Adding and Subtracting: Decimals
Adding and subtracting decimals involves understanding how to align numbers based on their positions relative to the decimal point. The decimal system expresses numbers in a base-ten format, where digits to the left represent nonnegative powers of ten and those to the right represent negative powers. When performing operations, it is essential to stack the decimals vertically, ensuring that their decimal points align. This alignment allows for accurate addition or subtraction, making it easier to compare digits in corresponding columns. In situations where the decimals have different numbers of digits after the decimal point, zeros can be added to the top number to ensure they match. While calculators can handle complex decimal calculations, developing mental estimation skills for everyday tasks—such as budgeting, measuring, or cooking—can be highly beneficial. Understanding how to manage decimals accurately is an essential skill in mathematics and practical life situations.
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Adding and Subtracting: Decimals
The decimal system is a way to express numbers in a base-ten format. "Deci" comes from the Latin word for "ten," indicating the centrality of powers of ten in the decimal system. Each decimal, or number written in the decimal system format, can be expressed using a decimal point (similar to a period). Digits to the left of the decimal point represent nonnegative powers of ten, and digits to the right of the decimal point represent negative powers of ten.
For example, in the decimal 203.76, the 2 represents 2 × 102, the 3 represents 3 × 100, the 7 represents 7 × 10−1, and the 6 represents 6 × 10−2. Each digit in a decimal can be thought of in terms of how many copies of that power of ten it contributes. In other words, in 203.76 there would be 2 hundreds, 3 ones (or "units"), 7 tenths, and 6 hundredths.
Although European scientists and mathematicians began using decimal arithmetic widely in the 1400s, it wasn’t until the 1800s that it became the preferred method for the commerce sector. However, once the ease of decimal arithmetic for financial calculations caught on, it grew deep roots. In fact, even after the advent of computers made binary arithmetic more logical for financial calculations, many companies continued to use an additional set of computations done with decimals, simply because their clients viewed them as more trustworthy. Thomas Jefferson has been credited with implementing the decimal-based currency system that we have today. He argued that a decimal system would be preferable, because it would be easy for students to learn and use decimal arithmetic. Interestingly, Jefferson was away when a similar decision was made regarding a decimal system for weights and measures; had he been in the country, the United States might be using the metric system today.
Overview
When adding or subtracting decimals, digits should be compared based on their position within the decimal. In other words, a digit in the tens column of one decimal should be added or subtracted from a digit in the tens column of another decimal, if possible. One easy way to maintain this alignment is to position the decimals in a vertical stack with all of the decimal points lined up vertically. In decimal subtraction, there are typically only two decimals in this stack, with zeros added at the end of the top number if necessary to have the same number of places past the decimal point as the bottom number. In decimal addition, there can be any number of decimals in the stack, as long as the decimals are arranged so that the decimal points align vertically, as shown below.
Although extensive decimal calculations can be managed with calculators or spreadsheets, being able to mentally estimate decimal addition and subtraction can be quite useful in many common everyday practices such as shopping, do-it-yourself projects around the house (measuring), and cooking.
Bibliography
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