Introduction to Significant Figures & Scientific Notation

Significant Figures • Scientist use significant figures to determine how precise a measurement is • Significant digits in a measurement include all of the known digits plus one estimated digit

For example… • Look at the ruler below • Each line is 0.1cm

• You can read that the arrow is on 13.3 cm • However, using significant figures, you must estimate the next digit • That would give you 13.30 cm

Let’s try this one • Look at the ruler below • What can you read before you estimate?

• 12.8 cm • Now estimate the next digit… • 12.85 cm

The same rules apply with all instruments • The same rules apply • Read to the last digit that you know • Estimate the final digit

Let’s try graduated cylinders • Look at the graduated cylinder below • What can you read with confidence?

• 56 ml • Now estimate the last digit • 56.0 ml

One more graduated cylinder • Look at the cylinder below… • What is the measurement?

• 53.5 ml

Rules for Significant figures Rule #1 • All non zero digits are

ALWAYS

significant • How many significant digits are in the following numbers?

•274 •3 Significant Figures •25.632

•5 Significant Digits •8.987

•4 Significant Figures

Rule #2 • All zeros between significant digits are

ALWAYS

significant • How many significant digits are in the following numbers?

504 3 Significant Figures 60002 5 Significant Digits 9.077

4 Significant Figures

Rule #3 • All

FINAL

zeros to the right of the decimal

ARE

significant • How many significant digits are in the following numbers?

32.

0 19.00

0 105.002

0 3 Significant Figures 5 Significant Digits 7 Significant Figures

Rule #4 • All zeros that act as place holders are

NOT

significant • Another way to say this is: zeros are only significant if they are between significant digits OR are the very final thing at the end of a decimal

For example How many significant digits are in the following numbers?

0.0002

6.02 x 10 23 100.000

150000 800 1 Significant Digit 3 Significant Digits 6 Significant Digits 2 Significant Digits 1 Significant Digit

Rule #5 • All counting numbers and constants have an infinite number of significant digits • For example: 1 hour = 60 minutes 12 inches = 1 foot 24 hours = 1 day

How many significant digits are in the following numbers?

0.0073

100.020

2500 7.90 x 10 -3 670.0

0.00001

18.84

2 Significant Digits 6 Significant Digits 2 Significant Digits 3 Significant Digits 4 Significant Digits 1 Significant Digit 4 Significant Digits

Rules Rounding Significant Digits Rule #1 • If the digit to the immediate right of the last significant digit is less that 5, do not round up the last significant digit.

• For example, let’s say you have the number 43.82 and you want 3 significant digits • The last number that you want is the 8 – 43.

8 2 • The number to the right of the 8 is a 2 • Therefore, you would not round up & the number would be 43.8

Rounding Rule #2 • If the digit to the immediate right of the last significant digit is greater that a 5, you round up the last significant figure • Let’s say you have the number 234.87 and you want 4 significant digits • 234.

8 7 – The last number you want is the 8 and the number to the right is a 7 • Therefore, you would round up & get 234.9

Rounding Rule #3 • If the number to the immediate right of the last significant is a 5, and that 5 is followed by a non zero digit, round up • 78.

6 57 (you want 3 significant digits) • The number you want is the 6 • The 6 is followed by a 5 and the 5 is followed by a non zero number • Therefore, you round up • 78.7

Rounding Rule #4 • If the number to the immediate right of the last significant is a 5, and that 5 is followed by a zero, you look at the last significant digit and make it even.

