Question

using significant figures
to assure that data is as precise and meaningful as possible, scientists report measured results in terms of significant figures. significant figures in a measurement consist of all of the digits known with certainty, plus the first uncertain or estimated digit. knowing which digits are significant prevents a scientist from reporting a value that is more precise than the equipment used to gather the data. for example, a scientist calculating the density of a substance with a mass of 5.90 grams and 3.89 milliliters might obtain the following on their calculator: 1.51670951156812 g/ml. however, this value has more digits than the values used in the calculation. an appropriate result would be 1.52 g/ml. to round values correctly, you first need to know which digits are significant.
rules for determining significant digits:
· nonzero integers are always significant. for example, 275 has three significant digits.
· zeros appearing between nonzero digits (captive zeros) are significant. for example, the value 40.7 has three significant figures. the value 87009 has five.
· zeros that precede nonzero digits (leading zeros) are not significant. for example, the value 0.095897 has five significant figures. the value 0.0009 has one significant figure.
· zeros at the end of a number (trailing zeros) are significant only if the number contains a decimal point. for example, the value 85.00 has four significant figures, and the value 9.000000000 has ten. the value 2000 has only one significant figure. however, the value written as 2000. (with a decimal) has four significant figures.
· exact numbers, such as conversion factors, counted numbers, and defined values, are assumed to have no uncertainty and therefore have infinite significant figures.
analyze write each measurement in the correct box according to the number of significant figures it has.
0.217 l 30.080 kg 0.00009 cm 23000 m 784.000 g 5.003 ml
mathematical operations using significant figures
calculating experimental results usually requires a variety of mathematical operations, such as adding, subtracting, multiplying, or dividing. calculations also frequently use several types of measurements, all with varying levels of accuracy. the uncertainty of the final result must be accounted for as precisely as possible. in order to do this, scientists have developed rules for determining the correct number of significant figures that should be included when reporting calculated results. in general, the final reported value should be no more precise than the least precise measurement used in the calculation.

Explanation:

0.217 L

• Leading zeros are not significant.
• Non - zero integers are significant.
• So, 0.217 has three significant figures.

30.080 kg

• Non - zero integers are significant.
• Captive zeros (between non - zero digits) are significant.
• Trailing zero in a number with a decimal is significant.
• So, 30.080 has five significant figures.

0.00009 cm

• Leading zeros are not significant.
• Only the non - zero digit 9 is significant.
• So, 0.00009 has one significant figure.

23000 m

• Without a decimal, trailing zeros are not significant.
• Only 2 and 3 are significant.
• So, 23000 has two significant figures.

784.000 g

• Non - zero integers are significant.
• Trailing zeros in a number with a decimal are significant.
• So, 784.000 has six significant figures.

5.003 mL

• Non - zero integers are significant.
• Captive zeros (between non - zero digits) are significant.
• So, 5.003 has four significant figures.

Answer:

• One significant figure: \(0.00009\space cm\)
• Two significant figures: \(23000\space m\)
• Three significant figures: \(0.217\space L\)
• Four significant figures: \(5.003\space mL\)
• Five significant figures: \(30.080\space kg\)
• Six significant figures: \(784.000\space g\)