Comparing Numbers in Scientific Notation

Learn to compare and order numbers in scientific notation: exponents first, then coefficients, and how many times bigger one number is.

Intermediate20 min18 problems
Practice Problems

Definition

To compare numbers in scientific notation a×10na \times 10^n (with 1≤a<101 \le a < 10), you almost never need to write out all the zeros. Follow three steps:
  1. 1.Normalize both numbers, so each coefficient is between 11 and 1010. For example, 45×103=4.5×10445 \times 10^3 = 4.5 \times 10^4.
  2. 2.Compare the exponents. The number with the greater exponent is the greater number: 3.1×106>9.8×1053.1 \times 10^6 > 9.8 \times 10^5, because 6>56 > 5.
  3. 3.If the exponents are equal, compare the coefficients: 4.7×108>4.2×1084.7 \times 10^8 > 4.2 \times 10^8, because 4.7>4.24.7 > 4.2.
The exponent wins first because each step up in the exponent multiplies the number by 1010, and a coefficient can never make up for that: even 9.99×1059.99 \times 10^5 is less than 1×1061 \times 10^6.
Negative exponents follow the same rule, you just compare them like negative numbers: −3>−5-3 > -5, so 10−3>10−510^{-3} > 10^{-5} (0.001>0.000010.001 > 0.00001).
How many times bigger? When the coefficients are equal, divide the powers of ten by subtracting the exponents:
108105=108−5=103=1000\frac{10^8}{10^5} = 10^{8-5} = 10^3 = 1000
Each difference of 11 in the exponent is one order of magnitude (a factor of 1010).

Try it now

Which number is greater: 8.9×1068.9 \times 10^6 or 5.2×1075.2 \times 10^7?

Worked Examples

Which is greater: 3.1×1063.1 \times 10^6 or 9.8×1059.8 \times 10^5?
1

Check that both are normalized

Both coefficients, 3.13.1 and 9.89.8, are between 11 and 1010 → Ready to compare
2

Compare the exponents

6>56 > 5 → 10610^6 is ten times 10510^5
3

Decide (the coefficients do not matter here)

3.1×106=3,100,0003.1 \times 10^6 = 3{,}100{,}000 and 9.8×105=980,0009.8 \times 10^5 = 980{,}000 → 3.1×106>9.8×1053.1 \times 10^6 > 9.8 \times 10^5

Common Mistakes

Comparing coefficients first: thinking 9.8×105>3.1×1069.8 \times 10^5 > 3.1 \times 10^6 because 9.8>3.19.8 > 3.1

Why it's wrong: The power of ten decides the size. 10610^6 is ten times 10510^5, and no coefficient below 1010 can make up for that.

Correct: Compare exponents first: 6>56 > 5, so 3.1×106>9.8×1053.1 \times 10^6 > 9.8 \times 10^5. Look at the coefficients only when the exponents are equal.

Comparing without normalizing: thinking 45×103>4.6×10445 \times 10^3 > 4.6 \times 10^4 because 45>4.645 > 4.6

Why it's wrong: 45×10345 \times 10^3 is not in scientific notation, so its exponent does not tell its real size.

Correct: Normalize first: 45×103=4.5×10445 \times 10^3 = 4.5 \times 10^4. Then 4.5<4.64.5 < 4.6, so 45×103<4.6×10445 \times 10^3 < 4.6 \times 10^4.

Thinking 10−5>10−310^{-5} > 10^{-3} because 5>35 > 3

Why it's wrong: Negative exponents are compared like negative numbers: −5-5 is less than −3-3.

Correct: 10−5=0.0000110^{-5} = 0.00001 and 10−3=0.00110^{-3} = 0.001, so 10−3>10−510^{-3} > 10^{-5}. The exponent closer to zero wins.

Thinking 10610^6 is twice 10310^3, or 33 more than 10310^3

Why it's wrong: The exponents differ by 33, but each step in the exponent multiplies by 1010, it does not add.

Correct: 106103=106−3=103\frac{10^6}{10^3} = 10^{6-3} = 10^3, so 10610^6 is 10001000 times 10310^3.

Interactive Visual

Which is bigger? Normalize first, then compare exponents, then coefficients

Compare
45×10345 \times 10^{3}?4.6×1044.6 \times 10^{4}

Which sign goes in the box?

10⁴10⁵10⁶AB
Powers of 10: each step to the right is 10 times bigger.

Sizes in the universe: tap from smallest to largest (meters)

Compare

Tap the numbers from smallest to largest.

Some are not in scientific notation yet: normalize them first.

10⁻²10⁰10²10⁴10⁶10⁸10¹⁰10¹²BDAC
Powers of 10: each step to the right is 10 times bigger.

For scale

10⁻⁵10⁻³10⁻¹10¹10³10⁵10⁷10⁹1234
Powers of 10: each step to the right is 10 times bigger.
1virus (m):1×10−71 \times 10^{-7}= 0.0000001
2human hair, width (m):8×10−58 \times 10^{-5}= 0.00008
3Mount Everest, height (m):8.85×1038.85 \times 10^{3}= 8850
4Earth to Moon (m):3.84×1083.84 \times 10^{8}= 384000000

Interactive Sandbox

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Practice Problems

18 problems
Problem 1 of 18
Easy
Which number is greater: 8.9×1068.9 \times 10^6 or 5.2×1075.2 \times 10^7?

