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Comparing numbers in scientific notation

Two numbers are each written as a few digits times a power of ten. Which is bigger, and by how much?

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Comparing numbers in scientific notation

When you reach for it

Scientific notation is built for numbers too large or too small to read at a glance: the mass of a planet, the width of a cell. Writing a number that way is half the point. The other half is that two such numbers can then be compared almost without arithmetic. Written out in full, the masses of the Earth and the Moon in kilograms have 25 and 23 digits, and deciding which is larger means counting them. In scientific notation they are about

6 \times 10^{24} \text{ kg} \qquad\text{and}\qquad 7.3 \times 10^{22} \text{ kg}

and the exponents settle it at once: the Earth is heavier, even though 7.3 is more than 6.

Reach for this whenever you need to say which of two measurements is larger, put several in order, or estimate roughly how many times larger one is than another.

The procedure

Every number in this section is positive and written in scientific notation, a \times 10^{n} with 1 \le a < 10 and n an integer. Call a the leading part and n the exponent.

Why the exponent decides

The condition 1 \le a < 10 traps the number between two neighboring powers of ten. Multiplying all three parts of 1 \le a < 10 by the positive number 10^{n} keeps both inequalities pointing the same way:

10^{n} \le a \times 10^{n} < 10^{n+1}

So every number with exponent 3 lies from 10^{3} up to, but not including, 10^{4}; every number with exponent 4 lies from 10^{4} up to 10^{5}; and so on. These stretches do not overlap, and each one sits wholly above the one before it.

10^{2}
10^{3}
10^{4}
10^{5}
10^{6}
\text{exponent } 3
\text{exponent } 4
8 \times 10^{3}
3 \times 10^{4}
Figure 1Every number with exponent 3 lies between 10^{3} and 10^{4}, so all of them sit to the left of every number with exponent 4. That is why 8 \times 10^{3} < 3 \times 10^{4} although 8 > 3.

That is the whole reason the exponent is checked first. Take 8 \times 10^{3} and 3 \times 10^{4}. The first is less than 10^{4} and the second is at least 10^{4}, so

8 \times 10^{3} < 10^{4} \le 3 \times 10^{4}

and the leading parts 8 and 3 never came into it. The argument works for any two exponents. If m > n, then m \ge n + 1 because both are integers, and

a \times 10^{n} < 10^{n+1} \le 10^{m} \le b \times 10^{m}

Of two positive numbers in scientific notation, the one with the larger exponent is the larger number, whatever the leading parts are.

Negative exponents obey the same rule, compared as integers on the number line: since -3 > -5, any number with exponent -3 is larger than any with exponent -5, just as 10^{-3} = 0.001 is larger than 10^{-5} = 0.00001. (Zero and negative exponents says what a negative exponent means.)

When the exponents match

If the exponents are equal, a \times 10^{n} and b \times 10^{n} are the two leading parts multiplied by the same positive number, and multiplying by a positive number does not change which of two numbers is larger. So the leading parts decide, and they are compared like any decimals, place by place from the left:

4.83 \times 10^{6} > 4.8 \times 10^{6} \quad\text{because}\quad 4.83 > 4.80

To put several numbers in order, sort them by exponent first, then sort each group that shares an exponent by its leading parts.

How many times larger

“Which is larger” has a sharper form: how many times larger. That is a division: the leading parts are divided and the exponents subtracted (the quotient rule from the Laws of exponents; calculating in scientific notation covers it in full):

\frac{b \times 10^{m}}{a \times 10^{n}} = \frac{b}{a} \times 10^{m-n}

Even before dividing, the exponents pin the answer down. Each leading part is at least 1 and less than 10, so the quotient \displaystyle \frac{b}{a} is more than \displaystyle \frac{1}{10} and less than 10. Multiplying by 10^{m-n} gives the rule: if the exponents differ by d, the larger number is more than 10^{d-1} times and less than 10^{d+1} times the smaller. Exponents two apart mean somewhere between 10 and 1000 times. When the larger number also has the larger (or an equal) leading part, \displaystyle \frac{b}{a} \ge 1 and the ratio is at least 10^{d}; when its leading part is smaller, the ratio is less than 10^{d}.

Estimating with one digit

Exact leading parts are often more than a comparison needs. Rounding each leading part to one digit gives an estimate such as 2 \times 10^{8} in place of 1.65 \times 10^{8}, and estimates like that can be compared and divided in your head. If a leading part rounds up to 10, a power of ten moves into the exponent: 9.8 \times 10^{-3} \approx 10 \times 10^{-3} = 1 \times 10^{-2}.

