Chapter 5 · Exponents and scientific notation

Compare, Estimate, and Find How Many Times as Much

Use scientific notation to compare numbers by size, estimate very large and very small quantities, and find how many times as much one quantity is as another.

18 min read

Scientific notation writes a number as a×10na \times 10^n. In that form, 10n10^n is a power of 10, the exponent nn is an integer, and the coefficient aa is the number that multiplies the power of 10. For a nonzero number, standard scientific notation uses a coefficient whose size is at least 11 and less than 1010. Every quantity in this lesson is positive, so here 1≤a<101 \le a < 10.

In this lesson you will use that form for three jobs. You will compare numbers, estimate very large and very small quantities, and say how many times as much one quantity is as another.

Each step up of 1 in the exponent multiplies the value by 10. For example, 10610^6 is 10 times 10510^5, and 10−3=0.00110^{-3} = 0.001 is 10 times 10−4=0.000110^{-4} = 0.0001. So the powers of 10 split the positive numbers into power-of-ten ranges. Each range starts at a power of 10 and stops just before the next one.

Which range holds a×10na \times 10^n? Start from 1≤a<101 \le a < 10 and multiply every part by 10n10^n. Because 10n10^n is positive, both inequality signs keep their direction. Also, 10×10n=10n+110 \times 10^n = 10^{n+1}. So a positive number in standard scientific notation is at least 10n10^n and less than 10n+110^{n+1}:

10n≤a×10n<10n+1when 1≤a<10 and n is an integer10^{n} \le a \times 10^{n} < 10^{n+1} \quad \text{when } 1 \le a < 10 \text{ and } n \text{ is an integer}
The exponent tells which power-of-ten range a number is in
NumberWritten outAt leastLess than
9.9×1059.9 \times 10^{5}
990,000990{,}000
105=100,00010^{5} = 100{,}000
106=1,000,00010^{6} = 1{,}000{,}000
1.2×1061.2 \times 10^{6}
1,200,0001{,}200{,}000
106=1,000,00010^{6} = 1{,}000{,}000
107=10,000,00010^{7} = 10{,}000{,}000
7.5×10−47.5 \times 10^{-4}
0.000750.00075
10−4=0.000110^{-4} = 0.0001
10−3=0.00110^{-3} = 0.001
1.2×10−31.2 \times 10^{-3}
0.00120.0012
10−3=0.00110^{-3} = 0.001
10−2=0.0110^{-2} = 0.01

Key concept

The exponent picks the power-of-ten range

For a positive number a×10na \times 10^n in standard scientific notation, the exponent picks the power-of-ten range. The coefficient places the number inside that range. Each step up of 1 in the exponent multiplies by 10. So when nn is an integer and kk is a whole number, going from 10n10^n up to 10n+k10^{n+k} takes kk steps. That multiplies by 10k10^k, not by kk.

10n+k=10k×10n10^{n+k} = 10^{k} \times 10^{n}

To compare two numbers in scientific notation, first check three conditions. Both numbers must be positive, both must be in standard scientific notation, and measurements must use the same unit. Then:

  • If the exponents differ, the number with the greater exponent is greater. It sits in a higher range, and every number in a higher range is greater than every number in a lower one. In the table, 9.9×1059.9 \times 10^5 has the larger coefficient, yet it is less than 1.2×1061.2 \times 10^6.
  • If the exponents are equal, the number with the greater coefficient is greater. Both coefficients are multiplied by the same positive power of 10.

With negative exponents, −3>−7-3 > -7, so 10−310^{-3} is greater than 10−710^{-7}. If a coefficient is not at least 1 and less than 10, rewrite the number first. For example, 45×103=4.5×10445 \times 10^3 = 4.5 \times 10^4, which is greater than 2×1042 \times 10^4. That holds even though 45×10345 \times 10^3 shows a smaller exponent. The rule is only for positive numbers. For negative numbers, a greater size means a lesser number.

Worked example

Order three small numbers

Order these numbers from least to greatest: 4.7×10−54.7 \times 10^{-5}, 8.1×10−68.1 \times 10^{-6}, 2.3×10−52.3 \times 10^{-5}.

  1. Check the three conditions. All three numbers are positive. Each coefficient (4.7, 8.1 and 2.3) is at least 1 and less than 10, and there are no units to match. So the comparison rule applies.

  2. Compare exponents first: −6 < −5, because −6 is farther left on the number line. A number with exponent −6 is less than 10⁻⁵, and a number with exponent −5 is at least 10⁻⁵. So 8.1 × 10⁻⁶ is the least, even though 8.1 is the largest coefficient.

