C1.8; E1.8 - Writing numbers in standard form

C1.8; E1.8 - Writing numbers in standard form

Standard form separates a number's digits from its size. You will write large and small positive numbers using powers of ten, convert them back, and compare their magnitudes. This lesson is for Core and Extended, including non-calculator work.

Powers of ten and scale

Which is larger, 8.1×10−48.1\times10^{-4} or 3.6×10−33.6\times10^{-3}? The first has the larger front number, but the powers of ten tell a different story.

Recall that 103=100010^3=1000, while 10−3=11000=0.00110^{-3}=\frac1{1000}=0.001. A positive integer power of ten scales up; a negative integer power scales down. Index zero gives 100=110^0=1.

[DIAGRAM: asset_name: Powers of ten across zero; asset_slug: powers_ten_scale; caption: Each step right divides by ten, including the steps through one.; description: A sequence of nine equally spaced labelled boxes showing ten to the fourth equals 10000 down to ten to the negative fourth equals 0.0001. Each arrow between neighbouring boxes is labelled divide by ten. The diagram is an operation sequence, not a linear number line.]
Each step right divides by ten, including the steps through one.

The equal spaces in this picture show the same operation each time, not equal differences between the values. Passing from 101=1010^1=10 to 100=110^0=1 and then 10−1=0.110^{-1}=0.1 continues the same division pattern.

A power of ten

For a positive integer nn, 10n10^n is 11 followed by nn zeros, and 10−n=110n10^{-n}=\frac1{10^n}. Also, 100=110^0=1.

For example, 102=10010^2=100 and 10−2=0.0110^{-2}=0.01. A negative index does not make the value negative.

Counting the index is safer than counting only the zeros you can see after the decimal point.

Worked example 1: a positive power of ten

Question. Write 10410^4 as an ordinary number.

Plan. Use four factors of ten.

104=10×10×10×10four factors=10 000multiply\begin{aligned} 10^4&=10\times10\times10\times10 && \text{four factors}\\ &=10\,000 && \text{multiply} \end{aligned}

Answer. 10 00010\,000.

Check. Dividing 10 00010\,000 by ten four times gives 11.

Change only the sign of the index. We need the reciprocal of the same power.

Worked example 2: a negative power of ten

Question. Write 10−410^{-4} as an ordinary number.

Plan. Write a reciprocal, then use decimal place value.

10−4=1104negative index means reciprocal=110 000evaluate the denominator=0.0001one ten-thousandth\begin{aligned} 10^{-4}&=\frac1{10^4} && \text{negative index means reciprocal}\\ &=\frac1{10\,000} && \text{evaluate the denominator}\\ &=0.0001 && \text{one ten-thousandth} \end{aligned}

Answer. 0.00010.0001.

Check. 0.0001×10 000=10.0001\times10\,000=1, as reciprocals must give.

Watch out

10−4=0.0000110^{-4}=0.00001 is one place too small. In 0.00010.0001, the 11 is in the fourth place after the decimal point. There are only three zeros between the point and that 11.

Use the reciprocal and check the place.

Writing standard form

To describe 73 00073\,000, you can say “7.37.3 lots of ten thousand”. The digits sit in 7.37.3, and the scale sits in 10410^4. This is standard form.

Standard form

A positive number in standard form is written

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

where nn is an integer. The front number AA is called the coefficient.

For example, 7.3×1047.3\times10^4 is in standard form. The equal expression 73×10373\times10^3 is not, because 7373 is too large. The coefficient may equal 11, but it must be smaller than 1010.

Choose the coefficient first: put its decimal point after the first nonzero digit. Then choose the power of ten that returns it to the original value. Large numbers need a positive scale; positive numbers below 11 need a negative scale. A number such as 6.26.2 can be written 6.2×1006.2\times10^0, since 100=110^0=1. Zero itself cannot meet the coefficient condition.

Worked example 3: writing a large number

Question. Write 73 00073\,000 in standard form.

Plan. Use the coefficient 7.37.3, then find its scale.

73 000=7.3×10 000coefficient is between one and ten=7.3×10410 000=104\begin{aligned} 73\,000&=7.3\times10\,000 && \text{coefficient is between one and ten}\\ &=7.3\times10^4 && 10\,000=10^4 \end{aligned}

Answer. 7.3×1047.3\times10^4.

Check. Multiplying 7.37.3 by ten four times gives 73, 730, 7300, 73 00073,\ 730,\ 7300,\ 73\,000.