The power of ten 10^{n} in a number’s scientific notation is called its order of magnitude, its size to the nearest factor of ten. Two quantities with orders of magnitude 10^{5} and 10^{8} are said to differ by three orders of magnitude: roughly a factor of 1000, within the limits just found.

Worked examples

Example 1

Comparing two numbers

Which number is larger in each pair? (a) 4.2 \times 10^{7} and 9.8 \times 10^{6}; (b) 6.05 \times 10^{-4} and 6.5 \times 10^{-4}; (c) 2 \times 10^{-3} and 8 \times 10^{-5}.

Solution

(a) The exponents are 7 and 6, and 7 > 6, so 4.2 \times 10^{7} is larger. The leading part 9.8 is bigger than 4.2, but that cannot help: 9.8 \times 10^{6} is less than 10^{7}. Written out, the numbers are 42{,}000{,}000 and 9{,}800{,}000.

(b) The exponents are both -4, so the leading parts decide. Comparing 6.05 and 6.50 from the left, the ones digits agree and the tenths are 0 against 5. So 6.5 \times 10^{-4} is the larger.

(c) The exponents are -3 and -5. On the number line -3 lies to the right of -5, so 2 \times 10^{-3} is the larger, although 8 > 2. Written out: 0.002 against 0.00008.

Example 2

Putting masses in order

The table gives approximate masses.

ObjectMass (kg)
the Earth6 \times 10^{24}
the Moon7.3 \times 10^{22}
Venus4.9 \times 10^{24}
Mars6.4 \times 10^{23}
a blue whale1.5 \times 10^{5}
an ant3 \times 10^{-6}

List the objects from lightest to heaviest.

Solution

Sort by exponent first. The exponents are 24, 22, 24, 23, 5, -6, and in increasing order the groups are: exponent -6, the ant; exponent 5, the blue whale; exponent 22, the Moon; exponent 23, Mars; exponent 24, the Earth and Venus.

Only the last group needs its leading parts: 4.9 < 6, so Venus is lighter than the Earth. From lightest to heaviest:

ant, blue whale, Moon, Mars, Venus, Earth.

Mars has the largest leading part of the four bodies, 6.4, and is still lighter than both Venus and the Earth, because its exponent is one smaller.

Example 3

When a number is not in scientific notation

Put these in order from least to greatest: 0.45 \times 10^{6}, 48 \times 10^{4}, 5.1 \times 10^{5}, 490{,}000.

Solution

Only 5.1 \times 10^{5} is in scientific notation, and the exponent rule rests on 1 \le a < 10, so every number is rewritten first.

  • 0.45 \times 10^{6} = 4.5 \times 10^{5}: the leading part is multiplied by 10 and the power of ten divided by 10, which leaves the number unchanged.
  • 48 \times 10^{4} = 4.8 \times 10^{5}, the same move the other way.
  • 490{,}000 = 4.9 \times 10^{5}.

Now all four have exponent 5, and the leading parts 4.5, 4.8, 5.1, 4.9 decide:

0.45 \times 10^{6} < 48 \times 10^{4} < 490{,}000 < 5.1 \times 10^{5}

Read as first printed, the exponents would have put 0.45 \times 10^{6} at the top. It is the smallest of the four.

Example 4

How many times heavier

(a) The Earth’s mass is about 6 \times 10^{24} kg and the Moon’s about 7.3 \times 10^{22} kg. About how many times as heavy as the Moon is the Earth? (b) A blue whale has a mass of about 1.5 \times 10^{5} kg and an ant about 3 \times 10^{-6} kg. How many times as heavy as the ant is the whale?

Solution

(a) The exponents differ by 2, so the answer is between 10 and 1000 times. The Earth’s leading part, 6, is smaller than the Moon’s, 7.3, so the ratio is below 10^{2} = 100. Dividing the leading parts and subtracting the exponents:

\frac{6 \times 10^{24}}{7.3 \times 10^{22}} = \frac{6}{7.3} \times 10^{2} \approx 0.82 \times 100 = 82

The Earth is about 80 times as heavy as the Moon. Since the masses were themselves rounded, a round answer is all the data supports.

(b) Subtracting a negative exponent adds: 5 - (-6) = 11. So

\frac{1.5 \times 10^{5}}{3 \times 10^{-6}} = \frac{1.5}{3} \times 10^{11} = 0.5 \times 10^{11} = 5 \times 10^{10}

The whale is about 5 \times 10^{10} times as heavy as the ant, fifty billion times. The exponents are 11 apart, and the whale’s leading part is the smaller one, so the ratio had to fall between 10^{10} and 10^{11}; it does.

