Chapter 5 · Exponents and scientific notation

Choosing Units of Appropriate Size

Learn to pick a measurement unit that fits the size of a quantity, so very large and very small measurements are easy to read, write and compare.

12 min read

Would you say that a new pencil is about 0.000190.00019 km long? It is true, but the number is hard to read. Most people would say about 1919 cm instead.

A measurement is a number together with a unit. A unit is a fixed amount that you count with, such as a meter or a second. The number tells you how many of that unit match the thing you measure.

You can write one length in many ways. The pencil is about 1919 cm long. It is also about 0.190.19 m, about 190190 mm and about 0.000190.00019 km long. All four statements name the same length.

In this lesson you learn to choose a unit of appropriate size: a unit that makes the number in a measurement easy to read, say and compare.

19 cm=0.19 m=190 mm=0.00019 km19\ \text{cm} = 0.19\ \text{m} = 190\ \text{mm} = 0.00019\ \text{km}

Why do the numbers differ when the pencil stays the same? A millimeter is smaller than a centimeter, so more millimeters fit along the pencil: 190190 of them instead of 1919. A kilometer is far longer than the pencil, so only a tiny part of one fits: 0.000190.00019 of a kilometer.

So a smaller unit gives a larger number, and a larger unit gives a smaller number. This holds for any positive measurement, whenever one unit is a fixed number of times as large as the other. Every pair of units in this lesson is like that. The length itself never changes. Only the way you write it changes.

That tells you how to choose. Pick a unit that is close to the size of the thing you measure, so the number is neither huge nor tiny. Sometimes two units both work: 1919 cm and 190190 mm are both easy to read.

Key concept

Match the unit to the size of the quantity

A measurement is a number times a unit. When you change the unit, the number changes, but the quantity stays the same.

A unit of appropriate size is close to the size of the quantity, so the number is easy to read and compare. A useful target is a number between 1 and 1,000. This is a guide, not a law. When no unit you may use reaches the target, choose the unit that comes closest. When you compare several measurements, write them all in one unit.

To see the size of a nonzero measurement, write it in scientific notation: a coefficient (the number in front) that is at least 1 and less than 10, times a power of 10, followed by the unit. Only then does the power of 10 show how large the quantity is compared with that unit.

To choose a unit, you need to know how big each unit is. The table lists six metric units of length. The meter (m) is the reference unit in this lesson. Each other unit in the table equals the meter times a power of 1010. The symbol μm\mu\text{m} uses the Greek letter mu and is read micrometer.

These unit facts are exact, because they are definitions: one kilometer is exactly 10310^{3} m. Measurements are different. The pencil is only about 1919 cm long. So keep the word about with a measured value in every unit you use.

Six metric units of length, each compared with the meter
UnitSize of one unitOne meter equalsRoughly the size of
kilometer (km)
10310^{3} m
10−310^{-3} km
the distance you walk in 12 minutes
meter (m)
11 m
11 m
one long step
centimeter (cm)
10−210^{-2} m
10210^{2} cm
the width of a fingernail
millimeter (mm)
10−310^{-3} m
10310^{3} mm
the thickness of a dime
micrometer (μm\mu\text{m})
10−610^{-6} m
106 μm10^{6}\ \mu\text{m}
the length of a small bacterium
nanometer (nm)
10−910^{-9} m
10910^{9} nm
the width of a sugar molecule

Rewriting a measurement in a new unit. The third column of the table gives equal amounts, such as 1 m=103 mm1\ \text{m} = 10^{3}\ \text{mm}. Because the two amounts are equal, you can swap one for the other and the length stays the same. For example:

4×10−2 m=4×10−2×103 mm=4×101 mm=40 mm4 \times 10^{-2}\ \text{m} = 4 \times 10^{-2} \times 10^{3}\ \text{mm} = 4 \times 10^{1}\ \text{mm} = 40\ \text{mm}

Multiplying by 10310^{3} means multiplying by 1010 three times, so the exponent (the raised number on the 1010) goes up by 33: here −2+3=1-2 + 3 = 1. Multiplying by a negative power such as 10−310^{-3} means dividing by 1010 three times, so the exponent goes down by 33.