    8.1×10−6<10−5≤2.3×10−58.1 \times 10^{-6} < 10^{-5} \le 2.3 \times 10^{-5}

  3. The other two numbers share the exponent −5, so compare their coefficients: 2.3 < 4.7. Multiplying both sides by the same positive number, 10⁻⁵, keeps the inequality pointing the same way.

    2.3×10−5<4.7×10−52.3 \times 10^{-5} < 4.7 \times 10^{-5}

  4. Check by writing each number out as a decimal.

    0.0000081<0.000023<0.0000470.0000081 < 0.000023 < 0.000047

Answer

From least to greatest, the order is 8.1×10−68.1 \times 10^{-6}, 2.3×10−52.3 \times 10^{-5}, 4.7×10−54.7 \times 10^{-5}.

An estimate is a simpler value close to the exact one. The symbol ≈\approx means is approximately equal to. To estimate a very large or very small positive quantity:

  1. Write it in standard scientific notation.
  2. Round the coefficient to the nearest whole number. Keep the exponent and the unit. Rounding changes the value by at most half of 10n10^n.
  3. If the coefficient rounds up to 10, rewrite 10×10n10 \times 10^n as 1×10n+11 \times 10^{n+1}. That keeps the coefficient less than 10.

The result is a single digit times a power of 10. For example, light in a vacuum travels exactly 299,792,458 meters per second. Since 299,792,458=2.99792458×108≈3×108299{,}792{,}458 = 2.99792458 \times 10^8 \approx 3 \times 10^8, light travels about 3×1083 \times 10^8 m/s. Once you round, write ≈\approx instead of ==, because an estimate gives up some precision to be simpler.

Worked example

Estimate a very small number

Estimate 0.0000009680.000000968 as a single digit times a power of 10.

  1. Write the number in standard scientific notation. The first nonzero digit, 9, is in the seventh place after the decimal point. So the power of 10 is 10⁻⁷, and the coefficient is 9.68, which is at least 1 and less than 10.

    0.000000968=9.68×10−70.000000968 = 9.68 \times 10^{-7}

  2. Round the coefficient to the nearest whole number: 9.68 is closer to 10 than to 9. Keep the exponent. Switch from = to ≈, because rounding changes the value slightly.

    9.68×10−7≈10×10−79.68 \times 10^{-7} \approx 10 \times 10^{-7}

  3. A coefficient of 10 is not less than 10, so rewrite it. Multiplying 10⁻⁷ by 10 moves the exponent one step up, which gives 10⁻⁶.

    10×10−7=10−7+1=1×10−610 \times 10^{-7} = 10^{-7+1} = 1 \times 10^{-6}

  4. Check that the estimate keeps the size: 1 × 10⁻⁶ is 0.000001, which is close to 0.000000968.

    1×10−6=0.0000011 \times 10^{-6} = 0.000001

Answer

The estimate is 0.000000968≈1×10−60.000000968 \approx 1 \times 10^{-6}, or about one millionth.

Saying how many times as much one quantity is as another is a multiplicative comparison. Let AA and BB be positive quantities in the same unit. Then AA is kk times as much as BB when A=k×BA = k \times B, so k=A÷Bk = A \div B. The units cancel, so kk has no unit. In this lesson you divide the larger quantity by the smaller one, so kk is at least 1.

Write the larger quantity as a×10ma \times 10^m and the smaller as b×10nb \times 10^n, both in standard scientific notation. Divide the coefficients and the powers of 10 separately. That works because dividing by b×10nb \times 10^n is the same as dividing by bb and then by 10n10^n. Neither divisor is 0: b≥1b \ge 1, and 10n10^n is positive.

For the powers of 10, count steps. By the comparison rule, the larger number cannot have the smaller exponent, so m≥nm \ge n. Getting from 10n10^n up to 10m10^m takes m−nm - n steps, each a factor of 10. So 10m÷10n=10m−n10^m \div 10^n = 10^{m-n}. If a÷ba \div b is less than 1, rewrite the answer in standard scientific notation.

a×10mb×10n=ab×10m−n(b≠0)\dfrac{a \times 10^{m}}{b \times 10^{n}} = \dfrac{a}{b} \times 10^{m-n} \qquad (b \neq 0)

Worked example

How many times as long?

