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Key concepts for Chapter 2 2.1 All measurements have uncertainties. Measurements are never exact. 2.2 The uncertainty in measurement is reflected / indicated by using significant figures (or significant digits). - the digits (s.f.) for a given measurement tells the reader that the last digit (right-most) is the digit where there is uncertainty and that all other digits are certain. - The number of significant figures for a number is the total number of digits the number has, counting from left to right. o EXCEPT in the follow 2 cases. If a number is less than 1 (i.e. 0.123, 0.0555 etc) but greater than 0, all zeroes before (to the left of) the first non-zero digit are not significant. ex. 0.0023 has 2 s.f. 0.1507 has 4 s.f. If a number is a whole number and ends with 0s but without a decimal point shown, all zeroes to the right of the last (right most) non-zero digit are not significant. Ex. 7000 has 1 s.f. 20700 has 3 s.f. The same two rules apply when the numbers are negative. - exact numbers have infinite number of significant figures o exact numbers are those that are counted and not measured(2 eggs, 3 years, 5 cars) o exact numbers are those that are definitions (i.e. 1 meter = 1000 millimeters) 2.3 Rounding of significant figures is the same as normal mathematical rounding. - if the 1st non-significant digit is less than 5, drop all non s.f. digits - if the 1st non-significant digit is 5 or more, drop all non s.f. digits and increase the last significant digit by 1 - if a calculation involves multiple steps, wait until all calculations are done before rounding to minimize error due to rounding. 2.4 Adding and Subtracting significant figures - perform the calculation normal, - examine each number and determine where the uncertain digits are for each of the number. o Determine the unit/digit (i.e. 10s, 1s, tenths, hundredths, etc) with the highest uncertainty (left most) - transfer that digit down to answer o so if the digit with the highest uncertainty is in the 10s digit, then the uncertainty in the answer will be in the 10s digit - round off non-significant digits in the answer 2.5 Multiplying and dividing significant figures - perform calculations normally - examine each number for the number of s.f. each number has o determine what is the least number of s.f. all the numbers have - the least number of s.f. is the number of s.f. the answer will have - round off non-significant digits in the answer 2.6 Exponential numbers - are numbers with a superscript o which is defined as the number multiplied by itself a number of times equal to the superscript. So, 63 = 6x6x6 102 = 10x10 24 = 2x2x2x2 - know the conversion of powers of 10s. 2.7 Scientific Notation - a method used to standardize writing of numbers, especially useful when writing really large or small numbers - always has the format of; #.### x 10n Start with a non-zero digit, followed by a decimal point, followed by the other significant digit(s) i.e.. 6.23 5.11 2 (exception, if there is only 1 s.f., decimal point is not needed) That number is then multiplied by the correct powers of 10. i.e. 105 10-7 102 0 (exception, 10 , which equals 1, is not used) 2.8 Unit equation and unit factors - unit equation is just an equation that relates two equivalent quantities o example: 1 dollar = 100 pennies - unit factor (also known as conversion factor) is simple two fractions that can be written using the two quantities o example: quantities A & B -> A/B and B/A 2.9 Unit analysis problem solving (ex. How many days in 2.5 years) - write down the unit asked for in the answer o = days - write down the units related to the answer that are given o 2.5 years - Find the unit equation relating the units o Years and days 1 year = 365 days - Write the conversion factors o 1 year / 365 days 365 days / 1 year - Plug in the conversion factor into the equation and find that will give the right units in the answer. o 2.5 years x (1 year/365 days) = 2.5 yr2 / 365 days WRONG units o 2.5 years x (365 days / 1 year) = 912.5 days CORRECT units - round off non significant digits in answer o 912.5 days 912 days (2.5 years is an exact number, it’s not measured) 2.10 Percentages (%) - are defined as the number parts (of a given quantity or quantities) per 100 parts of the sum (total sample). - Mathematically it is: % = (part(s) / sum ) x 100 o Example: 2 apples and 8 oranges in a basket, what percentage do the apples make up (Part / sum ) * 100 = % [2 apples / (2 apples + 8 oranges)] * 100 = (2 / 10) *100 = 0.2*100 = 20%