Significant Figures & Scientific Notation
Measurements are important in science (particularly chemistry!) Quantity that contains both a number and a unit Must be able to say how correct a measurement is No measurement is perfect and exact!
2 types: Exact The amount of money in your account Known with certainty Approximate Weight, height Anything MEASURED No measurement is perfect!
ALL measurements have some amount of uncertainty or error All certain digits and the first uncertain digit in a measurement are considered significant Example: A ruler can measure to the nearest 0.1 centimeter Ruler also allows estimation between each 0.1 cm division
A measurement between 40.1 cm and 40.2 cm is estimated at 40.16 cm The estimate has 4 significant figures 3 digits are certain 1 digit is uncertain
Five basic rules All non-zero digits are significant Example: 1457 Zeros appearing between non-zero digits are significant Example: 1.008 Zeros in a decimal occurring before the first non-zero digit are not significant Example: 0.0025 Zeros in a decimal occurring after the first non-zero digit are always significant Example: 4.2079 Exact numbers (numbers obtained by counting or from definitions) are assumed to have unlimited number of significant figures Never limits the number of significant figures in a calculation
Place the number in an outline of the United States If decimal point is PRESENT, begin counting with the first non-zero digit from the PACIFIC side of the USA If a decimal point is ABSENT, begin counting the first non-zero digit from the ATLANTIC side of the USA Pacific (Decimal Present) Atlantic (Decimal Absent)
25 cm 305 cm 0.00123 in 400 g 400. m 0.94600 ml 12 in = 1 ft 2 3 3 1 3 5 Exact number so infinite # of sig figs
Addition and subtraction Solution must have the same number of decimal places as the least accurate number i.e. Solution is rounded to the least number of decimal places in the data Solution with correct number of significant figures: 2540. Be sure to LINE UP DECIMAL POINT when adding/subtracting
Multiplication and division Solution must have the same number of significant digits as the least accurate number i.e Solution is rounded to the least number of significant figures in the data 4.8069 must be rounded to 2 significant figures 4.8069 becomes 4.8
Rounding Carry all digits significant or extra through all steps in the calculation. Only the final value is rounded to the correct number of significant figures. To round, look at the digit following the one to be rounded. If it is 5 or more, round up; if it is less than 5, round down.
In chemistry (and all sciences), the SI system of units is used for all measurements
In the metric system, prefixes are used to modify base units
Convert 2.3 cm to m
Numbers like 12,000 or 546 are written in standard form Scientific notation makes it easier to work with very large or very small numbers It is based on the power of ten and written as: N x 10 z N is the number and z is the exponent Example: 7.65 x 10-3 The exponent that the base ten is raised to shows the number of places left or right that the decimal place needs to be moved
Proper use includes: A number with only one digit to the left of the decimal point and as many as needed to the right A number between and Example: 2.3456 1 digit to the left of the decimal point 4 digits to the right of the decimal point Number ten (10) raised to an exponent The exponent is the number of place holders needed to change the number from a conventional number to a number with only one digit to the left of the decimal point If the decimal point is moved to the left (number greater than 1), the exponent will be a positive number Example: If the decimal point is moved to the right (number less than 1), the exponent will be a negative number Example:
Convert each standard number into scientific notation 2.78 x 10 6 9.34 x 10 2 7.32 x 10 5 2.78 x 10 8 9.34 x 10-2 1.6 x 10 3 7.3 x 10-5 1.6 x 10-3 9.34 x 10 4 7.32 x 10-5
Addition and subtraction Convert each number so that they all have the same exponent Add or subtract the numbers The exponents for each base number 10 will not change Put the given exponent on the 10 If the answer does not have just one digit to the left of the decimal point, convert the answer to the correct scientific notation If you move the decimal to the right, add -1 to the exponent If you move the decimal to the left, add +1 to the exponent General formulas:
Multiplication Multiply the base numbers normally and then add the exponents together Division Divide the base numbers normally and then subtract the exponents General formulas:
Have fun measuring and happy calculating!