One length is 6×10−46 \times 10^{-4} m and another is 3×10−73 \times 10^{-7} m. How many times as long is the first length as the second?

  1. Both lengths are positive and in meters, and the first is greater because −4 > −7. Divide the first by the second. The meters cancel, and you can divide the coefficients and the powers of 10 separately.

    6×10−4 m3×10−7 m=63×10−410−7\dfrac{6 \times 10^{-4}\,\text{m}}{3 \times 10^{-7}\,\text{m}} = \dfrac{6}{3} \times \dfrac{10^{-4}}{10^{-7}}

  2. Divide the coefficients. The divisor 3 is not 0, so the division is defined.

    63=2\dfrac{6}{3} = 2

  3. Divide the powers of 10 by counting the steps from 10⁻⁷ up to 10⁻⁴. Going from −7 to −6 to −5 to −4 takes 3 steps, and each step is a factor of 10. Subtracting the exponents gives the same count.

    10−410−7=10−4−(−7)=10−4+7=103\dfrac{10^{-4}}{10^{-7}} = 10^{-4-(-7)} = 10^{-4+7} = 10^{3}

  4. Multiply the two results. The coefficient 2 is already at least 1 and less than 10, so no rewriting is needed.

    2×103=2,0002 \times 10^{3} = 2{,}000

  5. Check by multiplying back: 2,000 times the shorter length should equal the longer one.

    2,000×3×10−7 m=6,000×10−7 m=6×10−4 m2{,}000 \times 3 \times 10^{-7}\,\text{m} = 6{,}000 \times 10^{-7}\,\text{m} = 6 \times 10^{-4}\,\text{m}

Answer

The first length is exactly 2×103=2,0002 \times 10^3 = 2{,}000 times as long as the second.

Worked example

Estimate, then compare masses

The Sun's mass is about 1.989×10301.989 \times 10^{30} kg, and Earth's mass is about 5.972×10245.972 \times 10^{24} kg. Estimate each mass as a single digit times a power of 10. Then estimate how many times as much mass the Sun has as Earth.

  1. Round each coefficient to the nearest whole number, keeping the exponents and the unit.

    1.989×1030 kg≈2×1030 kg,5.972×1024 kg≈6×1024 kg1.989 \times 10^{30}\,\text{kg} \approx 2 \times 10^{30}\,\text{kg}, \quad 5.972 \times 10^{24}\,\text{kg} \approx 6 \times 10^{24}\,\text{kg}

  2. Both masses are positive and in kilograms, and the Sun's mass is greater because 30 > 24. Divide the Sun's mass by Earth's mass. The kilograms cancel.

    2×1030 kg6×1024 kg=26×10301024\dfrac{2 \times 10^{30}\,\text{kg}}{6 \times 10^{24}\,\text{kg}} = \dfrac{2}{6} \times \dfrac{10^{30}}{10^{24}}

  3. Divide the coefficients and subtract the exponents. Two sixths is one third, which is a repeating decimal, so round it to two decimal places.

    26=13≈0.33,10301024=1030−24=106\dfrac{2}{6} = \dfrac{1}{3} \approx 0.33, \qquad \dfrac{10^{30}}{10^{24}} = 10^{30-24} = 10^{6}

  4. The coefficient 0.33 is less than 1, so rewrite it as 3.3 × 10⁻¹. Then 10⁻¹ × 10⁶ is one step down from 10⁶, which is 10⁵.

    0.33×106=3.3×10−1×106=3.3×1050.33 \times 10^{6} = 3.3 \times 10^{-1} \times 10^{6} = 3.3 \times 10^{5}

  5. Compare with the unrounded values: 1.989 ÷ 5.972 ≈ 0.333, which gives about 333,000. The estimate of 330,000 is close.

    1.9895.972≈0.333,0.333×106=333,000\dfrac{1.989}{5.972} \approx 0.333, \qquad 0.333 \times 10^{6} = 333{,}000

Answer

The Sun has about 3.3×1053.3 \times 10^5 times as much mass as Earth, which is roughly 330,000330{,}000 times as much.

Takeaway: The exponent tells which power-of-ten range a number is in, and each step up of 1 in the exponent multiplies by 10. So for positive numbers in standard scientific notation, compare exponents first. Compare coefficients only when the exponents match. Estimate by rounding the coefficient to a single digit. Find how many times as much by dividing quantities in the same unit, where an exponent difference of kk means a factor of 10k10^k. Next, you will choose units of an appropriate size for measurements.

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Choosing Units of Appropriate Size

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