Symbols of addition 1 ppt

2. Continue dividing. Bring down the 4 from the ones place in the dividend. How many times does 3 divide 24? Write the answer, 8, directly above the 4 in the dividend. Since 3 ϫ 8 = 24, write 24 below the other 24 and subtract 24 – 24 = 0. 18 3ͤ54 ෆ –3↓ 24 –24 0 REMAINDERS If you get a number other than zero after your last subtraction, this number is your remainder. Example 9 divided by 4. 2 4ͤ9 ෆ – 8 1 1 is the remainder. The answer is 2 r1. This answer can also be written as 2 ᎏ 1 4 ᎏ since there was one part left over out of the four parts needed to make a whole. Working with Integers Remember, an integer is a whole number or its opposite. Here are some rules for working with integers: ADDING Adding numbers with the same sign results in a sum of the same sign: (positive) + (positive) = positive and (negative) + (negative) = negative When adding numbers of different signs, follow this two-step process: 1. Subtract the positive values of the numbers. Positive values are the values of the numbers without any signs. 2. Keep the sign of the number with the larger positive value. – THEA MATH REVIEW– 90 Example –2 + 3 = 1. Subtract the positive values of the numbers: 3 – 2 = 1. 2. The number 3 is the larger of the two positive values. Its sign in the original example was positive, so the sign of the answer is positive. The answer is positive 1. Example 8 + –11 = 1. Subtract the positive values of the numbers: 11 – 8 = 3. 2. The number 11 is the larger of the two positive values. Its sign in the original example was negative, so the sign of the answer is negative. The answer is negative 3. SUBTRACTING When subtracting integers, change all subtraction signs to addition signs and change the sign of the number being subtracted to its opposite. Then follow the rules for addition. Examples (+10) – (+12) = (+10) + (–12) = –2 (–5) – (–7) = (–5) + (+7) = +2 MULTIPLYING AND DIVIDING A simple method for remembering the rules of multiplying and dividing is that if the signs are the same when mul- tiplying or dividing two quantities, the answer will be positive. If the signs are different, the answer will be nega- tive. (positive) ϫ (positive) = positive = positive (positive) ϫ (negative) = negative = negative (negative) ϫ (negative) = positive = positive Examples (10)( – 12) = – 120 – 5 ϫ – 7 = 35 – ᎏ 1 3 2 ᎏ = –4 ᎏ 1 3 5 ᎏ = 5 (negative) ᎏᎏ (negative) (positive) ᎏᎏ (negative) (positive) ᎏᎏ (positive) – THEA MATH REVIEW– 91 Sequence of Mathematical Operations There is an order in which a sequence of mathematical operations must be performed: P: Parentheses/Grouping Symbols. Perform all operations within parentheses first. If there is more than one set of parentheses, begin to work with the innermost set and work toward the outside. If more than one operation is present within the parentheses, use the remaining rules of order to determine which operation to perform first. E: Exponents. Evaluate exponents. M/D: Multiply/Divide. Work from left to right in the expression. A/S: Add/Subtract. Work from left to right in the expression. This order is illustrated by the following acronym PEMDAS, which can be remembered by using the first let- ter of each of the words in the phrase: Please Excuse My Dear Aunt Sally. Example + 27 = + 27 = + 27 = 16 + 27 = 43 Properties of Arithmetic Listed below are several properties of mathematics: ■ Commutative Property: This property states that the result of an arithmetic operation is not affected by reversing the order of the numbers. Multiplication and addition are operations that satisfy the commuta- tive property. Examples 5 ϫ 2 = 2 ϫ 5 5a = a5 b + 3 = 3 + b However, neither subtraction nor division is commutative, because reversing the order of the numbers does not yield the same result. Examples 5 – 2 ≠ 2 – 5 6 ÷ 3 ≠ 3 ÷ 6 64 ᎏ 4 (8) 2 ᎏ 4 (5 + 3) 2 ᎏ 4 – THEA MATH REVIEW– 92 ■ Associative Property: If parentheses can be moved to group different numbers in an arithmetic problem without changing the result, then the operation is associative. Addition and multiplication are associative. Examples 2 + (3 + 4) = (2 + 3) + 4 2(ab) = (2a)b ■ Distributive Property: When a