Example 5

Estimating with one digit

The Pacific Ocean covers about 1.65 \times 10^{8} square kilometers, and Lake Superior about 8.2 \times 10^{4} square kilometers. (a) Without dividing, is the Pacific more than 1000 times as large? (b) Round each leading part to one digit and estimate how many times as large the Pacific is.

Solution

(a) Yes. The exponents differ by 4, and the Pacific’s leading part is the smaller one, so the ratio lies between 10^{3} and 10^{4}: more than 1000 times, guaranteed.

(b) To one digit, 1.65 rounds to 2 (its tenths digit is 6) and 8.2 rounds to 8. Then

\frac{2 \times 10^{8}}{8 \times 10^{4}} = \frac{2}{8} \times 10^{4} = 0.25 \times 10^{4} = 2500

The Pacific is a few thousand times as large. The given leading parts give \displaystyle \frac{1.65}{8.2} \times 10^{4} \approx 0.20 \times 10^{4} = 2000: the estimate is a little high, because rounding 1.65 up to 2 made it about a fifth larger, but it has the size right.

Example 6

One exponent apart, but nowhere near ten times

The universe is about 1.4 \times 10^{10} years old, and the Earth about 4.5 \times 10^{9} years old. The exponents differ by 1. Is the universe about ten times as old as the Earth?

Solution

A difference of 1 in the exponents only promises a ratio between 10^{0} = 1 and 10^{2} = 100. Here the larger number has the smaller leading part, 1.4 against 4.5, so the ratio is below 10^{1}. Dividing:

\frac{1.4 \times 10^{10}}{4.5 \times 10^{9}} = \frac{1.4}{4.5} \times 10^{1} \approx 0.31 \times 10 = 3.1

The universe is about 3 times as old as the Earth, not ten times. One order of magnitude apart is a rough description, and here the leading parts pull the ratio well below 10.

Boundaries

The rule needs true scientific notation. “Larger exponent, larger number” depends on 1 \le a < 10. With any other leading part it can fail: 0.3 \times 10^{5} has the larger exponent but is less than 8 \times 10^{4}. Rewrite first, as in Example 3.

The rule is for positive numbers. A negative number such as -2 \times 10^{5} is written as the negative of a number in scientific notation. Among negative numbers the one further from zero is the smaller, so the order reverses: -2 \times 10^{5} = -200{,}000 is less than -4 \times 10^{3} = -4000. Zero has no scientific notation at all, since no a with 1 \le a < 10 makes a \times 10^{n} equal to 0; it is simply less than every positive number.

The units must match. 4.5 \times 10^{3} meters and 2 \times 10^{5} centimeters cannot be compared by their exponents, because the powers of ten count different units. Converted, 2 \times 10^{5} centimeters is 2 \times 10^{3} meters, the shorter of the two.

Exponents alone give a ratio only to within a factor of ten either way. A difference of d in the exponents means more than 10^{d-1} times and less than 10^{d+1} times. To say more, the leading parts have to be divided, as in Example 6.

One-digit estimates are rough. Rounding a leading part to one digit can change it by up to half a unit, which is a large share of a small leading part: 1.5 rounds to 2, a third larger. An estimate settles which number is larger when they are far apart, and gives a ratio to about its first digit, as in Example 5.

“Order of magnitude” has a second meaning. Some books mean the power of ten a number is nearest to, measured in times. The number a \times 10^{n} is a times 10^{n}, and 10^{n+1} is \displaystyle \frac{10}{a} times the number; these are equal when a^{2} = 10, that is \displaystyle a = \sqrt{10} \approx 3.16. So in that sense 5 \times 10^{5} is nearest to 10^{6}, not 10^{5}, and the cut-off is 3.16, not 5. The two meanings agree whenever the leading part is below \displaystyle \sqrt{10}; it is worth saying which is meant.

Where this goes next

Measuring a size by its power of ten leads to Logarithms, and the scale in the figure, where each equal step multiplies by ten, is a logarithmic scale. The how-many-times question grows into calculating in scientific notation, and orders of magnitude are the working language of Estimating a huge quantity from small facts, where an enormous quantity is built from a chain of rough guesses and only the power of ten is expected to come out right.

Exercises

Which is larger

  1. Which is larger: 3.8 \times 10^{9} or 9.1 \times 10^{8}?
  2. Which is larger: 5.27 \times 10^{-3} or 5.3 \times 10^{-3}?
  3. Which is larger: 7 \times 10^{-8} or 1.2 \times 10^{-6}?
  4. Which is larger: 6.4 \times 10^{12} or 6.04 \times 10^{12}?
  5. Which is larger: 9.99 \times 10^{-1} or 1 \times 10^{0}?
  6. Which is larger, or are they equal: 2.5 \times 10^{4} or 25{,}000?

