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Teacher Guide: 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.

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Class quiz

10 questions on Scientific Notation. Students join with a name, you see everyone's score.

For Teachers

Learning Objectives
  • Compare two numbers in scientific notation by their exponents
  • Compare coefficients when the exponents are equal
  • Normalize a number before comparing it
  • Compare and order numbers with negative exponents
  • Estimate how many times larger one number is than another using powers of ten
Prerequisites
  • • Scientific notation basics: the form a×10na \times 10^n (lesson Scientific Notation)
  • • Writing numbers in scientific notation (lesson Writing Numbers in Scientific Notation)
  • • Comparing negative numbers on a number line
  • • Comparing decimals such as 4.5 and 4.6
Discussion Starters
  • 1. Why can you compare 3.1×1063.1 \times 10^6 and 9.8×1059.8 \times 10^5 without writing out any zeros?
  • 2. Is 9.99×1059.99 \times 10^5 or 1×1061 \times 10^6 greater? How would you convince a friend?
  • 3. Why is 10−310^{-3} greater than 10−510^{-5}, even though 5>35 > 3?
  • 4. A scientist says one star is three orders of magnitude brighter than another. What does that mean in numbers?
Common Misconceptions

The number with the larger coefficient is larger

Remediation: Write 9.8×1059.8 \times 10^5 and 3.1×1063.1 \times 10^6 in standard (decimal) form side by side: 980,000980{,}000 vs 3,100,0003{,}100{,}000. Ask which digit position each starts in.

A larger absolute value of a negative exponent means a larger number

Remediation: Place 10−110^{-1}, 10−210^{-2}, 10−310^{-3} as 0.10.1, 0.010.01, 0.0010.001 on a number line, and let students see the values shrink as the exponent goes down.

Numbers like 45×10345 \times 10^3 can be compared by their exponents as they are

Remediation: Show that 45×10345 \times 10^3 and 4.5×1044.5 \times 10^4 are the same number. Make "normalize first" the first line of every comparison.

Differentiation Ideas

For Struggling Students:

  • • Start with pairs that have the same coefficient and different exponents
  • • Let students write both numbers in standard (decimal) form to check each comparison
  • • Use a powers-of-ten ladder to place each number before comparing

For On-Level Students:

  • • Compare pairs with negative exponents and with equal exponents
  • • Normalize numbers like 38×10538 \times 10^5 or 0.7×10−30.7 \times 10^{-3} before comparing
  • • Order lists of four to six real measurements

For Advanced Students:

  • • Estimate how many times larger one quantity is, including unequal coefficients
  • • Find which pairs of real sizes (atom, cell, Earth, Sun) differ by more than ten orders of magnitude
  • • Explain why a normalized number with exponent nn always lies between 10n10^n and 10n+110^{n+1}
Standards Alignment
  • 8.EE.A.3 (CCSS.MATH.CONTENT.8.EE.A.3)

    Use numbers expressed as a single digit times an integer power of 10 to estimate very large or very small quantities, and to express how many times as much one is than the other

  • 8.EE.A.4 (CCSS.MATH.CONTENT.8.EE.A.4)

    Perform operations with numbers expressed in scientific notation and interpret scientific notation generated by technology

Lesson Resources
  • visualScientific Notation Comparator

    Normalize, compare exponents, then coefficients, and order sizes in the universe on a powers-of-ten ladder

  • activityUniverse Size Line

    Students order cards with sizes from atoms to galaxies, written in scientific notation, along a wall line

  • worksheetCompare and Order

    Pairs and lists to compare, including numbers that must be normalized first and negative exponents

Lesson Content

Everything students see: definition, examples, common mistakes, applications. Tap to open.

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).

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.

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

Why does the exponent matter more than the coefficient?

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.

What if one number is not in proper scientific notation?

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.

Is a number with a negative exponent always smaller than one with a positive exponent?

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.

What does "two orders of magnitude bigger" mean?

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.

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