1 Decimals Adding and Subtracting Decimals are a group of digits, which express numbers or measurements in units, tens, and multiples of 10. The digits for units and multiples of 10 are followed by a decimal point and then by the digits for tenths, hundredths etc. 1. ADDING MORE THAN TWO POSITIVE DECIMAL NUMBERS When adding more than two positive decimal numbers, the sum is positive. Align the decimal points. Decimals must always be lined up. Add the decimals as if they are whole numbers. To add decimal numbers: 1. Put the numbers in a vertical column aligning the decimal points. 1. Add each column of digits, starting on the right and working left. If the sum of a column is more than 10, "carry" digits to the next column on the left. 2. Place the decimal point in the answer directly below the decimal points in the terms. Example: 12.45 10.2 15.22 8.654 46.524 2. ADDING TWO NEGATIVE DECIMAL NUMBERS When adding two negative decimal numbers, the sum will be negative and smaller than either of the decimal numbers. Example (- 0.37) + (-0.29) = (- 0.66) 3. ADDING MORE THAN TWO NEGATIVE DECIMAL NUMBERS If you add more than two negative decimal numbers, the sum will be negative and smaller than each of the decimal numbers. Again, the order in which they are added does not matter. Example (-0.3) + (-0.8) + (-0.2) + (-0.7) + (-0.4) = ( -2.4) 4. ADDITION WITH DECIMAL NUMBERS WITH MIXED SIGNS
2 If you add decimal numbers with both positive and negative signs, the sign of the larger of the positive or negative sum will determine the sign. Example (0.5) + (-0.8) + (0.2) = (-0.1) Because 0.5 + 0.2 = 0.7 and 0.7 and 0.7 + (-0.8) = (-0.1) 5. READING DECIMAL NUMBERS ON A NUMBER LINE As with other numbers, decimal numbers can be read on a number line. The line here indicates that this number is (+1.3). This number line shows that decimal numbers can be negative or positive. Decimals Multiplying 1. MULTIPLYING TWO POSITIVE DECIMAL NUMBERS When multiplying two positive decimal numbers, the product is always positive and larger than either factor. Example (+ 0.5) x + (0.8) = (+ 0.40)
3 2. MULTIPLYING MORE THAN TWO POSITIVE DECIMAL NUMBERS When multiplying more than two decimal numbers, the product is always positive and larger than each of the factors. Example 0.9 x 0.5 x 0.2 Three decimal places. = 0.45 x 0.2 Three decimal places. = 0.090 Three decimal places. The number of decimal places in the product should be the total number of decimal places in the factors. 0.9 has one decimal place, 0.5 has one decimal place and 0.2 has one decimal place. The product must have three decimal places. 3. MULTIPLYING TWO NEGATIVE DECIMAL FACTORS When you multiply two negative decimal numbers, the product is always positive and larger than either of the factors. Example (-0.7) x ( 0.3) = (+0.21) 4. MULTIPLYING MORE THAN TWO NEGATIVE DECIMAL NUMBERS When you multiply more than two negative decimal numbers, the product is always negative if there is an odd number of factors. The product is positive if there is an even number of factors. Example (-0.3) x ( 0.2) x ( 0.5) = (- 0.030) There are three factors and 3 is an odd number (-0.3) x ( 0.2) x ( 0.5) x (-0.4) = (+ 0.0120) There are four factors and 4 is an even number. 5. MULTIPLYING DECIMAL NUMBERS WITH MIXED SIGNS When multiplying a series of decimal numbers, the product is negative if the number of negative factors is odd. The product is positive if the number of negative factors is even. Example (-0.6) x (+1.3) x ( 2.33) x (+52.1) = (+94.68654) There are two negative factors and 2 is an even number. 6. REVIEW To multiply decimal numbers, follow these steps:
4 1) Multiply the numbers just as if they were whole numbers. Line up the numbers on the right - do not align the decimal points. Starting on the right, multiply each digit in the top number by each digit in the bottom number, just as with whole numbers. Add the products. 2) Place the decimal point in the answer by starting at the right and moving a number of places equal to the sum of the decimal places in both numbers multiplied. Decimals Dividing 1. DIVIDING DECIMAL NUMBERS When dividing decimal numbers, write the numbers in standard division form. Move the decimal point over the same number of places in the divisor and in the dividend. Remember these steps: 1) If the divisor is not a whole number, move the decimal point to right to make it a whole number and move the decimal point in the dividend the same number of places. 2) Divide as usual. Keep dividing until the answer terminates or repeats.
5 3) Put decimal point directly above decimal point in the dividend. 4) Check your answer. Multiply quotient by divisor. Does it equal the dividend? Example 3. 552. Decimal has moved in the divisor and in the dividend. Then divide as you would with whole numbers. Answer: 184 2. DIVIDING TWO POSITIVE DECIMAL NUMBERS When dividing one positive decimal number by another positive decimal number, the answer or the quotient is positive. Example (+0.56) (+0.4) = (+1.4)
6 3. DIVIDING TWO NEGATIVE DECIMAL NUMBERS When you divide one negative decimal number by another negative decimal number, the answer or quotient is positive. Example (-6.7) (-0.32) = (+7.02) 4. Dividing Decimal Numbers with Mixed Signs When dividing one decimal number by another, if one is positive and the other is negative, the answer or quotient is negative. Example (+0.85) (-0.05) = (-17) A thermometer is like a vertical number line. The reading on this thermometer would be (+.7) degrees Celsius.
7 The reading on this thermometer would be (-.7) degrees Celsius. 6. SUBTRACTING TWO POSITIVE DECIMAL NUMBERS When you subtract two positive decimal numbers, the sign following the subtraction sign, must be changed. Example (+0.57) (+0.17) = 0.57 0.17 = 0.40 The + sign in front of the 0.17 was changed to a sign. 7. SUBTRACTING TWO NEGATIVE DECIMAL NUMBERS When you subtract two negative decimal numbers, the sign following the subtraction sign must be changed. Example (-0.89) -(-0.59) = (- 0.89) + (+0.59) = (-0.30) The sign in front of the 0.59 was changed to a + sign. 8. SUBTRACTING TWO DECIMAL NUMBERS WITH MIXED SIGNS When you subtract two decimal numbers with mixed signs, the sign following the subtraction sign must be changed. Example (-0.97) (+ 0.61) = (-0.97) + (-0.61) = (-1.58)
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