value is being multiplied by a sum or difference, multiply that value by each quantity within the parentheses. Then, take the sum or difference to yield an equivalent result. Examples 5(a + b) = 5a + 5b 5(100 – 6) = (5 ϫ 100) – (5 ϫ 6) This second example can be proved by performing the calculations: 5(94) = 5(100 – 6) = 500 – 30 470 = 470 ADDITIVE AND MULTIPLICATIVE IDENTITIES AND INVERSES ■ The additive identity is the value which, when added to a number, does not change the number. For all of the sets of numbers defined above (counting numbers, integers, rational numbers, etc.), the additive identity is 0. Examples 5 + 0 = 5 –3 + 0 = –3 Adding 0 does not change the values of 5 and –3, so 0 is the additive identity. ■ The additive inverse of a number is the number which, when added to the number, gives you the addi- tive identity. Example What is the additive inverse of –3? – THEA MATH REVIEW– 93 This means, “what number can I add to –3 to give me the additive identity (0)?” –3 + ___ = 0 –3 + 3 = 0 The answer is 3. ■ The multiplicative identity is the value which, when multiplied by a number, does not change the number. For all of the sets of numbers defined previously (counting numbers, integers, rational numbers, etc.) the multiplicative identity is 1. Examples 5 ϫ 1 = 5 –3 ϫ 1 = –3 Multiplying by 1 does not change the values of 5 and –3, so 1 is the multiplicative identity. ■ The multiplicative inverse of a number is the number which, when multiplied by the number, gives you the multiplicative identity. Example What is the multiplicative inverse of 5? This means, “what number can I multiply 5 by to give me the multiplicative identity (1)?” 5 ϫ ___ = 1 5 ϫ ᎏ 1 5 ᎏ = 1 The answer is ᎏ 1 5 ᎏ . There is an easy way to find the multiplicative inverse. It is the reciprocal, which is obtained by reversing the numerator and denominator of a fraction. In the above example, the answer is the reciprocal of 5; 5 can be written as ᎏ 5 1 ᎏ , so the reciprocal is ᎏ 1 5 ᎏ . Some numbers and their reciprocals: 4 ᎏ 1 4 ᎏ ᎏ 2 3 ᎏᎏ 3 2 ᎏ – ᎏ 6 5 ᎏ – ᎏ 5 6 ᎏ Note: Reciprocals do not change sign. ᎏ 1 6 ᎏ 6 Note: The additive inverse of a number is the opposite of the number; the multiplicative inverse is the reciprocal. – THEA MATH REVIEW– 94 Factors and Multiples FACTORS Factors are numbers that can be divided into a larger number without a remainder. Example 12 ÷ 3 = 4 The number 3 is, therefore, a factor of the number 12. Other factors of 12 are 1, 2, 4, 6, and 12. The com- mon factors of two numbers are the factors that both numbers have in common. Examples The factors of 24 = 1, 2, 3, 4, 6, 8, 12, and 24. The factors of 18 = 1, 2, 3, 6, 9, and 18. From the examples above, you can see that the common factors of 24 and 18 are 1, 2, 3, and 6. From this list it can also be determined that the greatest common factor of 24 and 18 is 6. Determining the greatest common factor (GCF) is useful for simplifying fractions. Example Simplify ᎏ 1 2 6 0 ᎏ . The factors of 16 are 1, 2, 4, 8, and 16. The factors of 20 are 1, 2, 4, 5, and 20. The common factors of 16 and 20 are 1, 2, and 4. The greatest of these, the GCF, is 4. Therefore, to simplify the fraction, both numerator and denominator should be divided by 4. ᎏ 1 2 6 0 ÷ ÷ 4 4 ᎏ = ᎏ 4 5 ᎏ MULTIPLES Multiples are numbers that can be obtained by multiplying a number x by a positive integer. Example 5 ϫ 7 = 35 The number 35 is, therefore, a multiple of the number 5 and of the number 7. Other multiples of 5 are 5, 10, 15, 20, etc. Other multiples of 7 are 7, 14, 21, 28, etc. – THEA MATH REVIEW– 95 . answer is positive. The answer is positive 1. Example 8 + 11 = 1. Subtract the positive values of the numbers: 11 – 8 = 3. 2. The number 11 is the larger of the two positive values. Its sign in. common. Examples The factors of 24 = 1, 2, 3, 4, 6, 8, 12 , and 24. The factors of 18 = 1, 2, 3, 6, 9, and 18 . From the examples above, you can see that the common factors of 24 and 18 are 1, 2, 3, and 6 number without a remainder. Example 12 ÷ 3 = 4 The number 3 is, therefore, a factor of the number 12 . Other factors of 12 are 1, 2, 4, 6, and 12 . The com- mon factors of two numbers are the factors