Exponential NotationSignificant Figures

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Exponential Notation

In physics as we often use numbers that are very small and very large. For example:

The mass of an electron = 0.000000000000000000000000000000911 kg

The mass of the moon = 7,400,000,000,000,000,000,000,000 kg

Now consider for a moment the problems of working with such values... 1) They would not fit on your calculator 2) It would be annoying to write so many zeros.

Fortunately we can avoid all those zeros by converting numbers into exponential (scientific) notation.

Exponential Numbers:

Numbers expressed in correct scientific notation contain only ___ number to the left of the decimal point.

1

Examples: Convert to exponential notation:

1) 135,000 = ?

etc 1010350,110500,13000,13551035.1000,135

2) 0.0055 = ?

etc 111 101055.010055.00055.0

3105.50055.0

3) 127,000,000,000 = ?

111027.1000,000,000,127

4) 0.0000000444 = ?| | 81044.4

To put exponential numbers on your calculator is also very simple:

for example if I wanted to enter 1.61 X 10-19, I would...

step 1) enter the number 61.1

step 2) press exponent button 0061.1

step 3) enter the exponent 1961.1

step 4) if a negative exponent is desired, press ± key

1961.1 Note: Some TI’s require pressing the ± key before entering the exponent

USING YOUR CALCULATOR:

EXPONENT PROBLEMS: Simply enter the number as demonstrated above and then treat it like any other number.

sample 1) 58 1045.41033.3

Enter as 3.33 EE 8 X 4.45 EE 5 =

1458 1048.11045.41033.3

Sample 2)

2922

126

1021.11028.1

1044.41050.3

RAISING TO A POWER

step 1) enter the number

step 2) press ____ button yx

step 3) enter the exponent

step 4) press ____ button =

1) 66 1029.95.14

2) 2045 1075.81072.1

TAKING A ROOT

METHOD ONE METHOD TWO

step 1) enter the number

step 2) press ____ button

step 3) press ____ button

2nd

yx

step 4) enter the exponent

step 5) press ____ button =

step 1) enter the number

step 2) press ____ button yx

step 3) enter the exponent

step 4) press ____ button 1/x

step 5) press ____ button =

Note: Some calculators will use SHIFT instead of 2nd.

1) ?5.583 88.3

Also: 88.35.585.58 313

2) ?1077.74 6 21028.50528.0

Also: 0528.01077.71077.7 41

64 6

Note: The fourth root is also just the square root taken twice…..

Extra Example: ?1098.5524

Most calculators will give an error or overflow message. This is how to solve this type of problem: Separate it into parts!

5245524 1098.51098.5

5243 101065.7

1231203 1065.7101065.7

++++ SIGNIFICANT FIGURES ++++

A. INTRODUCTION:

When making measurements or doing calculations, you should not keep more digits in a number than is justified. Today you will learn how to use the correct number of digits in a measured or calculated value.

What is a significant figure? The numerical value of every observed measurement is an approximation. Consider that the length of an object is recorded as 15.7 cm. By convention, this means that the length was measured to the nearest tenth of a centimeter and that its exact value lies between 15.65 and 15.75 cm, or possibly between 15.6 and 15.8 cm. If this measurement were reported to the nearest hundredth of a centimeter, it would have been recorded as 15.70 cm. The value 15.7 cm represents three significant figures (1,5,7), while the value 15.70 represents four significant figures (1,5,7,0).

A significant figure is one which is known to be reasonably reliable. In measurements, significant figures are all the values (digits) known for sure plus one estimated figure (the last one).

example: Make the measurement with correct significant figures...

9 cm

10 c

m

a b

For measurement a, it is known for certain that the measurement is between 9.2 and 9.3 cm. These are considered ____________ .exact

Next comes estimating the last digit: a = 9.23 cm

Last, there should be an estimate of the uncertainty, the limit to the estimation made: Use the ± notation to show the amount of uncertainty.

± 0.01 cm

9 cm

10 c

m

a b

For measurement b, it is known for certain that the measurement is between 9.8 and 9.9 cm. For writing the measurement, the value 9.8 is considered exact.

Next comes estimating the last digit: b = 9.88 cm

Lastly, show the uncertainty of the measurement.

cmb 01.088.9

9 cm

10 c

m

dc

This is the same ruler as the first example, and will have the same uncertainty as the previous examples, ± 0.01 cm.

Measurement c is not 9 cm, but 9.00 ± 0.01 cm.

Measurement d is not 9.7 cm, but 9.70 ± 0.01 cm.

e f

9 cm

10 c

m

This ruler is different from the previous examples. For the above measurement, say f, we know it is definitely greater than 9 and less than 10 cm. The estimated value will be the first digit after the decimal, so the uncertainty is ± 0.1 cm.

Measurement e is not 9 cm, but 9.0 ± 0.1 cm.

Measurement f is 9.7 ± 0.1 cm.

