FN9, HN9 - Standard form

FN9, HN9 - Standard form

Standard form separates a number's size from its leading digits. Learn to write and compare very large or small values, convert quantities to matching units, and calculate without losing powers of ten. These are the interpretation and calculation skills required by FN9 and HN9.

Writing standard form

How can 48 300 00048\,300\,000 and 0.000004830.00000483 both be written using the digits 4.834.83? A power of ten records the scale that separates them.

You already know that 103=100010^3=1000 and 10−3=1/1000=0.00110^{-3}=1/1000=0.001. Multiplying by the first makes a positive number a thousand times larger; multiplying by the second makes it a thousand times smaller.

Standard form

A positive number is in standard form when it is written as

A×10n,1≤A<10,A\times10^n,\qquad 1\le A<10,

where nn is an integer. The coefficient AA has one nonzero digit before its decimal point. Thus 4.83×1074.83\times10^7 is in standard form, but 48.3×10648.3\times10^6 and 0.483×1080.483\times10^8 are not, although all three have the same value.

First choose the coefficient between 11 and 1010. Then choose the power of ten that restores the original value. A number at least 1010 needs a positive index; a positive number below 11 needs a negative index. Numbers from 11 up to but not including 1010 use index zero: 6.2=6.2×1006.2=6.2\times10^0.

Worked example 1: a large number

Question. Write 48 300 00048\,300\,000 in standard form.

Plan. Use coefficient 4.834.83, then find the power of ten needed to recover the original number.

48 300 000=4.83×10 000 000separate digits and scale=4.83×107ten million is 107\begin{aligned} 48\,300\,000&=4.83\times10\,000\,000 &&\text{separate digits and scale}\\ &=4.83\times10^7 &&\text{ten million is }10^7 \end{aligned}

Answer. 4.83×1074.83\times10^7.

Check. Multiplying 4.834.83 by 1010 seven times gives 48.3,483,4830,…,48 300 00048.3,483,4830,\ldots,48\,300\,000. The coefficient satisfies 1≤4.83<101\le4.83<10.

Keep the digits, but change the value to a very small number. The power of ten must now shrink the coefficient.

Worked example 2: a small number

Question. Write 0.000004830.00000483 in standard form.

Plan. Use coefficient 4.834.83, then divide by ten repeatedly until the value matches.

0.00000483=4.83×0.000001the scale is one millionth=4.83×10−6one millionth is 10−6\begin{aligned} 0.00000483&=4.83\times0.000001 &&\text{the scale is one millionth}\\ &=4.83\times10^{-6} &&\text{one millionth is }10^{-6} \end{aligned}

Answer. 4.83×10−64.83\times10^{-6}.

Check. Multiplying the original number by 10610^6 gives 4.834.83, so multiplying 4.834.83 by 10−610^{-6} takes us back.

To read a value in standard form, carry out its power-of-ten multiplication. For example, 7.2×104=72 0007.2\times10^4=72\,000 and 7.2×10−4=0.000727.2\times10^{-4}=0.00072. Zero itself cannot satisfy the positive-coefficient definition because multiplying a positive coefficient by any power of ten never gives zero.

Watch out

0.00000483≠4.83×1060.00000483\ne4.83\times10^6. A positive index makes 4.834.83 larger; this number is smaller. Check the direction of the scaling, rather than memorising a direction for moving a decimal point.

Write a new small number with the required coefficient.

Ordering scales

Compare positive numbers in standard form by their powers of ten first. Only compare coefficients directly when those powers match.

For example, 9.8×103<1.1×1049.8\times10^3<1.1\times10^4, because every standard-form number with index 33 is below 10×103=10410\times10^3=10^4. Negative indices follow ordinary number order: −4<−3-4<-3, so a positive standard-form value with index −4-4 is smaller than one with index −3-3.

Quantities must also use the same unit. Since 1 cm=0.01 m=10−2 m1\,\mathrm{cm}=0.01\,\mathrm m=10^{-2}\,\mathrm m, converting centimetres to metres multiplies the number by 10−210^{-2}. The object does not change size; the number of units changes.

Worked example 3: comparing small values

Question. Write these lengths in increasing order:

6.1×10−4 m,5.9×10−3 m,8.2×10−4 m.6.1\times10^{-4}\,\mathrm m,\qquad 5.9\times10^{-3}\,\mathrm m,\qquad 8.2\times10^{-4}\,\mathrm m.

