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B1.1 · Represent and compare very large and small numbers using scientific notation

Learn to represent and compare very large and small numbers using scientific notation through clear examples and targeted practice.

Ontario Grade 8 Mathematics

Number

Representing and comparing numbers with scientific notation

A long row of zeros can make a number hard to read and copy. For example, 63,200,000 is a large number, while 0.000072 is a small positive number. Scientific notation gives a shorter way to write numbers like these. It uses place value and powers of 10. You will first connect the notation to decimal-point movement, then use it to represent and compare numbers.

What you will learn

  • Explain how place value connects to very large and very small numbers.
  • Write numbers in scientific notation and change them back to ordinary form.
  • Compare numbers written in scientific notation.

1. Place value and the basic form

A digit's place tells its value. In 4,500,000, the 4 represents four million. In 0.000045, the 4 is in the hundred-thousandths place. Zeros help show where the other digits belong.
Scientific notation writes a number as a coefficient multiplied by a power of 10. The coefficient is the number before the multiplication sign. It must be at least 1 and less than 10. An exponent is the small raised number that tells which power of 10 is used. For example, the exponent in 10310^3 is 3.
For a large number, move the decimal point left until the coefficient is at least 1 and less than 10. For a positive number smaller than 1, move the point right until the coefficient meets the same rule. Count each place the point moves.
a×10n,1≤a<10a\times10^n,\quad 1\leq a<10
  • Scientific notation has a coefficient multiplied by a power of 10.
  • The coefficient is at least 1 and less than 10.
  • A whole number has an understood decimal point at its right end.

2. What the exponent tells us

For large numbers, moving the decimal point left to make the coefficient also means the coefficient must be multiplied by a positive power of 10 to keep the original value. For example, 6.326.32 is much smaller than 63,200,000, so the power of 10 restores the size.
For small positive numbers, the exponent is negative. A negative exponent represents a reciprocal place value: 10−410^{-4} means one part out of 10410^4, or 1 divided by 10,000. It does not mean that a negative number is being multiplied. The number remains positive when its coefficient is positive.
To return to ordinary form, move the decimal point right for a positive exponent and left for a negative exponent. Move it as many places as the exponent indicates. Add zeros when needed. For example, 2.5×10−32.5\times10^{-3} becomes 0.0025.
10−n=110n10^{-n}=\frac{1}{10^n}
  • A positive exponent is used for numbers at least 10.
  • A negative exponent represents a reciprocal place value and is used for positive numbers smaller than 1.
  • When changing back to ordinary form, move the point in the direction opposite to the move used to make the coefficient.

3. Compare numbers in scientific notation

First make sure each number has a coefficient from 1 up to, but not including, 10. Then compare the exponents. With coefficients in this range, the number with the greater exponent is greater. This rule works for negative exponents too: 10−210^{-2} is greater than 10−310^{-3} because one hundredth is greater than one thousandth.
If the exponents are equal, compare the coefficients. The number with the greater coefficient is greater. For instance, when both exponents are 4, 6.2×1046.2\times10^4 is greater than 3.9×1043.9\times10^4 because 6.2 is greater than 3.9.
You can check a comparison by writing both numbers in ordinary form. This is a useful way to catch an error, especially when the exponents are negative.
a×10m>b×10nif m>na\times10^m>b\times10^n\quad\text{if }m>n
  • Compare exponents first.
  • If the exponents match, compare coefficients.
  • A negative exponent closer to zero represents a greater positive number than a more negative exponent, when the coefficients are in the required range.

4. A careful routine

To convert an ordinary number, locate its decimal point. Move it until the leading part is a coefficient from 1 to less than 10. Count the moves, then choose the exponent sign: a number at least 10 needs a positive exponent; a positive number less than 1 needs a negative exponent.
To convert back, use the exponent to move the point the other way. To compare, check the coefficient rule, compare exponents, and compare coefficients only when the exponents match. If you are unsure, check by writing each number in ordinary form.
  • Count every decimal-point move once.
  • Use the original number's size to choose the exponent sign.
  • Check that the final notation represents the original value.

Decimal-point movement and scientific notation

Ordinary numberCoefficientPoint moves to make coefficientScientific notation
72,0007.24 places left7.2×1047.2\times10^4
0.00363.63 places right3.6×10−33.6\times10^{-3}

Worked example

Represent a large number

Write 63,200,000 in scientific notation.
  1. Make the coefficient
    Start with the decimal point at the end of the whole number. Move it left until the leading part is at least 1 and less than 10. This gives a coefficient of 6.32.
    63,200,000→6.3263{,}200{,}000\rightarrow6.32
  2. Count and write
    The point moved seven places left. The original number is large, so use a positive exponent. Multiplying the coefficient by this power of 10 restores the original size.
    63,200,000=6.32×10763{,}200{,}000=6.32\times10^7
Answer: 6.32×1076.32\times10^7
Check: Moving the decimal point seven places right from 6.32 gives 63,200,000.

