Basic Addition and Subtraction

The number line is a model that illustrates the relationships among numbers. The complete number line is infinite and includes every real number—both positive and negative. A ruler, for example, is a portion of a number line that assigns a unit (such as inches or centimetres) to each number. Typically, number lines depict smaller numbers to the left and larger numbers to the right. For example, a portion of the number line centered on 0 might look like the following:

Because people learn about numbers in part through counting, they have a basic sense of how to order them. The number line builds on this sense by placing all the numbers (at least conceptually) from least to greatest. Whether a particular number is greater than or less than another is determined by comparing their relative positions. One number is greater than another if it is farther right on the number line. Likewise, a number is less than another if it is farther left on the number line. Symbolically, < means “is less than” and > means “is greater than.” For example, 5 > 1 and 9 < 25.

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BE CAREFUL!

When ordering negative numbers, think of the number line. Although –10 > –2 may seem correct, it is incorrect. Because –10 is to the left of –2 on the number line, –10 < –2.

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The Number Line Review

Addition

Addition is the process of combining two or more numbers. For example, one set has 4 members and another set has 5 members. To combine the sets and find out how many members are in the new set, add 4 and 5 to get the sum. Symbolically, the expression is 4 + 5, where + is the plus sign. Pictorially, it might look like the following:

To get the sum, combine the two sets of circles and then count them. The result is 9.

Another way to look at addition involves the number line. When adding 4 + 5, for example, start at 4 on the number line and take 5 steps to the right. The stopping point will be 9, which is the sum.

Counting little pictures or using the number line works for small numbers, but it becomes unwieldy for large ones—even numbers such as 24 and 37 would be difficult to add quickly and accurately. A simple algorithm enables much faster addition of large numbers. It works with two or more numbers.

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KEY POINT

The order of the numbers is irrelevant when adding.

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STEP BY STEP

Step 1. Stack the numbers, vertically aligning the digits for each place.

Step 2. Draw a plus sign (+) to the left of the bottom number and draw a horizontal line below the last number.

Step 3.  Add the digits in the ones place.

Step 4.  If the sum from Step 3 is less than 10, write it in the same column below the horizontal line. Otherwise, write the first (ones) digit below the line, then carry the second (tens) digit to the top of the next column.

Step 5. Going from right to left, repeat Steps 3–4 for the other places.

Step 6.  If applicable, write the remaining carry digit as the leftmost digit in the sum.

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Addition Review

Subtraction

Subtraction is the inverse (opposite) of addition. Instead of representing the sum of numbers, it represents the difference between them. For example, given a set containing 15 members, subtracting 3 of those members yields a difference of 12. Using the minus sign, the expression for this operation is 15 – 3 = 12. As with addition, two approaches are counting pictures and using the number line. The first case might involve drawing 15 circles and then crossing off 3 of them; the difference is the number of remaining circles (12). To use the number line, begin at 15 and move left 3 steps to reach 12.

Again, these approaches are unwieldy for large numbers, but the subtraction algorithm eases evaluation by hand. This algorithm is only practical for two numbers at a time.

STEP BY STEP

Step 1.  Stack the numbers, vertically aligning the digits in each place. Put the number you are subtracting from on top.

Step 2.  Draw a minus sign (–) to the left of the bottom number and draw a horizontal line below the stack of numbers.

Step 3.  Start at the ones place. If the digit at the top is larger than the digit below it, write the difference under the line. Otherwise, borrow from the top digit in the next-higher place by crossing it off, subtracting 1 from it, and writing the difference above it. Then add 10 to the digit in the ones place and perform the subtraction as normal.

Step 4.  Going from right to left, repeat Step 3 for the rest of the places. If borrowing was necessary, make sure to use the new digit in each place, not the original one.

Be Careful

When dealing with numbers that have units (such as weights, currencies, or volumes), addition and subtraction are only possible when the numbers have the same unit. If necessary, convert one or more of them to equivalent numbers with the same unit.

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Subtraction Review

Let’s Review!

• Numbers are positive and negative quantities and often appear in base-10 format.
• The number line illustrates the ordering of numbers.
• Addition is the combination of numbers. It can be performed by counting objects or pictures, moving on the number line, or using the addition algorithm.
• Subtraction is finding the difference between numbers. Like addition, it can be performed by counting, moving on the number line, or using the subtraction algorithm.