C1.8; E1.8 - Standard form

C1.8; E1.8 - Standard form

How can 61 million and 0.000061 share the same digits without a long trail of zeros? Standard form separates a number's significant digits from its scale. C1.8 and E1.8 cover recognising this form, converting numbers and calculating with them; the hand calculation methods are Extended content.

Normalised form

The number 6100000061000000 can be written as 6.1×100000006.1\times10000000, or 6.1×1076.1\times10^7. The first factor tells you the digits; the power of ten tells you their place value.

Standard form

A positive number in standard form is written as A×10nA\times10^n, where 1≤A<101\le A<10 and nn is an integer. There is exactly one non-zero digit before the decimal point in AA.

Positive indices scale up by factors of ten; negative indices scale down. For example, 10−3=0.00110^{-3}=0.001, so 6.1×10−3=0.00616.1\times10^{-3}=0.0061. For a number from 1 up to, but not including, 10, the same notation uses 100=110^0=1.

ExpressionStandard form?Reason
7.2×1047.2\times10^4Yes7.27.2 lies between 1 and 10
72×10372\times10^3NoThe first factor is at least 10
0.72×1050.72\times10^5NoThe first factor is below 1
10×10210\times10^2NoThe upper boundary is excluded

The last three expressions are valid numbers; they simply need rewriting to meet the standard-form convention. For instance, 72×103=7.2×10472\times10^3=7.2\times10^4: dividing the first factor by ten requires multiplying the power-of-ten factor by ten to preserve the value.

Key idea

Standard form has both a correct value and a fixed format. The condition 1≤A<101\le A<10 makes each positive number's standard form unique.

Check the format before calculating anything.

Conversions

To convert into standard form, choose a first factor from 1 to below 10, then find the power of ten that restores the original place value. To convert out, evaluate that power of ten and multiply.

Worked example 1: a large number

Question. Write 84000008400000 in standard form.

Plan. Use 8.4 as the first factor and find the scale from 8.4 to the original number.

8400000=8.4×1000000six place-value steps=8.4×106\begin{aligned} 8400000&=8.4\times1000000 &&\text{six place-value steps}\\ &=8.4\times10^6 \end{aligned}

Answer. 8.4×1068.4\times10^6.

Check. 8.4×106=8.4×1000000=84000008.4\times10^6=8.4\times1000000=8400000. The first factor is in the required range.

Now keep the same first factor but make the original number small. The index must become negative, because the scale factor is below 1.

Worked example 2: a small number

Question. Write 0.0000840.000084 in standard form, then write 3.7×10−43.7\times10^{-4} as an ordinary decimal.

Plan. Count place-value steps from the first non-zero digit; use a negative index when scaling down.

0.000084=8.4×0.00001=8.4×10−53.7×10−4=3.7÷10000=0.00037\begin{aligned} 0.000084&=8.4\times0.00001=8.4\times10^{-5}\\ 3.7\times10^{-4}&=3.7\div10000=0.00037 \end{aligned}

The 8 in 0.0000840.000084 is in the hundred-thousandths column, so it needs five downward steps from the units column.

Answer. 8.4×10−58.4\times10^{-5} and 0.000370.00037.

Check. Multiplying 0.0000840.000084 by 100000100000 gives 8.4; multiplying 0.000370.00037 by 1000010000 gives 3.7. Both reverse the conversions.

Counting zeros alone is unreliable: the zero before a decimal point does not represent another downward place-value step.

Watch out

A negative index does not make the number negative. 3.7×10−43.7\times10^{-4} is positive and smaller than 1; −3.7×104-3.7\times10^4 is a different, negative number.

Check both the index sign and the number of steps.

Calculations

Enter standard-form numbers with the calculator's power-of-ten entry key, which may be labelled EXP or a similar symbol, or enter each complete number in brackets. An entry such as 3.6×1053.6\times10^5 must be treated as one factor. A display such as “9E2” means 9×1029\times10^2, not 9×29\times2.

For Core, calculations with standard form are assessed only on calculator papers. Both tiers use the calculator methods below; the labelled hand methods also prepare Extended students for non-calculator questions.

Worked example 3: multiplying standard-form numbers

Question. Calculate (3.6×105)(2.5×10−3)(3.6\times10^5)(2.5\times10^{-3}). Give your answer in standard form.

Plan. Enter the two complete factors, then check that the result is normalised.