Plan. Compare indices first, then compare coefficients where the indices match.

The two values with index −4-4 come first because −4<−3-4<-3. Between those two, 6.1<8.26.1<8.2.

Answer.

6.1×10−4 m<8.2×10−4 m<5.9×10−3 m.6.1\times10^{-4}\,\mathrm m <8.2\times10^{-4}\,\mathrm m <5.9\times10^{-3}\,\mathrm m.

Check. In ordinary decimals the lengths are 0.00061 m0.00061\,\mathrm m, 0.00082 m0.00082\,\mathrm m, 0.0059 m0.0059\,\mathrm m, in the same order.

Now express one of the lengths in centimetres. Its apparent index changes, so convert it before comparing.

Worked example 4: matching the units

Question. Write these lengths in increasing order:

6.1×10−4 m,5.9×10−3 m,8.2×10−2 cm.6.1\times10^{-4}\,\mathrm m,\qquad 5.9\times10^{-3}\,\mathrm m,\qquad 8.2\times10^{-2}\,\mathrm{cm}.

Plan. Convert the centimetre length to metres, then compare all three values.

8.2×10−2 cm=8.2×10−2×10−2 mone centimetre is 10−2 m=8.2×10−4 madd indices when multiplying\begin{aligned} 8.2\times10^{-2}\,\mathrm{cm} &=8.2\times10^{-2}\times10^{-2}\,\mathrm m &&\text{one centimetre is }10^{-2}\,\mathrm m\\ &=8.2\times10^{-4}\,\mathrm m &&\text{add indices when multiplying} \end{aligned}

We now have the same metre values as in the previous example.

Answer. From smallest to largest:
6.1×10−4 m, 8.2×10−2 cm, 5.9×10−3 m6.1\times10^{-4}\,\mathrm m,\ 8.2\times10^{-2}\,\mathrm{cm},\ 5.9\times10^{-3}\,\mathrm m.

Check. 8.2×10−2 cm=0.082 cm8.2\times10^{-2}\,\mathrm{cm}=0.082\,\mathrm{cm}. Dividing by 100100 gives 0.00082 m0.00082\,\mathrm m, between the other two lengths.

The same principle handles large quantities. For example, 2.4×103 km=2.4×106 m2.4\times10^3\,\mathrm{km}=2.4\times10^6\,\mathrm m, because one kilometre contains 10310^3 metres.

Watch out

The coefficient alone does not show the larger value: 8.2>5.98.2>5.9, but 8.2×10−4<5.9×10−38.2\times10^{-4}<5.9\times10^{-3}. Compare standard-form indices after matching units.

Use the given conversion to order another mixed-unit set.

Calculating in standard form

Multiplication and division act separately on coefficients and powers of ten. Addition and subtraction need matching powers first, just as fractions need matching denominators.

For multiplication, (2×103)(4×105)=(2×4)×103+5=8×108(2\times10^3)(4\times10^5)=(2\times4)\times10^{3+5}=8\times10^8. For division, divide the coefficients and subtract the divisor's index. After either operation, adjust the coefficient into the standard-form range. For example, 24×106=2.4×10724\times10^6=2.4\times10^7: dividing the coefficient by ten needs a power of ten ten times larger to preserve the value.

For addition or subtraction, rewrite one value at the other's scale: 4×103=0.4×1044\times10^3=0.4\times10^4, so 3×104+4×103=(3+0.4)×1043\times10^4+4\times10^3=(3+0.4)\times10^4. We add the coefficients, not the indices.

Worked example 5: four operations at one scale

Question. The numbers are 3×1043\times10^4 and 5×1035\times10^3. Work out their product, the first divided by the second, their sum and their positive difference. Give each answer in standard form.

Plan. Use index laws for the product and quotient; write 5×103=0.5×1045\times10^3=0.5\times10^4 for the sum and difference.

