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Numbers in Science
Scientists put only the digits they can reasonably be certain of in their numbers. They might say, for example, that they measured "10." cm (note the presence of the decimal point). This is actually different from saying that they measured "10" cm. The use of the decimal point indicates that the scientist is sure of both digits to some reasonable degree -- it is "10 point something", not 11 or 9, even though rounding both of these numbers to one digit gives 10.
The number "10." is said to have two significant digits, or significant figures, the 1 and the 0. The number 1.0 also has two significant digits. So does the number 130, but 10 and 100 only have one "sig fig" as written. Zeros that only hold places are not considered to be significant.
So, how does a scientist indicate that two of the digits in 100 are significant?? We can't put in a decimal point alone to make 100. because that would indicate 3 digits. What should we do?
Scientists use scientific notation to handle this problem. Scientific notation makes sure that everything but the first digit of a number is after the decimal place and therefore either certain or not used. Here are some numbers in scientific notation to study:
See the differences? In the first and second example, the zeros are really only place holders. In the third example, the extraneous decimal place is used to mean we are certain of all three digits. In the fourth example the extra zeros (on the right!) are used to indicate that we have extra certainty. In the fifth example, we have finally seen how to represent 100 with exactly two significant figures.
Here is one way to type these numbers into your TI 83 calculator:
= 2.183 ∗ 10 ^ 3 - 1.1 ∗ 10 ^ -2
When you are using your calculator, typing "something times ten to the something" over and over again gets to be a pain. Most calculators have an "EE" button, to help you out. EE means "times ten to the", so that:
= 2.183E3 - 1.1E-2
Note that when you type the EE key, most calculators simply display "E"! Do not be alarmed by this. This is not the E that means error.
Be careful! It's easy to make the following common mistake: Remember that EE -- times ten to the -- is not the same as ^ -- "to the"!
Our next question needs to be, "What happens to the number of sig figs when we perform calculations?"
When scientists are calculating with significant figures, the precision of the result should reflect the uncertainty of the numbers that went into it. If you add 0.25 to 100, given that 100 only has one sig fig and therefore we don't trust even the two zeros, can you really say you have 100.25? That would be saying that you were sure the first 100 was, in fact, 100.00. If the 100 had five sig figs, why didn't someone say so? What are the rules, exactly?
When you are adding or subtracting with significant figures, look where the right-most significant figures is in each number. The number that has its right-most sig fig in the higher place governs what is significant in the result. Watch:
1.0750 + 32,110.31 = 32,111.3850 last sig fig: .000x .0x use the .0x (highest place) = 32,111.39
Be careful when the numbers are in scientific notation:
= 99.6x103 + 2.51x103
= 102.11x103 ≈ 102.1x103 (for one decimal place)
What did we do??
Multiplication and division also have rules about this. When you are multiplying or dividing, you look at how many significant figures are in each number. The fewest of these is the number of significant figures the result will have. For example:
3.212x104 * 2.51x103 = 80621200 = 8.06x107 (4 sig figs) (3 sig figs) (3 sig figs)
Notice that the fewest number of sig figs is three so we put three in the result.
Exponentiation (including square root) works like multiplication -- since it is based on multiplication.
Try It Out:
The first three problems below deal with the following situation: You want to find out how many minutes are in a year.
Problem 1:1. There are 365.24 days in a year and exactly 1440 minutes in a day. How many significant digits are in 365.24? How many are in 1440.?
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