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Numbers in Science

Scientific Notation

When scientists measure a quantity, they actually measure two pieces of information--the value they think they have measured, and the uncertainty. This can be stated as "We measured ten, plus or minus one", and often scientists do use these terms. However, this notation gets cumbersome fast. We need a quick, generally accepted method by which we can indicate the precision of our measurements.

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:

(1) 1000 1x103 -- so one sig fig
(2) 0.001 1x10-3 -- so one sig fig
(3) 100. 1.00x102 -- so three sig figs
(4) 0.00100 1.00x10-3 -- so three sig figs
(5) 100 (with two sig figs) 1.0x102 -- so two sig figs

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.

Scientific Notation on Your Calculator

Here is one way to type these numbers into your TI 83 calculator:

2.183x103 - 1.1x10-2
= 2.183 ∗ 10 ^ 3 - 1.1 ∗ 10 ^ -2
= 2182.989

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:

2.183x103 - 1.1x10-2
= 2.183E3 - 1.1E-2
= 2182.989

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"!

4E1 equals 40 (since 4 ∗ 101), not 4 (which would be 41 = 4).

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?

Adding and Subtracting

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:

9.96x104 + 2.51x103
= 99.6x103 + 2.51x103
= 102.11x103 ≈ 102.1x103 (for one decimal place)
= 1.021x105

What did we do??

  1. The easiest way to add anything (if we don't have a calculator) is to first "line up the digits" appropriately; with numbers in scientific notation that means we need to rewrite them so that the powers of 10 match. Remember the rule of thumb:

    to move the point left one digit, increase the power by 1
    to move the point right one digit, decrease the power by 1

  2. We then added the digits (99.6 + 2.51), and noted that the right-most significant figure would be in the tenths place in the result.

  3. Lastly we put the number back in appropriate scientific notation, with one digit left of decimal ( with 1.021x105 rather than 102.1x103)

Multiplying and Dividing

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.?

Check your work.

Problem 2:

2. (a) Write 365.24 and 1440. in scientific notation. (b) Write both numbers in EE, or calculator notation.

Check your work.

Problem 3:

3. Multiply 365.24 and 1440. (days per year by minutes per day). Write the result in both regular and scientific notation. How many of the digits are significant?

Check your work.

Problem 4:

4. How many seconds are in a year?

Check your work.

[Numbers and their Properties] [Numbers in Science] [Ratios and Proportions]
[Units, Dimensions, and Conversions] [Percents] [Simple Statistics] [Logarithms]

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