Product=(3×5)×104+3multiply coefficients; add indices=15×107=1.5×108adjust the coefficientQuotient=(3÷5)×104−3divide coefficients; subtract indices=0.6×101=6×100adjust the coefficientSum=(3+0.5)×104=3.5×104add at one scaleDifference=(3−0.5)×104=2.5×104subtract at one scale\begin{aligned} \text{Product}&=(3\times5)\times10^{4+3} &&\text{multiply coefficients; add indices}\\ &=15\times10^7=1.5\times10^8 &&\text{adjust the coefficient}\\ \text{Quotient}&=(3\div5)\times10^{4-3} &&\text{divide coefficients; subtract indices}\\ &=0.6\times10^1=6\times10^0 &&\text{adjust the coefficient}\\ \text{Sum}&=(3+0.5)\times10^4=3.5\times10^4 &&\text{add at one scale}\\ \text{Difference}&=(3-0.5)\times10^4=2.5\times10^4 &&\text{subtract at one scale} \end{aligned}

Answer. 1.5×108, 6×100, 3.5×104, 2.5×1041.5\times10^8,\ 6\times10^0,\ 3.5\times10^4,\ 2.5\times10^4, respectively.

Check. In ordinary numbers, 30 000×5000=150 000 00030\,000\times5000=150\,000\,000, 30 000÷5000=630\,000\div5000=6, 30 000+5000=35 00030\,000+5000=35\,000, and 30 000−5000=25 00030\,000-5000=25\,000.

Change only the second number's index to 55. It is now larger than the first; the quotient drops below 11. We take the larger minus the smaller for the positive difference.

Worked example 6: a larger second value

Question. The numbers are 3×1043\times10^4 and 5×1055\times10^5. Work out their product, the first divided by the second, their sum and their positive difference. Give each answer in standard form.

Plan. Combine powers for multiplication and division; use 3×104=0.3×1053\times10^4=0.3\times10^5 for addition and subtraction.

Product=15×104+5=1.5×1010multiply, then adjustQuotient=0.6×104−5=6×10−2divide, then adjustSum=(0.3+5)×105=5.3×105use the same powerDifference=(5−0.3)×105=4.7×105use the same power\begin{aligned} \text{Product}&=15\times10^{4+5}=1.5\times10^{10} &&\text{multiply, then adjust}\\ \text{Quotient}&=0.6\times10^{4-5}=6\times10^{-2} &&\text{divide, then adjust}\\ \text{Sum}&=(0.3+5)\times10^5=5.3\times10^5 &&\text{use the same power}\\ \text{Difference}&=(5-0.3)\times10^5=4.7\times10^5 &&\text{use the same power} \end{aligned}

Answer. 1.5×1010, 6×10−2, 5.3×105, 4.7×1051.5\times10^{10},\ 6\times10^{-2},\ 5.3\times10^5,\ 4.7\times10^5, respectively.

Check. The quotient is 0.060.06, and 500 000×0.06=30 000500\,000\times0.06=30\,000, restoring the first number. The sum is 530 000530\,000 and the difference 470 000470\,000, which each differ from 500 000500\,000 by 30 00030\,000. Direct multiplication gives 15 000 000 00015\,000\,000\,000, matching the product.

On a calculator, a display such as 6E−26\mathrm E{-2} means 6×10−26\times10^{-2}. The letter is display notation for the power of ten, not another factor. Enter standard-form values using the calculator's power-of-ten key or bracket each complete value when typing the multiplication explicitly. For a division, divide by the whole second value: (3×104)÷(5×105)(3\times10^4)\div(5\times10^5). Write 6×10−26\times10^{-2} as the final mathematical answer, and check that the result has a sensible size.

Exam tip

If the question asks for standard form, finish with 1≤A<101\le A<10. A correct value such as 15×10715\times10^7 still needs rewriting as 1.5×1081.5\times10^8. Show mathematical operations when a method is required; calculator key names alone do not show the calculation.

The opening numbers have the same coefficient, but scales 10710^7 and 10−610^{-6}. Standard form keeps that difference visible while letting us calculate with each value.

Quick Recap

  • Standard form is A×10nA\times10^n, with 1≤A<101\le A<10 and integer nn.
  • Match units, compare indices, then compare coefficients if the indices match.
  • Multiply or divide coefficients and use the corresponding index law; adjust the result's coefficient.
  • For addition and subtraction, write both values with the same power of ten first.

The opening values are 4.83×1074.83\times10^7 and 4.83×10−64.83\times10^{-6}: the powers record their very different sizes.