How To Math Properties


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1 CLOSURE a + b is a real number; when you add 2 real numbers, the result is also a real number. and 5 are both real numbers, and the sum, 8, is also a real number. a b is a real number; when you subtract two real numbers the result is also a real number. 4 and 11 are both real numbers, , and the difference, 7, is also a real number. (a) (b) is a real number; when you multiply 2 real numbers, the result is also a real number. 10 and are both real numbers, (10)( ) 0, and the product, 0, is also a real number. a / b is a real number when b 0; when you divide two real numbers, the result is also a real number unless the denominator (divisor) is zero. 20 and 5 are both real numbers, 20 / 5 4, and the quotient, 4, is also a real number. VOCABULARY TOTAL or SUM is the answer to an addition problem. The numbers added are called addends. In , where 5 and 9 are addends and 14 is the total or sum. DIFFERENCE is the answer to a subtraction problem. The number subtracted is called the subtrahend. The number from which the subtrahend is subtracted is called the minuend. In , where 25 is the minuend, 8 is the subtrahend, and 17 is the difference. PRODUCT is the answer to a multiplication problem. The numbers multiplied are each called a factor. In , 15 and 6 are factors and 90 is product. QUOTIENT is the answer to a division problem. The number being divided is called the dividend. The number that you are dividing by is called the divisor. If there is a number remaining after the division process has been completed, that number is called the remainder. In , which may also be written as 5 45 or 45/5, where 45 is the dividend, 5 is the divisor and 9 is the quotient. An EXPONENT indicates the number of times the base is multiplied by itself; that is, used as a factor. In 5, 5 is the base and is the exponent, or power, and 5 (5)(5)(5) 125, notice that the base, 5, was multiplied by itself times. PRIME NUMBERS are natural numbers greater than 1 having exactly two factors, itself and one. EXAMPLES: 7 is prime because the only two natural numbers that multiply to equal 7 are 7 and 1; 1 is prime because the only two natural numbers that multiply to equal 1 are 1 and 1. COMPOSITE NUMBERS are natural numbers that have more than two factors. EXAMPLES: 15 is a composite number because 1,, 5, and 15 all multiply in some combination to equal 15; 9 is composite because 1,, and 9 all multiply in some combination to equal 9. The GREATEST COMMON FACTOR (GCF) or greatest common divisor (GCD) of a set of numbers is the largest natural number that is a factor of each of the numbers in the set; that is, the largest natural number that will divide into all of the numbers in the set without leaving a remainder. EXAMPLE. The greatest common factor (GCF) of l2, 0, and 42 is 6 because 6 divides evenly into 12, 0, and 42 without leaving remainders. 1
2 The LEAST COMMON MULTIPLE (LCM) of a set of numbers is the smallest natural number that can be divided (without remainders) by each of the numbers in the set. The least common multiple of 2,, and 4 is 12 because although 2,, and 4 divide evenly into many numbers including 48, 6, 24, and 12, the smallest is 12. The DENOMINATOR of a fraction is the number in the bottom; that is, the divisor of the indicated division of the fraction. In 5 / 8, 8 is the denominator and also the divisor in the indicated division. The NUMERATOR of a fraction is the number in the top; that is, the dividend of indicated division of the fraction. In / 4, is the numerator and also the dividend in the indicated division. FUNDAMENTAL THEOREM OF ARITHMETIC The Fundamental Theorem of Arithmetic (sometimes called the Prime Factorization Theorem) states that every composite number can be expressed as a unique product of prime numbers. EXAMPLES:15 ()(5), where 15 is composite and both and 5 are prime; 72 (2)(2)(2)()(), where 72 is composite and both 2 and are prime, notice that 72 also equals (8)(9), but this does not demonstrate the theorem because neither 8 nor 9 are prime numbers. ORDER OF OPERATIONS DESCRIPTION: The order in which addition, subtraction, multiplication, and division are performed determines the answer. ORDER (PEMDAS) 1. Parentheses: Any operations contained in parentheses are done first, if there are any. This also applies to these enclosure symbols { }, [ ], and absolute value symbols. 2. Exponents: Exponent expressions are simplified second, if there are any.. Multiplication and Division: These operations are done next in the order in which they are found, going left to right; that is, if division comes first going left to right, then it is done first. 4. Addition and Subtraction: These operations are done next in the order in which they are found going left to right; that is, if subtraction comes first, going left to right, then it is done first. DECIMAL NUMBERS The PLACE VALUE of each digit in a base ten number is determined by its position with respect to the decimal point. Each position represents multiplication by a power of ten. In 24, means 00 because it is times 10 2 ( ). 2 means 20 because it is 2 times 10 1 ( ), and 4 means 4 times one because it is 4 times 10 (10 I). There is an invisible decimal point to the right of the 4. In 5.82, 5 means 5 times one because it is 5 times 10 (10 1), 8 means 8 times one tenth because it is 8 times 10 1 ( / 10 ), and 2 means 2 times one hundredth because it is two times 10 2 ( / 100 ). 2
