2.2 Measured Numbers and Significant Figures Length is measured by observing the marked lines at the end of a ruler. The last digit in your measurement is an estimate, obtained by visually dividing the space between the marked lines. Learning Goal Identify a number as measured or exact; determine the number of significant figures in a measured number.

Measured Numbers Measured numbers are the numbers obtained when you measure a quantity such as your height, weight, or temperature. To write a measured number, observe the numerical values of the marked lines. estimate the value of the number between the marks. The estimated number is the final number in your measured number.

Measured Numbers for Length The lengths of the objects are measured as A. 4.5 cm. B. 4.55 cm. C. 3.0 cm.00

Significant Figures In a measured number, the significant figures (SFs) are all the digits, including the estimated digit. Significant figures are used to represent the amount of error associated with a measurement. are all nonzero digits and zeros between digits. are not zeros that act as placeholders before digits. are zeros at the end of a decimal number. Core Chemistry Skill Counting Significant Figures

Measured Numbers: Significant Figures A number is a significant figure (SF) if it is/has

Scientific Notation and Significant Zeros When one or more zeros in a large number are significant, they are shown clearly by writing the number in scientific notation. In this book, we place a decimal point after a significant zero at the end of a number. For example, if only the first zero in the measurement 300 m is significant, the measurement is written as 3.0 × 102 m.

Scientific Notation and Significant Zeros Zeros at the end of large standard numbers without a decimal point are not significant. 400 000 g is written with one SF as 4 × 105 g. 850 000 m is written with two SFs as 8.5 × 105 m. Zeros at the beginning of a decimal number are used as placeholders and are not significant. 0.000 4 s is written with one SF as 4 × 10−4 s. 0.000 0046 g is written with two SFs as 4.6 × 10−6 g.

Study Check Identify the significant and nonsignificant zeros in each of the following numbers, and write them in the correct scientific notation. A. 0.002 650 m B. 43.026 g C. 1 044 000 L

Solution Identify the significant and nonsignificant zeros in each of the following numbers, and write them in the correct scientific notation. A. 0.002 650 m is written as 2.650 × 10−3 m. four SFs The zeros preceding the 2 are not significant. The digits 2, 6, 5 are significant. The zero in the last decimal place is significant. B. 43.026 g is written as 4.3026 × 101 g. five SFs The zeros between nonzero digits or at the end of decimal numbers are significant.

Solution Identify the significant zeros and nonsignificant zeros in each of the following numbers, and write them in the correct scientific notation. C. 1 044 000 L is written as 1.044 × 106 L. four SFs The zeros between nonzero digits are significant. The zeros at end of a number with no decimal point are not significant.

Exact Numbers Exact numbers are not measured and do not have a limited number of significant figures. not used to find the number of significant figures in a calculated answer. numbers obtained by counting. 8 cookies definitions that compare two units. 6 eggs definitions in the same measuring system. 1 qt = 4 cups 1 kg = 1 000 g

Exact Numbers Examples of exact numbers include the following:

Study Check Identify the numbers below as measured or exact, and give the number of significant figures in each measured number. A. 3 coins B. The diameter of a circle is 7.902 cm. C. 60 min = 1 h

Solution Identify the numbers below as measured or exact, and give the number of significant figures in each measured number. 3 coins is a counting number and, therefore, is an exact number. The diameter of a circle is 7.902 cm. This is a measured number and the zero is significant, so it contains four SF. The relationship 60 min = 1 h is exact by definition.