Standard form

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レッスンノート

What is standard form?

  • Standard form is a way of writing very large or very small numbers using powers of 10.
  • A number in standard form is written as A × 10n, where A is a number with 1 ≤ A < 10 and n is an integer.
  • In the expression A × 10n, A is called the coefficient (or significand) and n is called the exponent.
  • Standard form is also called scientific notation or standard index form.
  • It is useful because it avoids writing long strings of zeros.
  • On a scientific calculator, standard form is usually shown in SCI mode.
  • The exponent n can be positive, negative, or zero.
  • If the number is negative, a minus sign is placed before A, for example −3.5 × 102.

Writing large numbers in standard form

  • To write a large number in standard form, move the decimal point left until you have a number between 1 and 10.
  • The number of places you moved the decimal point becomes the positive exponent.
  • For example, 350 = 3.5 × 102 because the decimal point moves 2 places left.
  • Another example: 6,022,000,000,000,000,000,000,000 = 6.022 × 1023.
  • If there is no decimal point shown, it is at the end of the number.
  • The exponent tells you how many places the decimal point has moved.

Writing small numbers in standard form

  • To write a small number (between 0 and 1) in standard form, move the decimal point right until you have a number between 1 and 10.
  • The number of places you moved the decimal point becomes the negative exponent.
  • For example, 0.5 = 5 × 10-1 because the decimal point moves 1 place right.
  • Another example: 0.00000000000000000000000000000000000016 = 1.6 × 10-35.
  • A negative exponent means the original number is less than 1.
  • The more places you move the decimal point, the larger the absolute value of the negative exponent.

Converting from standard form to ordinary numbers

  • To convert from standard form to an ordinary number, look at the exponent.
  • If the exponent is positive, move the decimal point in A that many places to the right.
  • If the exponent is negative, move the decimal point in A that many places to the left.
  • For example, 3.5 × 102 = 350.
  • For example, 5 × 10-1 = 0.5.
  • If the exponent is 0, the number is just A, for example 7 × 100 = 7.

Ordering numbers in standard form

  • Numbers in standard form can be compared easily by looking at their exponents.
  • A number with a larger exponent is larger than a number with a smaller exponent (when the numbers are positive).
  • If the exponents are the same, compare the coefficients A.
  • For example, 3 × 104 > 2 × 104 because 3 > 2.
  • For example, 2 × 105 > 9 × 104 because 5 > 4.
  • For negative numbers, the rules reverse: a number with a larger exponent is more negative (smaller).
  • Subtracting exponents gives an estimate of the number of orders of magnitude between two numbers.

Multiplying numbers in standard form

  • To multiply two numbers in standard form, multiply the coefficients and add the exponents.
  • Formula: (A × 10m) × (B × 10n) = (A × B) × 10m+n.
  • For example, (3 × 104) × (2 × 105) = (3 × 2) × 104+5 = 6 × 109.
  • If the product of the coefficients is not between 1 and 10, adjust it and change the exponent.
  • For example, (5 × 103) × (4 × 102) = 20 × 105 = 2 × 106.
  • Always check that the final coefficient is between 1 and 10.

Dividing numbers in standard form

  • To divide two numbers in standard form, divide the coefficients and subtract the exponents.
  • Formula: (A × 10m) ÷ (B × 10n) = (A ÷ B) × 10m−n.
  • For example, (8 × 106) ÷ (2 × 103) = (8 ÷ 2) × 106−3 = 4 × 103.
  • If the quotient of the coefficients is not between 1 and 10, adjust it and change the exponent.
  • For example, (3 × 104) ÷ (6 × 102) = 0.5 × 102 = 5 × 101.
  • Always check that the final coefficient is between 1 and 10.

Standard form in science contexts

  • Standard form is used in science to write very large distances, such as distances in space.
  • For example, the distance from the Earth to the Sun is about 1.5 × 1011 metres.
  • It is also used for very small sizes, such as the sizes of cells.
  • For example, a typical cell might be about 1 × 10-5 metres across.
  • Scientists, mathematicians, and engineers use standard form to simplify calculations with extreme numbers.
  • Standard form helps avoid errors when counting zeros.

Calculator and computer notation

  • Scientific calculators often display numbers in standard form using SCI mode.
  • Computers and calculators may use E notation, where E means 'times ten raised to the power of'.
  • For example, 6.022 × 1023 is written as 6.022E23 or 6.022e23.
  • For example, 1.6 × 10-35 is written as 1.6E-35 or 1.6e-35.
  • Sometimes the positive power is shown with a plus sign, like 6.022E+23.
  • E notation is common in computer output but is discouraged in published documents by some style guides.

Key points to remember

  • Standard form is A × 10n with 1 ≤ A < 10 and n an integer.
  • Positive exponent means a large number (≥ 10).
  • Negative exponent means a small number (between 0 and 1).
  • To convert to ordinary form, move the decimal point according to the exponent.
  • To multiply, multiply coefficients and add exponents.
  • To divide, divide coefficients and subtract exponents.
  • Always adjust the coefficient to be between 1 and 10 if needed.

スライド

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練習問題

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  1. 1.Which of these numbers is written correctly in standard form?

    Easy
    • A3.5 × 102
    • B35 × 101
    • C350 × 100
    • D0.35 × 103

  2. 2.Write 0.5 in standard form.

    Easy
    • A5 × 10-1
    • B5 × 101
    • C0.5 × 100
    • D1 × 10-2

  3. 3.What is 6.022 × 1023 written in E notation?

    Medium
    • A6.022E23
    • B6.022E-23
    • C6022E23
    • D6.022E+24

  4. 4.Which of the following are written in correct standard form? (select all that apply)

    Medium
    • A2.5 × 103
    • B10 × 104
    • C1.0 × 100
    • D0.5 × 102
    • E9.99 × 10-5

  5. 5.In standard form, the exponent must be an integer.

    Easy

    True or false?

  6. 6.Match each ordinary number to its standard form.

    Medium
    • 0.0002
    • 2000
    • 0.02
    • 200
    • 2 × 10-4
    • 2 × 103
    • 2 × 10-2

  7. 7.Arrange these numbers in standard form in ascending order (smallest first).

    Medium
    • 3.5 × 102
    • 3.5 × 10-1
    • 3.5 × 100
    • 3.5 × 103

  8. 8.Arrange these values in standard form in ascending order (smallest first).

    Medium
    • 3.1 × 10-2
    • 9.4 × 10-3
    • 1.2 × 100
    • 5.0 × 10-1

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