• 2.5

3 50 (want 3 significant digits) • The number to the right of the digit you want is a 5 followed by a 0 • Therefore you want the final digit to be even • 2.5

4

Say you have this number • 2.5

2 50 (want 3 significant digits) • The number to the right of the digit you want is a 5 followed by a 0 • Therefore you want the final digit to be even and it already is • 2.5

2

Let’s try these examples… 20 0 .99

1 8 .22

13 5 .50

0.00

2 99 9 8 .59

(want 3 SF) (want 2 SF) (want 3 SF) (want 1 SF) (want 2 SF) 201 18 136 0.003

99

Scientific Notation • Scientific notation is used to express very large or very small numbers • It consists of a number between 1 & 10 followed by x 10 to an exponent • The exponent can be determined by the number of decimal places you have to move to get only 1 number in front of the decimal

Large Numbers • If the number you start with is greater than 1, the exponent will be positive • Write the number 39923 in scientific notation • First move the decimal until 1 number is in front – 3.9923

• Now at x 10 – 3.9923 x 10 • Now count the number of decimal places that you moved (4) • Since the number you started with was greater than 1, the exponent will be positive • 3.9923 x 10 4

Small Numbers • If the number you start with is less than 1, the exponent will be negative • Write the number 0.0052 in scientific notation • First move the decimal until 1 number is in front – 5.2

• Now at x 10 – 5.2 x 10 • Now count the number of decimal places that you moved (3) • Since the number you started with was less than 1, the exponent will be negative • 5.2 x 10 -3

Scientific Notation Examples Place the following numbers in scientific notation: 99.343

4000.1

0.000375

0.0234

94577.1

9.9343 x 10 1 4.0001 x 10 3 3.75 x 10 -4 2.34 x 10 -2 9.45771 x 10 4

Going from Scientific Notation to Ordinary Notation • You start with the number and move the decimal the same number of spaces as the exponent.

• If the exponent is positive, the number will be greater than 1 • If the exponent is negative, the number will be less than 1

Going to Ordinary Notation Examples Place the following numbers in ordinary notation: 3 x 10 6 6.26x 10 9 5 x 10 -4 8.45 x 10 -7 2.25 x 10 3 3000000 6260000000 0.0005

0.000000845

2250

Significant Digits Calculations

Significant Digits in Calculations • Now you know how to determine the number of significant digits in a number • How do you decide what to do when adding, subtracting, multiplying, or dividing?

Rules for Addition and Subtraction • When you add or subtract measurements, your answer must have the same number of decimal places as the one with the fewest • For example: 20.4 + 1.322 + 83 = 104.722

Addition & Subtraction Continued • Because you are adding, you need to look at the number of decimal places 20.

4 + 1.

322 + 83 = 104.722

(1) (3) (0) • Since you are adding, your answer must have the same number of decimal places as the one with the fewest • The fewest number of decimal places is 0 • Therefore, you answer must be rounded to have 0 decimal places • Your answer becomes • 105

Addition & Subtraction Problems 1.23056 + 67.

809 = 69.03956  69.040

23.67 – 500 = 40.

08 + 32.064 = 22.9898 + 35.

453 = 95.

00 – 75.

00 = - 476.33  -476 72.1440  72.14

58.4428  58.443

20  20.00

Rules for Multiplication & Division • When you multiply and divide numbers you look at the

TOTAL

number of significant digits

NOT

just decimal places • For example: 67.50 x 2.54

= 171.45

Multiplication & Division • Because you are multiplying, you need to look at the

total

number of significant digits not just decimal places 67.50 x 2.54 = 171.45 (4) (3) • Since you are multiplying, your answer must have the same number of significant digits as the one with the fewest • The fewest number of significant digits is 3 • Therefore, you answer must be rounded to have 3 significant digits • Your answer becomes • 171

Multiplication & Division Problems 890.15 x 12.3 = 10948.845  1.09 x 10 4 3.916977  3.9170

88.132 / 22.500 = 141.954  142 (48.12)(2.95) = 3.5376  3.538

58.30 / 16.48 = 30.47123  30.47

307.15 / 10.08 =

More Significant Digit Problems 18.36 g / 14.20 cm 3 = 1.293 g/cm 3 105.40 °C –23.20 °C = 82.20 °C 324.5 mi / 5.5 hr 21.8 °C + 204.2 °C 460 m / 5 sec = 59 mi / hr = 226.0 °C = 90 or 9 x 10 1 m/sec