You already estimate very large and very small quantities as a single digit times a power of 1010. That power of 1010 is your clue for choosing a unit.

Worked example

A very small length

A human hair is about 7×10−57 \times 10^{-5} m wide. Which unit of length is of appropriate size for this width: the millimeter, the micrometer or the nanometer? Rewrite the width in that unit.

  1. Read the size. The coefficient 7 is at least 1 and less than 10, so this nonzero measurement is in scientific notation and its power of 10 shows its size: 7 hundred-thousandths of a meter. A meter is far too large a unit for this width, so test the smaller units.

    7×10−5 m=0.00007 m7 \times 10^{-5}\ \text{m} = 0.00007\ \text{m}

  2. Test millimeters. The table says that one meter equals exactly 10 to the power 3 millimeters, so swap m for that many mm. The width stays the same because the two amounts are equal. Multiplying by 10 three times raises the exponent by 3: -5 + 3 = -2.

    7×10−5 m=7×10−5×103 mm=7×10−2 mm=0.07 mm7 \times 10^{-5}\ \text{m} = 7 \times 10^{-5} \times 10^{3}\ \text{mm} = 7 \times 10^{-2}\ \text{mm} = 0.07\ \text{mm}

  3. Test micrometers. The table says that one meter equals exactly 10 to the power 6 micrometers, so swap m for that many micrometers. Multiplying by 10 six times raises the exponent by 6: -5 + 6 = 1.

    7×10−5 m=7×10−5×106 μm=7×101 μm=70 μm7 \times 10^{-5}\ \text{m} = 7 \times 10^{-5} \times 10^{6}\ \mu\text{m} = 7 \times 10^{1}\ \mu\text{m} = 70\ \mu\text{m}

  4. Test nanometers. The table says that one meter equals exactly 10 to the power 9 nanometers, so swap m for that many nm. Multiplying by 10 nine times raises the exponent by 9: -5 + 9 = 4.

    7×10−5 m=7×10−5×109 nm=7×104 nm=70,000 nm7 \times 10^{-5}\ \text{m} = 7 \times 10^{-5} \times 10^{9}\ \text{nm} = 7 \times 10^{4}\ \text{nm} = 70{,}000\ \text{nm}

  5. Compare the three numbers with the target of 1 to 1,000. The number 0.07 is less than 1, so the millimeter is too large a unit. The number 70,000 is far above 1,000, so the nanometer is too small a unit. The number 70 is between 1 and 1,000, so the micrometer fits.

Answer

The micrometer is the unit of appropriate size: a human hair is about 70 μm70\ \mu\text{m} wide, which is 7×101 μm7 \times 10^{1}\ \mu\text{m} in scientific notation.

Worked example

A very large length

Earth's diameter is about 1.27×1071.27 \times 10^{7} m. Choose the unit of appropriate size from the table and rewrite the diameter in that unit.

  1. Read the size. The coefficient 1.27 is at least 1 and less than 10, so the power of 10 shows the size of this nonzero measurement: more than ten million meters. The meter is far too small a unit, so take the largest unit in the table, the kilometer.

    1.27×107 m=12,700,000 m1.27 \times 10^{7}\ \text{m} = 12{,}700{,}000\ \text{m}

  2. Swap the unit. The table says that one meter equals exactly 10 to the power -3 kilometers, so swap m for that many km. Multiplying by 10 to the power -3 divides by 10 three times, so the exponent falls by 3: 7 - 3 = 4.

    1.27×107 m=1.27×107×10−3 km=1.27×104 km1.27 \times 10^{7}\ \text{m} = 1.27 \times 10^{7} \times 10^{-3}\ \text{km} = 1.27 \times 10^{4}\ \text{km}

  3. Write the number out and check the direction. A kilometer is larger than a meter, so the number must shrink. It does: from 12,700,000 to 12,700. The digits 1, 2 and 7 stay the same, so the measurement is exactly as precise as before.

    1.27×104 km=12,700 km1.27 \times 10^{4}\ \text{km} = 12{,}700\ \text{km}

  4. Judge the result. The number 12,700 is above the target of 1,000, but no unit in the table is larger than the kilometer. So the kilometer comes closest, and 12,700 km is much easier to read than 12,700,000 m.

