount for one. The accuracy mark is for the value written as $A \times 10^{n}$ , e.g. $6.25 \times 10^{7}$ , with $A$ rounded to 3 significant figures when it is not exact. A display copy such as 6.25E7 is not standard form.
Multiplying and dividing standard-form numbers without a calculator: A 2-mark Work out on the non-calculator paper, with 'Give your answer in standard form' Method: the $A$ parts combined and the powers combined, e.g. $28 \times 10^{4}$ , or the correct ordinary number $280 000$ . Accuracy: the final answer corrected to $1 ⩽ A < 10$ , e.g. $2.8 \times 10^{5}$ . For division, the power is the first minus the second, so $10^{3} \div 10^{- 4} = 10^{7}$ .
Adding or subtracting standard-form numbers with different powers: A 2-mark Work out, with or without a calculator, where the two powers of ten are different Method: both numbers as ordinary numbers, e.g. $520 000 + 80 000$ , or both with the same power, e.g. $5.2 \times 10^{5} + 0.8 \times 10^{5}$ . Accuracy: the sum in standard form, $6 \times 10^{5}$ . Adding the powers scores nothing.
Ordering numbers given partly in standard form and partly as ordinary numbers: A 2-mark list of four or five numbers, some with negative powers, to write in order of size Method: every number written in one form, e.g. all as ordinary numbers. Accuracy: the correct order, written with the numbers as the question gave them and in the direction asked for (smallest first or largest first).
Converting a single number into or out of standard form: A 1-mark Write: a large or small ordinary number to put in standard form, or a standard-form number to write as an ordinary number Exactly $A \times 10^{n}$ with $1 ⩽ A < 10$ : $6.3 \times 10^{6}$ , not $63 \times 10^{5}$ . For an ordinary number, the right digits with the right number of zeros: $2.06 \times 10^{- 4} = 0.000 206$ .

Words to know

Standard form: A number written as $A \times 10^{n}$ , where $1 ⩽ A < 10$ and $n$ is an integer, e.g. $5.2 \times 10^{7}$ .
Power (index): The small raised number in $10^{n}$ . It says how many times 10 is multiplied (positive power) or divided (negative power).
Negative power of ten: $10^{- n} = \frac{1}{10^{n}}$ , a positive number less than 1, e.g. $10^{- 3} = 0.001$ .
Ordinary number: A number written out in full with its digits and zeros, e.g. $52 000 000$ or $0.000 63$ .
Significant figures: The digits of a number counted from the first non-zero digit. In standard form, rounding to 3 significant figures means rounding $A$ , e.g. $2.53125 \times 10^{11} \to 2.53 \times 10^{11}$ .

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