0 cm

1 cm

g h

This is the same ruler as the first example, and will have the same uncertainty as the previous examples, ± 0.01 cm.

The measurement g is between 0 and 0.1 cm. g = 0.03 ± 0.01 cm.

The measurement h is appears to be 0.9 cm, or: h = 0.90 ± 0.01 cm.

B. ZEROS: Any nonzero digit reported will be assumed significant. Zeros may be significant or they may merely serve to locate the decimal point.

1. Zeros between numbers (“captive zeros”) are always significant.

examples: 1.007 4 sig figs assumed to mean 1.007 ± 0.001

107.08 5 sig figs

1001 4 sig figs assumed to mean 1001 ± 1

2. Zeros in front of numbers (“leading zeros”) are not significant!

These zeroes are used only to serve as markers for the placement of the decimal point.

examples: 0.7 g 1 sig fig assumed to mean 0.7 ± 0.1 g

Note: It makes no difference if the above measurement is written as 0.7 g, or 700 mg, or even as 0.0007 kg. Only the size of the measuring unit is changed, but the amount of accuracy is still the same: 1 part in 7

0.006 cm 1 sig fig

0.00705 kg 3 sig fig

Again, think in terms of uncertainty: 0.00705 ± 0.00001 kg has the same meaning as 7.05 ± 0.01 g

3. Zeros behind numbers (“trailing zeros”) may or may not be significant:

(a) Zeros behind numbers are significant if there is a decimal point .

examples: 1.200 12.0 120.000 120.

4 sig figs 3 sig figs 6 sig figs WRONG!!

Note: Writing a number with a decimal point but no digits after the decimal point is incorrect mathematical usage. There must always be some digit after the decimal point. If the number becomes ambiguous when written in the above form, use scientific notation. (see the next rule)

3. (b) Zeros behind numbers in front of the decimal point may or may not significant. (You cannot tell which figure was the estimated one)

examples: 120 As written, one cannot tell if this is 120 ± 10 or 120 ± 1

2 to 3 sig figs

1,200 2, 3, or 4 sig figs

1,000,000 1 to 7 sig figs

(1) Exponential notation is used to indicate how many sig figs there are forsure in numbers like these. (a) Write the number 100,000 with ...

1 sig fig 3 sig figs 5 sig figs

5101 51000.1 5100000.1 120. should be written as 1.20 x 102

4. How many significant figures are there in each of the following? (a) 107.0 (b) 0.0084 (c) 15,000

(d) 0.0750 (e) 0.060070 g

4 sig figs 2 sig figs 2 to 5 sig figs

3 sig figs 5 sig figs

Rules for significant figures

C. ROUNDING OFF: A number is rounded off to the desired number of significant figures by dropping one or more digits to the right. You should round off ________ you have solved for the answer. When the first digit dropped is less than 5, the last digit retained remains the same; when it is more than five, or when it is 5 followed by digits not all zeros, 1 is added to the last digit retained. When it is 5 followed by zeros, 1 is added to the last digit retained if that digit is odd but not when it is even. (not always done)

after

Mosig’s KISS rule {Keep It Simple ……}If 5 or more, round up. Otherwise, round down.

examples: round off to 3 sig figs: a. 1.456 b. 1.3548 c. 1.34501

d. 1.2249 e. 1.225 f. 1.335

g. 1.88500 h. 1.555002

1.46 1.35 1.35

1.22 1.22 or 1.23

simple

1.34

1.88 or 1.89

simple

1.56

D. ADDITION AND SUBTRACTION: The answer should be rounded off after adding or subtracting, so as to retain digits only as far as the first column containing estimated figures. (remember that the last figure is estimated)

Hint: Look for the number with its last digit in the left most position relative to the decimal point. That will be the place to round the solution after adding or subtracting.

examples: 12.2 14+ 0.03626.236

The number 14 has its last digit the furthest to the left. This number dominates the uncertainty.

Round the answer to this same decimal location, in this case, the one’s position.

The answer should be 26. Note: The answer has two sig figs, but that is due to rounding to the one’s place, and not to the least number of digits!

Add: 17.17 6.2+ 11.080

The number 6.2 has its last digit the furthest to the left. Round the answer to the tenth’s place.

34.450

The correct answer is 34.4 or 34.5. Either rounding system is ok.

Add: 107.42 9.759+ 333

The number 333 has its last digit the furthest to the left. Round the answer to the one’s place.

450.179

Rounding to the one’s place gives 450, but that is ambiguous. Use scientific notation!

21050.4

E. MULTIPLICATION AND DIVISION: The answer should be rounded off to contain only as many sig figs as the number with the fewest sig figs.

examples:

?52.024.17 9648.8

This number has the least sig figs!

Round the answer to only 2 sig figs.

0.9

?5.324.118 84.413Round the answer to only 2 sig figs.

2101.4

?58.7

40.14007.1

91.1

Now do your homework!