Answer

The kilometer is the best choice in the table: Earth's diameter is about 12,70012{,}700 km, which is 1.27×1041.27 \times 10^{4} km in scientific notation.

Worked example

One unit for several masses

Here are three masses in kilograms: a housefly, about 1.2×10−51.2 \times 10^{-5} kg; a grain of rice, about 2×10−52 \times 10^{-5} kg; a honeybee, about 1×10−41 \times 10^{-4} kg. You may use kilograms, grams or milligrams, where 1 kg=103 g1\ \text{kg} = 10^{3}\ \text{g} and 1 g=103 mg1\ \text{g} = 10^{3}\ \text{mg} exactly. Choose one unit of appropriate size for all three masses and rewrite them in that unit. Then order the masses from least to greatest.

  1. Read the sizes. Each coefficient is at least 1 and less than 10, so the powers of 10 show the sizes of these nonzero measurements. Every mass is a tiny part of a kilogram, so the kilogram is too large a unit. Use one smaller unit for all three, because you can compare numbers directly only when their units match.

  2. Test grams on the largest mass, the honeybee. One kilogram equals exactly 10 to the power 3 grams, so swap kg for that many g. The exponent rises by 3: -4 + 3 = -1. Even this largest number is less than 1, so the gram is still too large a unit for all three.

    1×10−4 kg=1×10−4×103 g=1×10−1 g=0.1 g1 \times 10^{-4}\ \text{kg} = 1 \times 10^{-4} \times 10^{3}\ \text{g} = 1 \times 10^{-1}\ \text{g} = 0.1\ \text{g}

  3. Find how many milligrams equal one kilogram. Swap kg for 10 to the power 3 grams, then swap g for 10 to the power 3 milligrams. Both swaps use exact facts, so the result is exact. Multiplying by 10 three times and then three more times gives the exponent 3 + 3 = 6.

    1 kg=103 g=103×103 mg=106 mg1\ \text{kg} = 10^{3}\ \text{g} = 10^{3} \times 10^{3}\ \text{mg} = 10^{6}\ \text{mg}

  4. Rewrite the housefly. Swap kg for 10 to the power 6 milligrams. The mass stays the same because the two amounts are equal, and the exponent rises by 6: -5 + 6 = 1.

    1.2×10−5 kg=1.2×10−5×106 mg=1.2×101 mg=12 mg1.2 \times 10^{-5}\ \text{kg} = 1.2 \times 10^{-5} \times 10^{6}\ \text{mg} = 1.2 \times 10^{1}\ \text{mg} = 12\ \text{mg}

  5. Rewrite the grain of rice and the honeybee the same way. The exponents become -5 + 6 = 1 and -4 + 6 = 2.

    2×10−5×106 mg=2×101 mg=20 mg,1×10−4×106 mg=1×102 mg=100 mg2 \times 10^{-5} \times 10^{6}\ \text{mg} = 2 \times 10^{1}\ \text{mg} = 20\ \text{mg}, \quad 1 \times 10^{-4} \times 10^{6}\ \text{mg} = 1 \times 10^{2}\ \text{mg} = 100\ \text{mg}

  6. Check and order. The numbers 12, 20 and 100 are all between 1 and 1,000. A milligram is smaller than a kilogram, so each number grew, as it must. The units match, so you can order the masses by their numbers.

    12 mg<20 mg<100 mg12\ \text{mg} < 20\ \text{mg} < 100\ \text{mg}

Answer

The milligram is the unit of appropriate size for all three masses. From least to greatest, they are the housefly at about 1212 mg, the grain of rice at about 2020 mg and the honeybee at about 100100 mg.

The takeaway. A measurement is a number and a unit working together. When the number is hard to read, change the unit, not the quantity. Choose a unit close to the size of what you measure, so the number lands between 1 and 1,000 when a unit you may use allows it. Swap equal amounts to rewrite the measurement, and check the direction: for a positive measurement, a smaller unit gives a larger number. Keep the power of 10, the unit and the word about in view the whole way. When you write a nonzero measurement in scientific notation, use a coefficient that is at least 1 and less than 10.

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