3 PLACE VALUE 10 2, Hundreds 10 1, Ones 101, Tenths 10, Thousandths , Thousands 10 1, Tens 102, Hundredths 104, TenThousandths WRITING DECIMAL NUMBERS AS FRACTIONS 1. Write the digits that are behind the decimal point as the numerator (top) of the fraction. 2. Write the place value of the last digit as the denominator (bottom) of the fraction. Any digits in front of the decimal point are whole numbers. In 4.068, the last digit behind the decimal point is 8 and it is in the 1000ths place; therefore, becomes 4068 / Notice the number of zeros in the denominator is equal to the number of digits behind the decimal point in the original number. 4. If the decimal is a nonterminating decimal, like 0., then use the decimal without the decimal point as the numerator, and place it over that many 9 s. Example: 0. placed over four nines, / 9999, then reduce to 1 /. ADDITION Write the decimal numbers in a vertical form with the decimal points lined up one under the other, so digits of equal place value are under each other. ADD would become because there is an implied decimal point behind the SUBTRACTION Write the decimal numbers in a vertical form with the decimal points lined up one under the other. Write additional zeros after the last digit behind the decimal point in the minuend (number on top) if needed (both the minuend and the subtrahend should have an equal number of digits behind the decimal point). In ,40.06 only has two digits behind the decimal point, so it needs two more zeros because has four digits behind the decimal point; therefore, the problem becomes:
4 MULTIPLICATION Multiply Count the number of digits behind the decimal points in all factors. Count the number of digits behind the decimal point in the answer. The answer must have the same number of digits behind the decimal point as there are digits behind the decimal points in all the factors. It is not necessary to line the decimal points up in multiplication. In (.05)(.007), multiply the numbers and count the 5 digits behind the decimal points in the problem so you can put 5 digits behind the decimal point in the product (answer); therefore, (.05)( This process works because 0. times 0.2 can be written as fractions, / 10 times 2 / 10, which equals 6 / 100 which equals 0.06 as a decimal number  two digits behind the decimal points in the problem and two digits behind the decimal point in the answer. ABSOLUTE VALUE Definition: x x, if x > 0 or x 0 and x x if x < 0; that is, the absolute value of a number is always the positive value of that number, since it is a measure of distance from the yaxis of the coordinate plane. EXAMPLES: 6 6 and 6 6, the answer is positive 6 in both cases. ADDITION If the signs of the numbers are the same, ADD. The answer has the same sign as the numbers. EXAMPLES: ( 4) + ( 9) 1 and If the signs of the numbers are different, SUBTRACT. The answer has the sign of the larger number (ignoring the signs or taking the absolute value of the numbers to determine the larger number). EXAMPLES: ( 4) + (9) 5 and (4) + ( 9) 5. SUBTRACTION Subtraction is the addition of opposites. Change subtraction to addition of the opposite number; a b a + ( b); that is, change the subtraction sign to addition and also change the sign of the number directly behind the subtraction sign to the opposite. Then follow the addition rules above. EXAMPLES: (8) (12) (8) + ( 12) 4 and ( 8) (12) ( 8) + ( 12) 20 and ( 8) ( 12) ( 8) + (12) 4. Notice the sign of the number in front of the subtraction sign never changes. MULTIPLICATION AND DIVISION Multiply or divide, then follow these rules to determine the sign of the answer. 1. If the numbers have the same signs the answer is POSITIVE. 2. If the numbers have different signs the answer is NEGATIVE.. It makes no difference which number is larger when you are trying to determine the sign of the answer. EXAMPLES: ( 2)( 5) 10 and ( 7)() 21 and ( 2)(9) 18. DOUBLE NEGATIVE ( a) a that is, the sign in front of the parentheses changes the sign of the contents of the parentheses. EXAMPLES: ( ) + or () ; also, (5a 6) 5a
5 DIVISION Rule: Always divide by a whole number. How To Math Properties If the divisor is a whole number simply divide and bring the decimal point up into the quotient (answer) )0.16 If the divisor is a decimal number, move the decimal point behind 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 (answer) ).50 > 5. ) 50. This process works because both the divisor and the dividend are actually multiplied by a power of ten, that is 10, 100, 1000, or to move the decimal point..5 x x FRACTIONS REDUCING Divide numerator (top) and denominator (bottom) by the same number, thereby renaming it to an equivalent fraction in lower terms. This process may be repeated. EXAMPLE ADDITION Change to equivalent fractions with common denominator. EXAMPLE. To evaluate these steps: follow 1. Find the least common denominator by I determining the smallest number which can I be divided evenly (no remainders) by all of the numbers in the denominators (bottoms)., 4, and 6 divide evenly into Multiply the numerator and denominator of each fraction so the fraction value has not changed but the common denominator has been obtained Add the numerators and keep the same denominator because the addition of fractions is counting equal parts
