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 divide round off to the same number of figures as in the factor with the least number of figures significant figures x1.4 2 significant figures

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 decimal places decimal place decimal places

Significant Figures and Decimal Places decimal place decimal places decimal place significant figures x significant figures significant figures

I am a “ZERO” 1. Nonzero Integers: Nonzero integers are ALWAYS significant significant figures significant figures 2. Leading Zeros: Zeros which precede nonzero digits are NEVER significant si gnificant figures 02 1 si gnificant figure

3. Captive Zeros: Zeros which fall between nonzero digits are ALWAYS significant si gnificant figures sig nificant figure I am a “ZERO” 4. Trailing Zeros: Zeros at the right end of a number is significant ONLY with a decimal point si gnificant figure si gnificant figures significant figures significant figures

A Few Examples , x x ,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 N N N

Measurement Errors Measurements are central to science. The laws of science are discovered by measurements. Any law which is contradicted by a measurement 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(  x 2 +  y 2 ) 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 ( ) = 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 Can also be written in scientific notation: 4.47 x Geochemists write this as What is this ? Increased CO 2 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 10 9 One of these people might write it as What is this ? To speak to someone like a geochemist it might help to learn this language...?

Logarithms 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: = 4,600,000,000 These all follow from the basic definition of a logarithm. Log b N = x,If b x = N So a logarithm is nothing more than an exponent. Log b N = x is asking: b “to what power” equals N ?

Rules of Logarithms are also Rules of Exponents (10 a ) (10 b ) = 10 a+b 10 a / 10 b = 10 a-b (10 a ) b = 10 ab Exponents Logarithms log AB = log A + log B log (A/B) = log A - log B log A n = n log A 1. log 3 + log 4 = ? 2. log (2/3) = ? A Few Examples:

Logarithmic Notation There are many ways to express a number. Ex.) Distance from Earth to the Moon is 380,000 km. Scientific Notation: 3.8 x 10 5 Logarithmic Notation: (assumes log with base 10) Natural Logarithm (ln) : log e N = x The transcendental number e has a value of (to 6 figures ) The above equation asks: e “to what power” equals N ? If ln (380,000) = , then what is this in logarithmic notation ?

Moving from one Logarithmic Base to Another log b N = (log a N) / (log b a) You may remember another rule of logs to transform from one base (a) to another base (b): Do an example: convert to

What Does This Tell us About Scientific Notation ? Let's try to translate between scientific notation and logarithmic notation: Take the distance to the moon of 380,000 km. In Logarithmic notation this is written as: But in scientific notation this is 3.80 x 10 5 We an rewrite this according to rules for multiplying exponents. 380,000 = = x 10 5 = 3.80 x 10 5

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 = = x 10 5 = 3.80 x 10 5 This works because = 3.80 or log (3.80) =

Try an Example Write the age of the Earth in 1.) scientific notation and 2.) logarithmic notation 3.) Show the translation