Worked example

Represent a small decimal

Write 0.000072 in scientific notation.
  1. Make the coefficient
    Move the decimal point right until the number is at least 1 and less than 10. The coefficient is 7.2.
    0.000072→7.20.000072\rightarrow7.2
  2. Choose the exponent
    The point moved five places right. The original number is positive and smaller than 1, so use a negative exponent. That exponent represents the reciprocal place value needed to return to the small original number.
    0.000072=7.2×10−50.000072=7.2\times10^{-5}
Answer: 7.2×10−57.2\times10^{-5}
Check: Moving the decimal point five places left from 7.2 gives 0.000072.

Worked example

Compare two large numbers

Which number is greater: 5.8×1065.8\times10^6 or 6.1×1056.1\times10^5?
  1. Check the coefficients
    Both coefficients are at least 1 and less than 10, so the exponents can be compared directly.
    1≤5.8<10,1≤6.1<101\leq5.8<10,\quad1\leq6.1<10
  2. Compare the exponents
    The first exponent is 6 and the second is 5. The greater exponent represents a larger place-value range, so the first number is greater.
    6>56>5
  3. State the comparison
    The exponents differ, so there is no need to compare the coefficients. The first number is greater.
    5.8×106>6.1×1055.8\times10^6>6.1\times10^5
Answer: 5.8×1065.8\times10^6 is greater.
Check: In ordinary form, the numbers are 5,800,000 and 610,000. Since 5,800,000 is greater than 610,000, the comparison is correct.

Common mistakes and how to avoid them

Using a positive exponent for a positive number smaller than 1.
Correction: When you move the decimal point right to make the coefficient, use a negative exponent.
Leaving the coefficient as 12 or 0.4.
Correction: Adjust the point so the coefficient is at least 1 and less than 10. The exponent must also match the changed point.
Thinking a more negative exponent makes a positive number larger.
Correction: 10−510^{-5} is smaller than 10−210^{-2} because one hundred-thousandth is less than one hundredth.
Comparing coefficients before checking exponents.
Correction: Compare exponents first. Compare coefficients only when the exponents are equal.

Lesson summary

  • Scientific notation writes a number as a coefficient multiplied by a power of 10.
  • The coefficient is at least 1 and less than 10.
  • Positive exponents represent large numbers; negative exponents represent reciprocal place values for positive numbers smaller than 1.
  • To compare numbers in scientific notation, compare exponents first and coefficients next if the exponents match.

Check your understanding

Question 1

Which is the correct scientific notation for 0.00091?
  1. 9.1×10−49.1\times10^{-4}
  2. 9.1×1049.1\times10^4
  3. 0.91×10−30.91\times10^{-3}
  4. 91×10−591\times10^{-5}
Show answer and explanation
9.1×10−49.1\times10^{-4}
Move the decimal point four places right to get a coefficient of 9.1. The original number is positive and less than 1, so the exponent is negative. The other choices either use the wrong exponent or do not have a coefficient from 1 to less than 10.

Question 2

Which number is greater: 3.4×1083.4\times10^8 or 8.2×1078.2\times10^7?
  1. 3.4×1083.4\times10^8
  2. 8.2×1078.2\times10^7
  3. They are equal.
  4. There is not enough information.
Show answer and explanation
3.4×1083.4\times10^8
The coefficients are in the required range. Since 8 is greater than 7, the first number is greater.

Question 3

What ordinary number is represented by 2.5×10−32.5\times10^{-3}?
  1. 0.0025
  2. 0.025
  3. 250
  4. 2,500
Show answer and explanation
0.0025
A negative exponent means the decimal point moves left when returning to ordinary form. Move it three places left from 2.5 to get 0.0025.

Key terms

Coefficient
The number before the multiplication sign in scientific notation. It is at least 1 and less than 10.
Exponent
The small raised number that tells which power of 10 is used. A negative exponent represents a reciprocal place value; for example, 10−4=110410^{-4}=\frac{1}{10^4}.
Scientific notation
A way to write a number as a coefficient multiplied by a power of 10.
Place value
The value a digit has because of its position in a number.

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