Putting numbers in order

  1. Order from least to greatest: 4.1 \times 10^{5}, 3.9 \times 10^{6}, 8.8 \times 10^{4}, 4.01 \times 10^{5}.
  2. Order from least to greatest: 6 \times 10^{-2}, 5.9 \times 10^{-1}, 6.1 \times 10^{-3}, 6 \times 10^{-3}.
  3. Order from least to greatest: 0.7 \times 10^{-4}, 65 \times 10^{-6}, 6.8 \times 10^{-5}, 0.000071.
  4. These lengths are approximate: the diameter of the Earth, 1.3 \times 10^{7} m; the distance from the Earth to the Moon, 3.8 \times 10^{8} m; the diameter of the Sun, 1.4 \times 10^{9} m; a speck of dust Elio measures under a microscope, 5 \times 10^{-5} m; the height of Mount Everest, 8.8 \times 10^{3} m. Order them from shortest to longest.
  5. Which is longer: 4.5 \times 10^{3} meters or 2 \times 10^{5} centimeters? There are 10^{2} centimeters in a meter.

How many times larger

  1. How many times as large as 2 \times 10^{6} is 8 \times 10^{9}?
  2. The mass of Jupiter is about 1.9 \times 10^{27} kg, and the mass of the Earth about 6 \times 10^{24} kg. About how many times as heavy as the Earth is Jupiter?
  3. The mass of the Sun is about 2 \times 10^{30} kg. About how many times as heavy as the Earth, at 6 \times 10^{24} kg, is the Sun?
  4. One online video has been watched about 9 \times 10^{7} times, and another about 2 \times 10^{6} times. Show that the first has been watched more than 40 times as often.
  5. Under a microscope, Deepika measures a pollen grain as 2.4 \times 10^{-5} m across and a strand of spider silk on the same slide as 6 \times 10^{-6} m across. How many times as wide as the silk is the pollen grain?

Estimating

  1. Round each leading part to one digit, and write the estimate in scientific notation: (a) 2.9 \times 10^{8}; (b) 6.2 \times 10^{-4}; (c) 4.47 \times 10^{11}; (d) 680.
  2. Estimate how many times as large 6.8 \times 10^{9} is as 2.1 \times 10^{6}, rounding each leading part to one digit first. Then divide the given leading parts and compare.
  3. Using the distance to the Moon and the height of Mount Everest from Exercise 10, each rounded to one digit, estimate how many Everests stacked on top of each other would reach the Moon.
  4. (a) What are the orders of magnitude of 9.5 \times 10^{6} and 1.1 \times 10^{6}? (b) How many times as large is the first? (c) Explain why sharing an order of magnitude does not make two numbers nearly equal.

