# Basic Arithmetic, Rounding Numbers, and Significant Figures

*At the end of this chapter, the reader should be able to do the following:*

1. Perform basic arithmetic calculations, including addition, subtraction, multiplication, and division with positive and negative numbers.

2. Perform calculations that require multiple steps in the correct order.

3. Perform ratio and proportion calculations.

4. Convert numbers between their percentage and decimal forms.

5. State the rules for rounding numbers in the following situations:

a. The number to be rounded ends in a number less than 5

b. The number to be rounded ends in a number greater than 5

c. The number to be rounded ends in 5

6. State the rules for addition, subtraction, multiplication, and division using significant figures.

7. Perform calculations (incorporating rounding numbers) including addition, subtraction, multiplication, and division to achieve the correct result.

## BASIC ARITHMETIC

This chapter as well as Chapters 2 and 3are designed as a review of basic mathematical concepts. Students already proficient in these concepts may wish to briefly review them and begin at Chapter 4.

### Positive and Negative Numbers

A positive number is a number that has a value greater than zero; a negative number is a number with a value less than zero. Figure 1–1 is a number line that demonstrates this concept. A plus sign (+) is used to identify a positive number and a negative, or minus, sign (−) is used to identify a negative number. If only positive numbers are used in an equation, the (+) sign is usually omitted.

### Addition of Positive Numbers

The sum of two or more positive numbers will also be a positive number:

### Addition of Both Positive and Negative Numbers

The sum of an addition of both positive and negative numbers will be the sign of the larger number involved in the addition. If you think of the numbers on the number line as players in a “tug-of-war” game (Figure 1–2), the larger number will be able to pull the smaller number to the sign direction of the larger number; that is, if the larger number is positive, the smaller number is pulled to the positive direction. When a negative number is added to a positive number, it is actually **subtracted** from the positive number. For example: What is the sum of −25 and +12? Because the larger number is a negative number, the sum will have a negative number. By convention, when placing the numbers in correct order to perform the calculation, the positive number is listed first followed by the negative number. The negative number is actually subtracted from the positive number to arrive at the sum.

The sum of these two numbers is −13, a negative number.

###### Example 1–5

What is the sum of −66 and +37?

When presented with an equation that contains both positive and negative numbers, first list the numbers in the equation so that the positive number(s) are listed first, followed by the negative number(s):

In this equation, the negative number was larger than the positive number, leading to the sum being a negative number. Remember: think of the numbers as being a “tug of war.” The sign of the larger number will determine the sign of the final sum (Figure 1–3). In this example, since the number −66 is a larger number than +37, the final sum of 29 is a negative number (i.e., −29).

###### Example 1–6

What is the sum of the following group of numbers?

When performing a calculation with both positive and negative numbers, always list the positive numbers first, followed by the negative numbers.

Then determine the sum of the positive numbers:

Next, determine the sum of the negative numbers:

Last, set up the final equation of the sums of the positive and negative numbers:

Now, solve for X using the rules for addition and subtraction of positive and negative numbers.

### Subtraction of Two or More Positive Numbers

If the difference is greater than zero, it remains a positive number. However, if a larger positive number is subtracted from a smaller positive number, the difference will have a value less than zero and will be a negative number.

### Subtraction of Two or More Negative Numbers

When two negative numbers are subtracted from each other, the remainder remains negative as long as it is less than zero. However, as with positive numbers, if a larger negative number is subtracted from a smaller negative number, the difference will have a value greater than zero and be a positive number. This is because when a negative number is subtracted from a positive or negative number, it is actually **added** to the positive or negative number because of the following rule:

#### Two Negatives Rule

Two negatives become a positive (Figure 1–4).

Whether the final result is a positive or negative number depends on how much “pull” there is on the number line by the numbers in the equation.

###### Example 1–9

In this equation, because the negative 14 is being subtracted from the negative 29, the double negatives convert to a positive (+) sign for the number 14.

This equation can be rearranged to have the positive number listed first:

The final answer is still a negative number because the (−14) did not have enough “pull” to pull the final result into the positive numbers.

Again, because two negatives make a positive, the sign associated with the number 27 is changed to a positive:

The equation can be rearranged so that the positive number is listed first:

In this case, the final result is a positive number because the number 27 had enough “pull” to pull the final result over the zero value in the number line.

What is the answer to the following problem?

The problem is rearranged to be:

What is the answer to the following problem?