SIGNIFICANT FIGURES Significant Figure Rules There are three

SIGNIFICANT FIGURES

Significant Figure Rules • There are three rules on determining how many significant figures are in a number: • Non-zero digits are always significant. • Any zeros between two significant digits are significant. • A final zero or trailing zeros in the decimal portion ONLY are significant.

Rule 1: Non-zero digits are always significant. • For example: 123 (3 sig figs) • 1234 (4 sig figs)

Rule #2 • Zeros between non-zero digits are significant • For example: 12. 507 (5 sig figs) • 1001 (4 sig figs. )

Rule 3 • Zeros to the left of the first non-zero digit are not significant. • For example: 1. 02 (3 sig figs) 0. 12 (2 sig figs) 0. 012 (2 sig figs)

Rule 4 • If a number ends in zeros to the right of the decimal point, those zeros are significant • For example: 2. 0 (2 sig figs) 2. 00 (3 sig figs) {This signifies greater accuracy. }

Rule 5 • If a number ends in zeros to the left of the decimal point, those zeros may or may not be significant. • For example: If we make a statement that the weight of an object is 120 g, how do we convey our knowledge of whether the balance was accurate to ± 1 g or ± 10 g? • The ambiguity can be removed by using scientific notation (see later)

Multiplication or Division • the result can have no more sig figs than the least accurate number. For example: • If an object has mass of 29. 1143 g and a volume of 25. 0 cm 3, then its density is given by Density = 29. 1143 g = 1. 164572 g cm-3 = 1. 16 g cm-3 25. 0 cm • least number of sig figs is three therefore answer is also three

Addition or Subtraction: • the result must be reported to the same number of decimal places as the number with the fewest decimal places. For example: • 19. 2 g 0. 4745 g 127 g SUM = 146. 6745 g = 147 g because one weight is known only to the nearest 1 g! • NOTE: Round off numbers only at the END of calculations; otherwise, errors may be inadvertently carried through.

Scientific notation • Do you know this number, 300, 000 m/sec. ? It's the Speed of light ! • Do you recognize this number, 0. 000 000 753 kg. ? This is the mass of a dust particle! • Scientists have developed a shorter method to express very large numbers. This method is called scientific notation. Scientific Notation is based on powers of the base number 10.

• The number 123, 000, 000 in scientific notation is written as : • The first number 1. 23 is called the coefficient. It must be greater than or equal to 1 and less than 10. The second number is called the base. The base number 10 is always written in exponent form. In the number 1. 23 x 10 11 the number 11 is referred to as the exponent

To write a number in scientific notation: • Put the decimal after the first digit and drop the zeroes • In the number 123, 000, 000 The coefficient will be 1. 23 • To find the exponent count the number of places from the decimal to the end of the number. • In 123, 000, 000 there are 11 places. Therefore we write 123, 000, 000 as:

• For small numbers we use a similar approach. Numbers less smaller than 1 will have a negative exponent • A millionth of a second is: • 0. 000001 sec. or 1. 0 X 10 -6

Adding and Subtracting using Scientific Notation • All exponents MUST BE THE SAME before you can add and subtract numbers in scientific notation. The actual addition or subtraction will take place with the numerical portion, NOT the exponent • Example #1: 1. 00 x 103 + 1. 00 x 102 • A good rule to follow is to express all numbers in the problem in the highest power of ten. • Convert 1. 00 x 102 to 0. 10 x 103, then add: • 1. 00 x 103 + 0. 10 x 103 = 1. 10 x 103

Example #2: • The significant figure issue is sometimes obscured when numbers are in scientific notation. For example, add the following four numbers: • (4. 56 x 106) + (2. 98 x 105) + (3. 65 x 104) + (7. 21 x 103) • When the four numbers are written in the highest power, we get: 4. 56 x 10 6 + 0. 298 x 106 +0. 0365 x 106 +0. 00721 x 106 = 4. 90171 x 106 with sig. figs the answer is 4. 90 X 106

Multiplication using scientific notation • Mutliplying: multiply the coefficients and add the base exponents • (3. 05 x 106) x (4. 55 x 10¯ 10) Here is the rearranged problem: • (3. 05 x 4. 55) x (106 x 10¯ 10) The exponent is easy, since 6 + -10= -4. • 3. 05 x 4. 55= 13 You now have 13 x 10¯ 4.

Division using scientific notation • Divide the coefficients and subtract the base exponents (3. 05 x 106) ÷ (4. 55 x 10¯ 10) Here is the rearranged problem: (3. 05 ÷ 4. 55) x (106 ÷ 10¯ 10) The exponent is easy, since 6 –(-10) is +16. Remember that to divide exponents, you subtract them. 3. 05 ÷ 4. 55= 0. 7. You now have 0. 7 x 1016. This can also be written as 7 X 10 15

Assignment • Sig. fig worksheet (do all the odds questions) • Scientific notation (do all the odd questions) • Hand in: due tomorrow’s class