# Mathematics Basics

## Significant Figures

Communicating information as values infers that something was measured. If reported correctly, how well the measurement was performed is, generally, communicated in the value reported for a single measurement.

The objective of this section is to survey significant figures rules, significant figure use in mathematical operations, and identifying when to communicate the accuracy of the measurement explicitly.

Four rules apply to reporting significant figures (sometimes called significant digits).

1. All non-zero numbers (1, 2, 3, 4, 5, 6, 7, 8, and 9) are always significant.
2. All zeroes between non-zero numbers are always significant.
3. All zeroes which are successively to the right of the decimal point and at the end of the number are always significant.
4. All zeroes which are to the left of a written decimal point and are in a number greater or equal to 10 are always significant.

## Determining the Correct Number of Significant Digits

### Addition and Subtraction

Look at all the numbers used and adjust your answer to the same as the least accurate number (least accurate place) added or subtracted using the rounding rules below.

### Multiplication or Division

Look at all the numbers used and adjust your answer to the same as the number used with the least number of signifiant digits (least significant digits) using the rounding rules below.

## Rounding

Once you have performed a calculation and determined the correct number of significant digits to keep, you must round off your answer. There are two steps involved in rounding off a number, and you are already likely to be familiar with the rounding process from your earlier math classes. The steps for rounding are:

1. Starting from the leftmost significant digit, move to the right until you have as many digits as you are allowed to keep. Then look to the immediate right and note the number present.
2. If the number to the right is a 5, 6, 7, 8, or 9, round the last significant digit up one. If the number to the right is a 4, 3, 2, 1, or 0, keep the last significant digit the same.

#### Examples:

Round 1034.56 to 4 significant digits.

1. Step 1: Start with 1 and keep 0, 3, and 4. To the immediate right of 4 is a 5.
2. Step 2: From the above rules, since we have a 5 to the immediate right, we round 4 up to 5.

ANSWER: 1035

Round 0.000343 to 1 significant digit.

1. Step 1: Start from the left and skip all of the 0's (they aren't significant). The first significant digit encountered is a 3. To the immediate right of the 3 is a 4.
2. Step 2: From the rules above, since we have a 4 to the immediate right, we leave the 3 alone.

ANSWER: 0.0003

Round 4589 to 3 significant digits.

1. Step 1: Start from the 4 at the left and keep the 5 and 8. To the immediate right of the 9 is another 9.
2. Step 2: From the rules above, since we have a 9, we must round up.

ANSWER: 4590

## Scientific Notation

Chemistry quantities and units are more specialized, or more refined, than everyday usages. Numbers that are very large, or very small, are frequently expressed in scientific notation such as the value for 1 mole of atoms (6.022 x 1023 atoms). Scientific notation utilizes the powers of 10 and this has advantages because writing out very large numbers is cumbersome especially when many zeros are involved.

Sometimes the term "order of magnitude" is used. While this can be any base unit the term is typically used in a base 10 context.

The value for a mole (6.022 x 1023) can be converted from scientific notation by moving the decimal 23 places to the right. The value 3.7 x 10-9 m can be converted from scientific notation by moving the decimal place 9 places to the left (0.0000000037 m).

For 2 x 10-3 the exponent is -3 and this is division becoming 2 ÷ (10 x 10 x 10) = 0.002.  Note that if a zero is to the right of the decimal a zero must proceed the decimal.

For 2 x 103 the exponent 3 is positive and multiplication becoming 2 x (10 x 10 x 10) = 2,000.

Two observations can be made about what is written above. First, we can sometimes write words rather than numbers. The example above, 2 x 103, is could have been stated as two thousand. Many readers will readily know and be able to have a mental image of two thousand written out in arabic numbers. The example, 3.7 x 10-9 m, is not so readily known as three point seven nanometers and also exceptable as 3.7 nm. See how crafty I was to use the symbol for meter (m) in the example above? You probably knew the symbol for meter, but may not have known the value for nanometer 10-9 m.

Second, be aware that when the valence state of ions is written on elements we often use superscript(s) just as exponents are superscripts. The difference here is that the plus or minus symbol comes after the number representing the valence state of an element or compound that is greater than one (Fe3+). When the valence state is one, the number is implied and only the positive or negative symbol is used in the superscript location (Na+ or Cl-).

Beyond convenience and efficiency, scientific notation offers a clean way to communicate information about significant figures.