Keep the digits 7373, but change the size of the original number. The coefficient stays the same.

Worked example 4: writing a small number

Question. Write 0.000730.00073 in standard form.

Plan. Use 7.37.3, then count the divisions by ten needed to reach the original.

0.00073=7.3÷10 000four divisions by ten=7.3×10−4use a negative power\begin{aligned} 0.00073&=7.3\div10\,000 && \text{four divisions by ten}\\ &=7.3\times10^{-4} && \text{use a negative power} \end{aligned}

Answer. 7.3×10−47.3\times10^{-4}.

Check. Starting with 0.000730.00073, multiplying by 10 00010\,000 gives 7.37.3. This reverses the scale.

Watch out

0.00073=7.3×1040.00073=7.3\times10^4 has the wrong sign of index: it would make 73 00073\,000. Before counting places, decide whether your power of ten must enlarge or reduce the coefficient.

Choose the coefficient before counting the scale.

Reading and comparing standard form

To read standard form, carry out the multiplication by its power of ten. For example, 4.6×103=46004.6\times10^3=4600, while 4.6×10−3=0.00464.6\times10^{-3}=0.0046. Writing ordinary numbers is a reliable way to compare magnitudes.

When every positive number is already in standard form, compare the indices first. A larger index gives a larger number. If the indices match, compare the coefficients. This works because a standard-form coefficient is at least 11 and smaller than 1010: all numbers with one index fit below the numbers with the next index.

Worked example 5: ordering by scale then coefficient

Question. Write these numbers in order of size, starting with the smallest:

2.8×105,7.3×104,5.1×1052.8\times10^5,\qquad7.3\times10^4,\qquad5.1\times10^5

Plan. Read the powers as ordinary numbers, then order them.

2.8×105=280 000multiply by one hundred thousand7.3×104=73 000multiply by ten thousand5.1×105=510 000same scale as the first73 000<280 000<510 000compare values\begin{aligned} 2.8\times10^5&=280\,000 && \text{multiply by one hundred thousand}\\ 7.3\times10^4&=73\,000 && \text{multiply by ten thousand}\\ 5.1\times10^5&=510\,000 && \text{same scale as the first}\\ 73\,000&<280\,000<510\,000 && \text{compare values} \end{aligned}

Answer. 7.3×104, 2.8×105, 5.1×1057.3\times10^4,\ 2.8\times10^5,\ 5.1\times10^5.

Check. The index 44 comes first. Of the two numbers with index 55, 2.8<5.12.8<5.1.

Change only the first number's index from 55 to 33. Its position changes even though its coefficient does not.

Worked example 6: changing one index

Question. Write these numbers in order of size, starting with the smallest:

2.8×103,7.3×104,5.1×1052.8\times10^3,\qquad7.3\times10^4,\qquad5.1\times10^5

Plan. Compare the indices and check by converting back.

2.8×103=2800a scale of one thousand7.3×104=73 000a scale of ten thousand5.1×105=510 000a scale of one hundred thousand2800<73 000<510 000increasing magnitude\begin{aligned} 2.8\times10^3&=2800 && \text{a scale of one thousand}\\ 7.3\times10^4&=73\,000 && \text{a scale of ten thousand}\\ 5.1\times10^5&=510\,000 && \text{a scale of one hundred thousand}\\ 2800&<73\,000<510\,000 && \text{increasing magnitude} \end{aligned}

Answer. 2.8×103, 7.3×104, 5.1×1052.8\times10^3,\ 7.3\times10^4,\ 5.1\times10^5.

Check. The indices increase 3,4,53,4,5, matching the order of the positive standard-form numbers.

The opening comparison uses exactly the same idea: −4<−3-4<-3, so 8.1×10−4<3.6×10−38.1\times10^{-4}<3.6\times10^{-3}. In ordinary notation, 0.00081<0.00360.00081<0.0036.

Watch out

Comparing coefficients alone can reverse the order. Also, compare indices only after checking that every coefficient satisfies 1≤A<101\le A<10. An expression such as 81×10−581\times10^{-5} needs rewriting first.

Try three new small values.

Quick Recap

  • Standard form is A×10nA\times10^n, with 1≤A<101\le A<10 and an integer index.
  • The coefficient carries the digits; the power of ten carries the scale.
  • Positive indices enlarge the coefficient; negative indices reduce it; index zero leaves it unchanged.
  • For positive numbers already in standard form, compare indices first, then coefficients if needed.

The opening larger number is 3.6×10−33.6\times10^{-3}, even though its coefficient is smaller.