Numbers

Scientific Notation In physics numbers can be very large and very small. Scientific notation uses powers of 10 to represent decimal places. Positive powers for large numbers: 456000 = 4.56 x 105 Negative powers for small numbers: 0.00753 = 7.53 x 10-3 Write a number in scientific notation with only one non-zero value to the left of the decimal place.

Order of Magnitude You Lecture Hall Faraday West NIU 5’9” = 1.75 m ≈ 100 m 14 m ≈ 101 m 80 m ≈ 102 m 2000 m ≈ 103 m Each of these lengths is different by about one order of magnitude

Map Lengths Huskie Stadium DeKalb City DeKalb County Illinois 100 m ≈ 102 m 4 km = 4000 m ≈ 103 m 30 km = 30,000 m ≈ 104 m 400 km ≈ 105 m Mapping software uses scaling factors, about two steps per order of magnitude

Accuracy The smallest unit on a measuring device sets the accuracy. In general, a measurement is only as accurate as the smallest unit. Significant figures are a guide to the accuracy of a measurement.

Significant Figures Any value is expressed in some number of digits. The number of digits (without left side zeroes) is the number of significant figures. With no decimal point, skip right side zeroes. 38 2 digits, 2 significant figures 5.06 3 digits, 3 significant figures 0.0041 5 digits, 2 significant figures 7,000. 4 digits, 4 significant figures 2,000 4 digits, 1 significant figure

Using Significant Figures Add or Subtract Keep the significant figures to decimal place of the least accurate value, rounding as needed. 4.361 + 14.2 = 18.6 12000 + 364 = 12000 Multiply or Divide Keep the same number of significant figures as the value with the fewest, rounding as needed. 4.361  14.2 = 61.9 12000  364 = 4.4  106

Absolute Uncertainty Measure 50.0 cm. There are three significant figures. The smallest figure suggests an accuracy of 0.1 cm. This is also equal to 1 mm. The absolute uncertainty has the same type of units as the measurement.

Percent Uncertainty Measure 50.0 cm. Compare 0.1 cm to 50.0 cm. The ratio is 0.1/50.0 = 0.002. Multiply by 100 % to get 0.2 %. The percent uncertainty has no units, and is either a pure number or a percent.