Significant Figures, Why do they Matter ?

“Truth in advertising”

When you give the result of a calculation, you communicate two things:

1) The number itself

2) How well that the number is known

If you measure the sinking velocity of a foram while out at sea on a ship and you give the result: 1 cm/sec

But you neglect to mention that your boat was in 20 ft seas:Your proper answer would correctly be 1cm/sec +/- 620 cm!

How well is your results known ???

A Few Rules on Significant Figures

1. When you multiply or divideround off to the same number of figures as in the factor with the least number of figures.

4.56 3 significant figures

x1.4 2 significant figures

6.384 6.4

A Few Rules on Significant Figures

2. When you add or subtract, it is the position of the digits that is important, NOT how many there are. Imagine (or actually do this!) that the various terms that are added or subtracted together are in a vertical column with the decimal points all lined up. Spot the term that goes least far to the right. Round off to that position.

12.11 2 decimal places 18.0 1 decimal place + 1.013 3 decimal places

31.123 31.1

Significant Figures and Decimal Places

20.0 1 decimal place+ 4.00 2 decimal places 24.0 1 decimal place

20.0 3 significant figuresx 4.0 2 significant figures 80. 2 significant figures

I am a “ZERO”

1. Nonzero Integers: Nonzero integers are ALWAYS significant.

3798 4 significant figures 26.5 3 significant figures

2. Leading Zeros: Zeros which precede nonzero digits are NEVER significant

0.0025 2 significant figures 02 1 significant figure

3. Captive Zeros: Zeros which fall between nonzero digits are ALWAYS significant

1.008 4 significant figures 3079 4 significant figure

I am a “ZERO”

4. Trailing Zeros: Zeros at the right end of a number issignificant ONLY with a decimal point.

100 1 significant figure100. 3 significant figures3300 2 significant figures 3.300 4 significant figures

A Few Examples

12

1900

1098

20,100

2.001 x 103

.0001

7.300

1.01 x 10-5

220,400

Identify the number of significant figures in each example:

How would you convert 35.9 m to cm – using significant figures ?

Determine Bob's average weight over the surface of the Earth from the Pole to the Equator.

Give your answer with the appropriate number of significant figures.

978.66 N

983.2 N

981.40 N

Measurement Errors

Measurements are central to science.

The laws of science are discovered by measurements.

Any law which is contradicted by ameasurement must be re-considered.

Measurements

Measurements are approximate – subject to uncertainties given by your measuring device.

Errors are not mistakes! - Errors cannot be avoided when making measurements. Mistakes can usually be avoided...

Measurements

When making a measurement, your accuracy is only good to ½ of the smallest graduation you have available.

What is the length of the gray bar above ?

Propagation of Errors

Making more than one measurement Combine your measurements using addition or multiplication.

Rule #1: When 2 measurements are combined by addition:

z = sqrt(

x2 +

y2)

Rule #2: When 2 measurements are combined by multiplication:

z/Z = sqrt((

x/x)2 + (

y/y)2)

Propagation of Errors

Keep all significant digits to the end.

The Factor-of-Two rule: If one measurement has much larger error than another, the smaller one will have a negligible effect on the final answer. If A has 1% error and B has 2% error then:

% error = sqrt ( 12 + 22 ) = 2.24 = 2%

Speaking Logarithmically

Logarithms Exponents Scientific Notation

BIG numbers and small numbers...how to deal with them.

Speaking Logarithmically

Solubility product of calcite is 0.00000000447 Can also be written in scientific notation:

4.47 x 10-9

Geochemists write this as 10-8.35 What is this ?

Increased CO2 absorbed by oceans

(Feely et al., 2004)

Speaking Logarithmically

The age of the Earth is 4,600,000,000 yrs

Can also be written in scientific notation: 4.6 x 109

One of these people might write it as 109.6 What is this ?

To speak to someone like a geochemist it might help to learn this language...?

Logarithms

109.6 has an exponent which is the logarithm of 4,600,000,000.

So: log (4,600,000,000) = 9.6

But the notation is accurate: 109.6

= 4,600,000,000

These all follow from the basic definition of a logarithm.

Logb N = x, If b

x = N

So a logarithm is nothing more than an exponent.

Logb N = x is asking: b “to what power” equals N ?

Rules of Logarithms are also Rules of Exponents

(10a) (10b) = 10a+b

10a / 10b = 10a-b

(10a)b = 10ab

Exponents Logarithms

log AB = log A + log B

log (A/B) = log A - log B

log An = n log A

1. log 3 + log 4 = ?

2. log (2/3) = ?

A Few Examples:

Logarithmic NotationThere are many ways to express a number.

Ex.) Distance from Earth to the Moon is 380,000 km.

Scientific Notation: 3.8 x 105

Logarithmic Notation: 105.6 (assumes log with base 10)

Natural Logarithm (ln) : loge N = x

The transcendental number e has a value of 2.71828 (to 6 figures)

The above equation asks: e “to what power” equals N ?

If ln (380,000) = 12.848, then what is this in logarithmic notation ?

Moving from one Logarithmic Base to Another

logb N = (log

a N) / (log

b a)

You may remember another rule of logs to transform fromone base (a) to another base (b):

Do an example: convert 105.5798 to 164.6399

What Does This Tell us About Scientific Notation ?

Let's try to translate between scientific notation and logarithmic notation:105.5798

Take the distance to the moon of 380,000 km.

In Logarithmic notation this is written as: 105.579

But in scientific notation this is 3.80 x 105

We an rewrite this according to rules for multiplying exponents.

380,000 = 105.5798 = 100.5798 x 105 = 3.80 x 105

What Does This Tell us About Scientific Notation ?

Let's try to translate between scientific notation and logarithmic notation

We an rewrite this according to rules for multiplying exponents.

380,000 = 105.5798 = 100.5798 x 105 = 3.80 x 105

This works because 100.5798 = 3.80

or log (3.80) = 100.5798

Try an Example

Write the age of the Earth in 1.) scientific notation and 2.) logarithmic notation

3.) Show the translation