Calculator entry is (3.6×105)×(2.5×10−3)(3.6\times10^5)\times(2.5\times10^{-3}), giving 900900.

Extended hand method: multiply the first factors and add the indices:

(3.6×105)(2.5×10−3)=(3.6×2.5)×105−3=9×102\begin{aligned} (3.6\times10^5)(2.5\times10^{-3})&=(3.6\times2.5)\times10^{5-3}\\ &=9\times10^2 \end{aligned}

Answer. 9×1029\times10^2.

Check. In ordinary numbers, 360000×0.0025=360000÷400=900360000\times0.0025=360000\div400=900. The result is smaller than the first factor, as multiplication by a positive number below 1 should be.

Division changes the index operation: subtract the denominator's index. Keep both complete numbers grouped in the calculator entry.

Worked example 4: dividing standard-form numbers

Question. Calculate 4.8×10−41.2×103\dfrac{4.8\times10^{-4}}{1.2\times10^3}. Give your answer in standard form.

Plan. Divide the complete numerator by the complete denominator and normalise the result.

Calculator entry is (4.8×10−4)÷(1.2×103)(4.8\times10^{-4})\div(1.2\times10^3), giving 0.00000040.0000004.

Extended hand method: divide the first factors and subtract the indices:

4.8×10−41.2×103=4.81.2×10−4−3=4×10−7\begin{aligned} \frac{4.8\times10^{-4}}{1.2\times10^3}&=\frac{4.8}{1.2}\times10^{-4-3}\\ &=4\times10^{-7} \end{aligned}

Answer. 4×10−74\times10^{-7}.

Check. Reverse the division: 0.0000004×1200=0.000480.0000004\times1200=0.00048, the original numerator.

Addition and subtraction join quantities of the same size. On a calculator, enter the whole expression. For hand work (Extended), first express both numbers with the same power of ten, just as fractions need a common denominator before adding.

Worked example 5: adding different powers of ten

Question. Calculate (6.2×106)+(7×105)(6.2\times10^6)+(7\times10^5). Give your answer in standard form.

Plan. Use a calculator, or rewrite the second number in millions before adding.

The calculator gives 69000006900000. For the Extended hand route,

6.2×106+7×105=6.2×106+0.7×106=(6.2+0.7)×106=6.9×106\begin{aligned} 6.2\times10^6+7\times10^5&=6.2\times10^6+0.7\times10^6\\ &=(6.2+0.7)\times10^6\\ &=6.9\times10^6 \end{aligned}

Answer. 6.9×1066.9\times10^6.

Check. 6200000+700000=69000006200000+700000=6900000, which is larger than either positive term.

Subtraction uses the same common-power idea. The first factor may then leave the allowed range, so check the format once more.

Worked example 6: subtracting and normalising

Question. Calculate (2.4×10−3)−(1.8×10−3)(2.4\times10^{-3})-(1.8\times10^{-3}). Give your answer in standard form.

Plan. Subtract the first factors because the powers of ten already match, then normalise.

The calculator gives 0.00060.0006. For the Extended hand route,

(2.4−1.8)×10−3=0.6×10−3=6×10−4increase the first factor tenfold\begin{aligned} (2.4-1.8)\times10^{-3}&=0.6\times10^{-3}\\ &=6\times10^{-4} &&\text{increase the first factor tenfold} \end{aligned}

Answer. 6×10−46\times10^{-4}.

Check. 0.0024−0.0018=0.00060.0024-0.0018=0.0006; adding 0.00180.0018 back restores 0.00240.0024.

A standard-form answer keeps its digits unless rounding is requested. Conversion alone is not an instruction to round.

Watch out

Do not add indices when adding numbers. 103+103=2×10310^3+10^3=2\times10^3, whereas 103×103=10610^3\times10^3=10^6. Match the rule to the operation.

Use a calculator for this check at Core; at Extended, also practise it by hand.

Quick Recap

  • Standard form is A×10nA\times10^n, with 1≤A<101\le A<10 and an integer index.
  • Positive powers scale up; negative powers scale down.
  • Group complete standard-form numbers when entering a calculation.
  • Extended hand methods use index laws for products/quotients and common powers for sums/differences.

The opening numbers are 6.1×1076.1\times10^7 and 6.1×10−56.1\times10^{-5}: the same first factor, with different scales.