# Properties of decimals - addition, substraction, multiplication, division, table

The place value of each digit in a base-10 number is determined by its position with respect to the decimal point; each position represents multiplication by a power of 10

EX: In 324, 3 means 300 because it is 3 times 10^{2} (10^{2} = 100), 2 means 20 because it is 2 times 10^{1} (10^{1} = 10), and 4 means 4 times 1 because it is 4 times 10^{0} (10^{0} = 1); because this is a whole number, the decimal point is to the right of the digit 4;

in 5.82, 5 means 5 times 1 because it is 5 times 10^{0} (10^{0} = 1), 8 means 8 times 0.1 because it is 8 times 10^{-1} (10^{-1} = 0.1 = 1/10), and 2 means 2 times 0.01 because it is 2 times 10^{-2} (10^{-2} = 0.01 = 1/100)

### Properties of decimals table

WRITING DECIMALS AS FRACTIONS |
• Write the digits that are to the right of the decimal point as the numerator (top) of the fraction • Write the place value of the last digit as the denominator (bottom) of the fraction; any digits to the left of the decimal point are whole numbers EX: In 4.068, the last digit to the right of the decimal point is 8 and it is in the 1,000ths place; therefore, 4.068 becomes 4 x 68/1000 Note that the number of zeros in the denominator is equal to the number of digits to the right of the decimal point in the original number |

ADDITION DECIMALS |
• Write the decimals in vertical form with the decimal points lined up one under the other so that digits of equal place value are under each other • Add EX: 23.045 + 7.5 + 143 + 0.034 would become because there is a decimal point after 143 |

SUBTRACTION |
• Write the decimals in vertical form with the decimal points lined up one under the other • Write additional zeros after the last digit to the right of the decimal point in the minuend (number on top) if needed (both the minuend and the subtrahend should have an equal EX: In 340.06 - 27.3057, 340.06 only has 2 digits to the right of the decimal point, so it needs 2 more zeros because 27.3057 has 4 digits to the right of the decimal point; there-fore, the problem becomes: |

MULTIPLICATION |
• Multiply the factors, ignoring any decimal points • Count the number of digits to the right of the decimal points in all factors • Count the number of digits to the right of the decimal point in the product (answer); the answer must have the same number of digits to the right of the decimal point as there are digits to the right of the decimal points in all the factors; it is not necessary to line the decimal points up in multiplication EX: In (3.05)(0.007), multiply the numbers (ignoring the decimal points) and count the 5 digits to the right of the decimal points in the problem so you can put 5 digits to the right of the decimal point in the product (answer); therefore, (3.05)(0.007) = 0.02135 |

DIVISION DECIMALS |
•To divide decimals, use this rule: always divide by a whole number • If the divisor is a whole number, simply divide and bring the decimal point up into the quotient (answer) • If the divisor is a decimal, move the decimal point to the right of the last digit and move the decimal point in the dividend the same number of places; divide and bring the decimal point up into the quotient • This process works because both the divisor and the dividend are actually multiplied by a power of 10 (i.e., 10, 100, 1,000, or 10,000) to move the decimal point |

SCIENTIFIC NOTATION |
• Scientific notation is a way to describe very large or very small numbers using powers of 10; a number written in scientific notation has two factors — one between 1 and 10 and one that is a power of 10; when the power of 10 has a positive exponent, the number is greater than 1; when the power of 10 has a negative exponent, the number is less than 1 |

WRITING LARGE NUMBERS |
• Find the first factor by writing the first digit followed by a decimal point and then writing the other digits up to when the remaining digits are 0; then write a power of 10 using a positive exponent that is one less than the number of digits in the original number EX: The number 57,000,000,000 written in scientific notation is 5.7 x 10 |

WRITING SMALL NUMBERS |
• Find the first factor by writing the first nonzero digit followed by a decimal point and then writing the remaining digits; then write a power of 10 using a negative exponent that is one more than the number of zeros to the right of the decimal point before the first nonzero digit in the original number EX: The number 0.00000652 written in scientific notation is 6.52 x 10 |

WRITING NUMBERS GIVEN IN SCIENTIFIC NOTATION |
• The power of 10 tells how many places to move the decimal point; fill with zeros as needed EX: The number 1.93 x 10 EX: The number 5.4 x 10 |

MULTIPLYING IN SCIENTIFIC NOTATION |
• Multiply the first factors; use the rules of exponents to combine the powers of 10; adjust the answer as needed to be in proper scientific notation form EX: (4.08 x 10 EX: (9.4 x 10 Note that because 78.396 is not between 1 and 10, the answer had to be adjusted |

DIVIDING IN SCIENTIFIC NOTATION |
• Divide the first factors; use the rules of exponents to combine the powers of 10; adjust the answer as needed to be in proper scientific notation form EX: (7.48 x 10 |