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Chemistry LibreTexts

1.4: Scientific Notation

  • Page ID
    213133
  • Learning Objectives
    • Express a large number or a small number in scientific notation.
    • Carry out arithmetical operations and express the final answer in scientific notation

    Chemists often work with numbers that are exceedingly large or small. For example, entering the mass in grams of a hydrogen atom into a calculator would require a display with at least 24 decimal places, many of which are occupied by placeholder zeroes, whose sole purpose is to hold a decimal place. A system called scientific notation avoids much of the tedium and awkwardness of manipulating numbers with large or small magnitudes by eliminating the placeholder zeroes. In scientific notation, these numbers are expressed in the format that is shown below.

    Coefficient × 10power

    The coefficient must be greater than or equal to 1 and less than 10 (1 ≤ coefficient < 10) and is determined by selecting all of the non-zero digits, 1-9, as well as any significant zeroes, in a given number.  A decimal is inserted so that the coefficient falls into the acceptable range.  The power is a positive or negative whole number. A positive power is used when the given number is "large," or greater than 1, and a negative power is used when the given number is "small," or less than 1.  The numerical value of the power is determined by how many places the decimal must shift from its location in the given number to where it must be inserted in the coefficient .  The number 10 is called the base because it is this number that is raised to the power. Although a base number may have values other than 10, the base number in scientific notation is always 10.

    Example \(\PageIndex{1}\): Expressing Numbers in Scientific Notation

    Convert each number to scientific notation.

    1. 637.8
    2. 0.000479
    3. 21,067,000,000

    Solutions

    Answer a:  6.378 × 102
    Explanation:  In order to derive a coefficient that is greater than or equal to 1 and less than 10 from the given number, all of the digits are selected, as they all have non-zero values, and a decimal is inserted between the 6 and the 3.  Therefore, the resultant coefficient has a value of 6.378.  Because the decimal point was moved two places and the given value is "large" (> 1), the power is 2.

    Answer b:  4.79 × 10−4
    Explanation:  In order to derive a coefficient that is greater than or equal to 1 and less than 10 from the given number, the digits "479" are selected.  The zeroes are not included because they are located before the first non-zero digit.  A decimal is inserted between the 4 and the 7, and the resultant coefficient has a value of 4.79.  Because the decimal point was moved four places and the given value is "small" (< 1), the power is −4.

    Answer c:  2.1067 × 1010
    Explanation:  In order to derive a coefficient that is greater than or equal to 1 and less than 10 from the given number, the digits "21067" are selected.  The zero is included because it is located between non-zero digits.  A decimal is inserted between the 2 and the 1, and the resultant coefficient has a value of 2.1067.  Because the decimal point was moved ten places and the given value is "large" (> 1), the power is 10.

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