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College Mathematics Notes Section 1.2 Page 1 of 8 Chapter 1: Pre-Algebra Section 1.2: Decimals Big Idea for this section: A decimal gives you another way to represent a fraction. It is based on a base-10 positional notation, so many of the rules you already know for the arithmetic of whole numbers apply to decimals. Big Skills for this section: You should be able to do arithmetic with decimals. That means be able to add, subtract multiply, and divide fractions, and convert back and forth between fractions and decimals. Section 1.2.1: The Decimal Representation (for Terminating Decimals) Positional notation as increasing factors of 10… Positional notation as decreasing factors of 10… once get past 1 10, you’re in the decimal range… The positional notation can be used to write a fraction equivalent: 1 1 1 7 5 10 100 1000 3 7 5 10 100 1000 300 70 5 1000 1000 1000 375 1000 3 125 8 125 3 8 0.375 3 Or you can use the trick of writing a fraction with the digits after the decimal in the numerator and the next larger power of 10 in the denominator: 5128 10,000 2 2 2 641 22225555 641 25555 641 1250 0.5128 375 1000 3 125 8 125 3 8 0.375 Practice: 1. Write 0.3125 as a fraction. College Mathematics Notes Section 1.2 Page 2 of 8 Section 1.2.2: General Conversion between Fractions and Decimals To convert from fraction to a decimal, you can try to write the fraction as an equivalent fraction with a denominator that is a power of 10. 1 2 3 4 7 16 1 3 Or you can do long division, remembering to put a decimal at the end if quotient and dividend. Stop when you get zero for a remainder… 3 4 7 16 Or you can punch it into your calculator. For some fractions, you never get a remainder of zero no matter how long you do the long division. These fractions have a decimal representation that repeats forever, and are called non-terminating or repeating decimals. You know you have a repeating decimal when the long-division pattern keeps on repeating. 1 3 College Mathematics Notes Section 1.2 A bar over the repeating pattern is frequently used to represent a repeating decimal. 1 3 1 11 Page 3 of 8 To convert a repeating decimal to a fraction… Practice: College Mathematics Notes Section 1.2 Page 4 of 8 Section 1.2.3: Rounding Decimals A place to round to will be specified for you with phrases like “round to two decimal places” or “round to the nearest hundredth”. The idea is that you will re-write the number, but only up to that place, and with the last digit rounded according to the following rules: o If the next digit is less than 5, leave the last digit alone o If the next digit is 5 or more, add 1 to the last digit. For an example, here is how to round the number 10,547.395 to different specified decimal places: 10,547.395 rounded to 2 places 1 place the nearest unit the nearest ten the nearest hundred the nearest thousand Decimal Place of Rounding hundredth’s place tenth’s place one’s place ten’s place hundred’s place thousand’s place Practice: 2. Round 1.2479 to the nearest thousandth. Result 10,547.40 10,547.4 10,547. 10,550 10,500 11,000 College Mathematics Notes Section 1.2 Page 5 of 8 Section 1.2.4: Adding and Subtracting Decimals To add: Line up the decimal points and add the digits from right to left, carrying as necessary. To subtract: Line up the decimal points and subtract the digits from right to left, borrowing as necessary. Practice: 3. 15.156 + 0.9957 = 4. 10.295 – 7.87 = 5. 158.34 – 77.473 = College Mathematics Notes Section 1.2 Page 6 of 8 Section 1.2.5: Multiplying Decimals Multiply as if there were no decimals present The answer will have as many decimal places as all the decimal places of both factors added up. Practice: 1. 2.5 1.7 = College Mathematics Notes Section 1.2 Page 7 of 8 Section 1.2.6: Dividing Decimals Move the decimal point the same number of places in the divisor and dividend until the divisor is a whole number. Then do standard long division, putting a decimal point in the quotient right above the decimal point in the dividend. Practice: 2. 3.82 1.2 = College Mathematics Notes Section 1.2 Section 1.2.7: Applications of the Arithmetic of Decimals Practice: 3. Page 8 of 8