Why It Matters

Science is full of numbers that are too big or too small to compare by eye:
  • Space: The Earth is about 1.27×1071.27 \times 10^7 m wide, while the distance from the Earth to the Sun is about 1.5×10111.5 \times 10^{11} m. The exponents tell you at once that the distance is about 10410^4 (ten thousand) times larger.
  • Biology: A red blood cell (7×10−67 \times 10^{-6} m) is much larger than a hydrogen atom (1×10−101 \times 10^{-10} m), because −6>−10-6 > -10.
  • Computing: A terabyte (101210^{12} bytes) holds 10001000 times as much as a gigabyte (10910^9 bytes).
  • Medicine: Doses and concentrations such as 5×10−45 \times 10^{-4} g or 8×10−68 \times 10^{-6} g must never be mixed up.
Comparing exponents first is fast and safe. Counting zeros in 0.00000700.0000070 and 0.000000010.00000001 is slow and easy to get wrong.

Real World Applications

Biology: Bacteria and Viruses

Microbiologists compare tiny lengths written with negative exponents to choose the right microscope.

Example:

A light microscope shows things down to about 2×10−72 \times 10^{-7} m. A bacterium of 2×10−62 \times 10^{-6} m is larger (−6>−7-6 > -7), so it can be seen. Most viruses are smaller and need an electron microscope.

1Try It Yourself

A bacterium is 2×10−62 \times 10^{-6} m long. A virus is 1×10−71 \times 10^{-7} m across.

How many times longer is the bacterium than the virus?

Step 1: Write the mathematical expression

Compare the powers of ten, then include the coefficients:

Computing: Storage Sizes

Storage is measured in bytes: a gigabyte is $10^9$ bytes and a terabyte is $10^{12}$ bytes. Comparing exponents tells you how many files fit.

Example:

A photo of 3×1063 \times 10^6 bytes is much smaller than a video of 3×1093 \times 10^9 bytes: same coefficient, and the exponents differ by 33, so the video is 10001000 times larger.

2Try It Yourself

A hard drive holds 2×10122 \times 10^{12} bytes. One movie file is 2×1092 \times 10^9 bytes.

How many movies fit on the drive?

Step 1: Write the mathematical expression

The coefficients are equal, so compare only the powers of ten:

Physics: Laser Wavelengths

Light wavelengths are tiny and often written in different forms, for example in nanometers ($10^{-9}$ m). To compare them, normalize first.

Example:

Blue light has a wavelength of about 4.5×10−74.5 \times 10^{-7} m and red light about 7×10−77 \times 10^{-7} m. Same exponent, so 7>4.57 > 4.5 shows red light has the longer wavelength.

3Try It Yourself

A lab sheet lists a red laser as 650×10−9650 \times 10^{-9} m and a green laser as 5.3×10−75.3 \times 10^{-7} m.

Which laser has the longer wavelength?

Step 1: Choose the right answer

Normalize 650×10−9650 \times 10^{-9}, then compare:

Key Takeaways

  • 1Normalize first: every coefficient must satisfy 1≤a<101 \le a < 10 (for example 45×103=4.5×10445 \times 10^3 = 4.5 \times 10^4)
  • 2Compare exponents first: the greater exponent gives the greater number
  • 3Only when the exponents are equal, compare the coefficients
  • 4Negative exponents compare like negative numbers: 10−3>10−510^{-3} > 10^{-5} because −3>−5-3 > -5
  • 5To order a list, sort by exponent, then by coefficient within the same exponent
  • 6With equal coefficients, 10m10^m is 10m−n10^{m-n} times as large as 10n10^n: each step in the exponent is one order of magnitude

Frequently Asked Questions

A normalized coefficient is always less than 1010, while one step up in the exponent multiplies by 1010. So 9.99×105=999,0009.99 \times 10^5 = 999{,}000 is still less than 1×106=1,000,0001 \times 10^6 = 1{,}000{,}000.
A normalized coefficient is always less than 1010, while one step up in the exponent multiplies by 1010. So 9.99×105=999,0009.99 \times 10^5 = 999{,}000 is still less than 1×106=1,000,0001 \times 10^6 = 1{,}000{,}000.
Normalize it before comparing. Move the decimal point until the coefficient is between 11 and 1010 and change the exponent to match: 0.8×105=8×1040.8 \times 10^5 = 8 \times 10^4, 45×103=4.5×10445 \times 10^3 = 4.5 \times 10^4.
Yes, for positive numbers. A normalized number with a negative exponent is less than 11 (for example 9.9×10−1=0.999.9 \times 10^{-1} = 0.99), and one with exponent 00 or more is at least 11.
It means about 102=10010^2 = 100 times bigger. The order of magnitude is roughly the exponent, so a difference of 22 in the exponents means a factor of about 100100.

Glossary

Scientific notation
A number written as a×10na \times 10^n with 1≤a<101 \le a < 10 and nn a whole number
Coefficient
The number aa in front of the power of ten, for example 4.64.6 in 4.6×1044.6 \times 10^4
Exponent
The power nn in 10n10^n; it is compared first
Normalize
Rewrite a number so its coefficient is between 11 and 1010: 45×103=4.5×10445 \times 10^3 = 4.5 \times 10^4
Order of magnitude
A factor of 1010; numbers whose exponents differ by 33 differ by about three orders of magnitude (10310^3)

Formula Card

Comparison order

normalize→exponents→coefficients\text{normalize} \rightarrow \text{exponents} \rightarrow \text{coefficients}

Make sure $1 \le a < 10$, compare the exponents, and compare the coefficients only if the exponents are equal.

How many times as large

10m10n=10m−n\frac{10^m}{10^n} = 10^{m-n}

With equal coefficients, the larger number is $10^{m-n}$ times the smaller one.

Today's challengeOne problem, three tries

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