6 SUBTRACTION a b a + b where c 0 c d c How To Math Properties Change to equivalent fractions with a common denominator. 1. Find the least common denominator by determining the smallest number which can be divided evenly by all of the numbers in the denominators (bottoms). 2. Multiply the numerator and denominator by the same number so the fraction value has not changed, but the common denominator has been obtained. EXAMPLE Subtract the numerators and keep the same denominator because subtraction of fractions is finding the difference between equal parts MULTIPLICATION a b a b where c 0 and d 0 c d c d Common denominators are NOT needed. 1. Multiply the numerators (tops) and multiply the denominators (bottoms) then reduce the answer to lowest terms OR  reduce any numerator (top) with any denominator (bottom) and then multiply the numerators and multiply the denominators. DIVISION a b a d a d where c 0, d 0, b 0 c d c b c b Common denominators are NOT needed. 1. Change division to multiplication by the reciprocal; that is, flip the fraction in back of the division sign and change the division sign to a multiplication sign, 6
7 4 2 9 How To Math Properties becomes Now follow the steps for multiplication of fractions as indicated above Mixed Numbers and Improper Fractions GENERAL COMMENTS Description of mixed numbers: Whole numbers followed by fractions; that is, a whole number added to a fraction. 4 ½ means 4 + ½ and ( 4 ½) means 4 ½ Improper fractions are fractions that have a numerator (top number) larger than the denominator (bottom number). Conversions 1. Mixed number to improper fraction: Multiply the denominator (bottom) by the whole number and add the numerator (top) to find the numerator of the improper fraction. The denominator of the improper traction is the same as the denominator in the mixed number x 2. Improper fraction to mixed number: Divide the denominator into the numerator and write the remainder over the divisor (the divisor is the same number as the denominator in the improper fraction). ADDITION Add the whole numbers / means 5 ) Add the fractions by following the steps for addition of fractions in the fraction section of this study guide. If the answer has an improper fraction, change it to a mixed number and add the resulting whole number to the whole number in the answer. 4 / / / / / 5 7
8 SUBTRACTION SUBTRACT THE FRACTIONS FIRST. 1. If the fraction of the larger number is larger than the fraction of the smaller number, then follow the steps of subtracting fractions in the fraction section of this study guide and then subtract the whole numbers. 7 5 / / / / 2. If that is not the case, then borrow ONE from the whole number and add it to the fraction (must have common denominators) before subtracting. 6 2 / / / / 7 5 / 7 5 / / 7 MULTIPLICATION AND DIVISION Change each mixed number to an improper faction and follow the steps for multiplying and dividing fractions. Ratio, Proportion, and Percent RATIO Definition: Comparison between two quantities. Forms: to 5, :5, /5, / 5 PERCENTS Definition: Percent means "out of 100" or "per 100.". Percents and equivalent fractions 1. Percents can be written as fractions by placing the number over 100 and simplifying or reducing. EXAMPLES: 0% 0 / 100 / % 4.5 / / / Fractions can be changed to percents by writing them with denominators of 100. The numerator is then the percent number % Percents and decimal numbers 1. To change a percent to a decimal number, movethedecimalpoint2 places to the left because percent means "out of 100" and decimal numbers with two digits behind the decimal point also means "out of 100." 45%.45 ; 125% 1.25 ; 6%.06 ;.5%.05 because the 5 (in.5%) was already behind the decimal point and is not counted as one of the digits in the "move two places" 2. To change a decimal number to a percent, move the decimal point two places to the right. 8
9 EXAMPLES: %;.2 20%; % PROPORTION Definition: Statement of equality between two ratios or fractions. Forms: is to 5 as 9 is to 15, :5::9:15, / 5 9 / 15 SOLVING PROPORTIONS Change the fractions to equivalent fractions with common denominators, set numerators (tops) equal to each other, and solve the resulting statement. EXAMPLES: ¾ n / 20 becomes 15 / 20 n / 20, so 15 n. And for n + / 7 10 / 14 becomes n + / 7 5 / 7, so n + 5, and n 2. Cross multiply and solve the resulting equation. NOTE: cross multiplication is used to solve proportions only and may NOT be used in fraction multiplication. Cross multiplication may be described as the product of the means being equal to the product of the extremes. EXAMPLES: n, 5n 21, n 21 5, n 4 1 / n + 2, n n 22 n 7 1 / 9
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