Scientific Notation and Significant Figures

Scientific Notation Scientist often have to manipulate very large or small numbers. For example, an electron’s mass is 0.000 000 000 000 000 000 000 000 000 000 910 938 356 kg. It becomes very troublesome to write so many zeroes in standard notation.

Scientific Notation To overcome this issue, we use scientific notation to represent these numbers. A more condensed version of the number system by eliminating all repetitive zeroes and writing it as a multiple of 10. Uses a number between 1 and 10 (must be less than 10) multiplied by a power in the form of 10n. Therefore, the mass of an electron can be expressed as 9.109 383 56 x 10-31 kg

Practice Change the following into scientific notation. 13 500 000 000 000 000 0.000 000 002 843 Change the following into standard notation. 6.4 x 10-4 1.5896 x 1010 Write the following proper scientific notation. 243 x 103 0.045 x 10-6 51.7 x 10-5 2986 x 10-7

Solutions Change the following into scientific notation. 1.350 000 000 000 000 0 x 1016 2.843 x 10-9 Change the following into standard notation. 0.000 64 15 896 000 000 Write the following proper scientific notation. 2.43 x 105 4.5 x 10-8 5.17 x 10-4 2.986 x 10-4

Significant Figures (or digits) It is equally important to report numbers efficiently (using scientific notation), accurately, and precisely. Every measurement is “inexact” because it contains some uncertainty, no matter what instrument you use. This uncertainty is estimated. We use significant figures to represent every measurement.

Significant Figures (or digits) You should always take measurements such that you read all the units using the device AND then estimate one more digit. For example, in the following ruler. D is 3.70 cm What is C?

Significant Figures (or digits) Therefore, the last digit given in any measurement is uncertain. And there is no point is recording any digits AFTER the one you estimated. Rules: All non-zero digits are significant (including the one uncertain digit). For example, 32.8 kg contains 3 significant figures.

Significant Digits Zeroes are more problematic! Example: What is the difference between 56 mm, 56.0 mm, and 56.00 mm? What about 0.0026 kg, 0.000026 kg, and 0.002060 kg?

Significant Digits Rules: All final zeroes AFTER the decimal point are significant. Zeroes between other significant digits are always significant. Zeroes to the right of a whole number are considered ambiguous so we avoid writing them.

Practice How many significant figures are in each of the following values? 218 g 22568 L 4755.50 cm3 108.63 420.07 mm 0.0001 m2 0.00530 kg 200.0 s Solutions 3 5 6 1 4

Calculations with Sig. Figs. Certain rules for significant digits apply when we perform calculations. Why? adding and subtracting The sum/difference is as precise as the LEAST precise value 24.51 g + 3.9 g = 28.41 g ≈ 28.4g multiplying and dividing The number of sig. figs. in the product/quotient is equal to the lesser number of sig. figs. possible. 2.6 s x 3.846 s = 9.9996 s2 ≈ 1.0 x 101 s2