# A Review of Math Fundamentals: Operations with Decimals

**Overview: What Are Decimals?**

The word decimal has a Latin root that means “10”. Any number written in base 10 (which is the numerical base that we generally use) is already expressed in a decimal value. The digits to the left of the decimal point are increasing powers of 10, and to the right of the decimal point are decreasing, negative powers of 10. For example, a number such as 225.321 is represented in expanded notation as (2 X 10^{2}) + (2 X 10^{1}) + (5 X 1) + (3 X 10^{-1}) + (2 X 10^{-2}) + (1 X 10^{-3}). Whole numbers, no matter what size they are, have an understood decimal point and zeros to the right of that decimal point.

**Addition of Decimals**

In order to add numbers with decimals, it is important to line up the decimal points so that digits in the same place values can be added, just as with any other whole numbers. For example, when adding 423.56 + 902.12, the 6 and 2 are added (8), then the 5 +1 (6), the 3 +2 (5), 2 +0 (2), 4+9 (13) to equal 1325.68. In the example, when the 9 and 4 were added to form 13, the 13 was regrouped for an additional increasing power of 10.

**Subtraction of Decimals**

Since subtraction is the inverse of addition, it is logical to line up the decimal points and then subtract each digit from the smallest value to the largest value. When solving 902.98 – 502.12, think 8-2 (6), 9-1 (8), 2-2 (0), 0-0 (0), 9-5 (4) to equal 400.86. Regrouping is the same as when subtracting whole numbers, if it is needed. If the problem were 902.54 – 213.82, think 4-2 (2), 15-8 (7), 11-3 (8), 9-1 (8) 8-2 (6), or 688.72. This problem has several instances of regrouping or “borrowing”, because the regrouping to solve the subtraction in the tenths place (15-8), creates a cascade effect making regrouping needed throughout the rest of the problem. Rather than 12-3 in the ones place, 10 tenths were borrowed, so that regrouping became 11-3. One ten was regrouped from the tens place, resulting in that subtraction being 9-1 rather than 10-1. Finally, 10 tens were regrouped from the hundreds place, leaving that subtraction as 8-2 rather than 9-2.

**Multiplication of Decimals**

When multiplying a number with a decimal point by another number, it is not necessary to line up the decimal points as in addition or subtraction. This is because the product is computed by multiplying the numbers as if there were no decimal points, and then adding the number of decimal places in the original numbers together to put the decimal point in the correct place in the product. For example, 3.2 x 4.3 = 13.76. This is because 3.2 X .3 equals .96 and 3.2 X 4 equals 12.8 . Adding .96 and 12.8 together equals 13.76. Using the shortcut, multiply 32 X 43 to equal 1376. Think that there is one decimal place in 3.2 plus one decimal place in 4.3 equals two decimal places. Moving the decimal point over two decimal places in 1376 gives the same answer, 13.76.

**Division of Decimals**

There are two different algorithms to consider when dividing by decimals. First, when dividing a decimal by a whole number, the decimal point in the quotient will be the same as in the dividend. For example, divide 12.69 by 3. The quotient, 4.23, has as many decimal places as the dividend, 12.69. If the divisor has a decimal point, the decimal point in the dividend is moved over the same number of spaces to the right, so that the divisor is a whole number without a decimal point. For example, divide 1.504 by .32. In order to make .32 a whole number (32), move the decimal to the right two spaces, and then do the same thing to the dividend. The dividend is then 150.4, divided by 32, gives a quotient of 4.7. This is true because 1.504/.32 is the same thing as 1.504 X 100/.32 X 100 equals 150.4/32.

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