Reasoning

  1. Explain why 1 \times 10^{7} is larger than 9.99 \times 10^{6}, even though 9.99 > 1.
  2. Is 0.3 \times 10^{5} larger or smaller than 8 \times 10^{4}? Explain why comparing the exponents alone does not settle it.
  3. Which is larger, -4 \times 10^{3} or -2 \times 10^{5}? Explain why the larger exponent does not win here.
  4. Mirela rounds 9.7 \times 10^{5} to one digit and writes 10 \times 10^{5}. Explain why that is not scientific notation, write it correctly, and say why the exponent changed.
  5. Explain why two positive numbers in scientific notation whose exponents differ by 3 are always more than 100 times apart and less than 10{,}000 times apart.
Answers
  1. 3.8 \times 10^{9}. The exponents are 9 and 8, and 9 > 8; the leading parts do not matter.
  2. 5.3 \times 10^{-3}. The exponents match, so compare 5.27 and 5.30 from the left: the ones digits agree, and the tenths are 2 against 3.
  3. 1.2 \times 10^{-6}. The exponents are -8 and -6, and -6 > -8.
  4. 6.4 \times 10^{12}. The exponents match, and 6.40 > 6.04 at the tenths place.
  5. 1 \times 10^{0}. The exponents are -1 and 0, and 0 > -1. Written out: 0.999 < 1.
  6. They are equal: 25{,}000 = 2.5 \times 10^{4}.
  7. 8.8 \times 10^{4}, 4.01 \times 10^{5}, 4.1 \times 10^{5}, 3.9 \times 10^{6}. Exponent 4 comes first and exponent 6 last; within exponent 5, 4.01 < 4.10.
  8. 6 \times 10^{-3}, 6.1 \times 10^{-3}, 6 \times 10^{-2}, 5.9 \times 10^{-1}. The exponents in increasing order are -3, -2, -1; within -3, 6 < 6.1.
  9. 65 \times 10^{-6}, 6.8 \times 10^{-5}, 0.7 \times 10^{-4}, 0.000071. Rewritten they are 6.5 \times 10^{-5}, 6.8 \times 10^{-5}, 7 \times 10^{-5} and 7.1 \times 10^{-5}, all with exponent -5.
  10. Speck of dust, Mount Everest, diameter of the Earth, distance to the Moon, diameter of the Sun. The exponents are -5, 3, 7, 8, 9, all different, so the leading parts are never needed.
  11. 4.5 \times 10^{3} meters. In meters the other length is 2 \times 10^{5} \div 10^{2} = 2 \times 10^{3} meters, and with matching exponents 4.5 > 2.
  12. 4 \times 10^{3} times, that is 4000: \displaystyle \frac{8}{2} \times 10^{9-6} = 4 \times 10^{3}.
  13. About 320 times, roughly 300: \displaystyle \frac{1.9}{6} \times 10^{27-24} \approx 0.32 \times 1000 = 320.
  14. About 330{,}000 times: \displaystyle \frac{2}{6} \times 10^{30-24} \approx 0.33 \times 10^{6} = 3.3 \times 10^{5}.
  15. \displaystyle \frac{9 \times 10^{7}}{2 \times 10^{6}} = 4.5 \times 10^{1} = 45, and 45 > 40.
  16. 4 times: \displaystyle \frac{2.4}{6} \times 10^{-5-(-6)} = 0.4 \times 10^{1} = 4.
  17. (a) 3 \times 10^{8}. (b) 6 \times 10^{-4}. (c) 4 \times 10^{11}: to one digit only the tenths digit 4 matters, so 4.47 rounds down to 4. (d) 680 = 6.8 \times 10^{2}, which rounds to 7 \times 10^{2}.
  18. About 3.5 \times 10^{3}, that is 3500: \displaystyle \frac{7}{2} \times 10^{9-6} = 3.5 \times 10^{3}. With the given leading parts, \displaystyle \frac{6.8}{2.1} \times 10^{3} \approx 3.2 \times 10^{3}, so the estimate is a little high but has the right size.
  19. About 44{,}000. The Moon’s distance rounds to 4 \times 10^{8} m and Everest to 9 \times 10^{3} m, and \displaystyle \frac{4}{9} \times 10^{8-3} \approx 0.44 \times 10^{5} = 4.4 \times 10^{4}. (Without rounding, \displaystyle \frac{3.8}{8.8} \times 10^{5} \approx 4.3 \times 10^{4}.)
  20. (a) Both are 10^{6}. (b) \displaystyle \frac{9.5}{1.1} \approx 8.6 times. (c) The order of magnitude ignores the leading part, which can be anything from 1 up to almost 10, so two numbers with the same exponent can be almost ten times apart.
  21. 9.99 \times 10^{6} is less than 10 \times 10^{6}, which is 10^{7}. Every number with exponent 6 is below 10^{7}, and 1 \times 10^{7} is 10^{7} itself. Written out: 9{,}990{,}000 < 10{,}000{,}000.
  22. Smaller. 0.3 \times 10^{5} is not in scientific notation, because 0.3 < 1; rewritten it is 3 \times 10^{4}, which is less than 8 \times 10^{4}. The exponent rule only applies once both leading parts are at least 1 and less than 10.
  23. -4 \times 10^{3}, which is -4000, is larger than -2 \times 10^{5} = -200{,}000. The exponent rule compares sizes, and among negative numbers the larger size is further left on the number line, so it is the smaller number.
  24. A leading part must be less than 10, and 10 is not. Since 10 = 10^{1}, 10 \times 10^{5} = 10^{1} \times 10^{5} = 1 \times 10^{6}: the extra ten joins the power of ten, which raises the exponent by one.
  25. Call the numbers a \times 10^{n} and b \times 10^{n+3}. The ratio is \displaystyle \frac{b}{a} \times 10^{3}. Since b \ge 1 and a < 10, \displaystyle \frac{b}{a} > \frac{1}{10}, so the ratio is more than \displaystyle \frac{1}{10} \times 10^{3} = 100. Since b < 10 and a \ge 1, \displaystyle \frac{b}{a} < 10, so the ratio is less than 10 \times 10^{3} = 10{,}000.
kg1128 · updated 2026-10-09 · CC-BY-SA-4.0