Chapter 2 Operations with Integers PDF

CHAPTER 2 OPERATIONS ON INTEGERS In this chapter, we will discuss How to add, subtract, multiply and divide sign numbers (integers) Prime and composite numbers Prime factorization How to find the GCF and LCM of numbers Target Skills Perform operations on integers. App...

CHAPTER 2 OPERATIONS ON INTEGERS In this chapter, we will discuss How to add, subtract, multiply and divide sign numbers (integers) Prime and composite numbers Prime factorization How to find the GCF and LCM of numbers Target Skills Perform operations on integers. Apply rules on signed numbers in word problems. Define prime, composites, factors and multiples based on given examples. List the prime factors of an integer Find the GCF and LCM of two or more integers.  TOPIC 1 RULES ON SIGN NUMBERS ADDITION Adding integers with the same sign Just add the addends and maintain their sign. Examples: 1. 81 + 14 = 95 (add then copy the positive sign) 2. − 5 + −5= (since both addends are negative, − 10 just add then copy the negative sign) 3. − 12 + −8 = − 20 Adding integers with unlike sign Subtract the smaller number from the larger number. Then copy the sign of the number with higher value. Examples: 1. 54 + −12 = 42(Subtract 12 from 54, then copy the sign of 54 since 54 > 12) 2. − 40 + 5 =− 35(After subtracting 5 from 40, copy negative since 40 is negative, and 40 > 5) 3. − 12 + 8 =− 4 (follow negative sign) 4. − 4 + 65 = 61 (follow the positive sign) Try!!! 1. 89 + −45 44 2. 657 + 72 729 3. −62 + 21 − 4. −218 + −221 − 5. 67 + −90 − SUBTRACTION Recall: Subtrahend minuend − = difference Minuend – where we subtract from Subtrahend – what we take away from minuend Difference – Resulting value after subtracting Subtracting integers Change the sign of the subtrahend, then proceed as to addition of integers Example: 1. 56 − 23 = 56 + −23 = 2. 125 − −34 = 125 + 34 = 3. −87 − 34 = −87 + −34 = 4. −144 − −21 = −144 + 21 − = − MULTIPLICATION Multiplying Integers Multiply the integers a. If the multiplying numbers have unlike sign, the sign of the product is negative 8 −3 =− 24 −33 2 =− 66 b. If the multiplying numbers have the same sign, the sign of the product is positive −12 −12 = 144 4 50 = 200 Examples: 1. 13 −3 =− 2. −6 −3 = 3. 10 −12 =− 4. −7 9 =− 5. −5 −5 = DIVISION Dividing Integers Divide the integers a. If the dividing numbers have unlike sign, the sign of the product is negative 81 ÷ −3 =− 27 33 −33 ÷ 2 =− 2 b. If the dividing numbers have the same sign, the sign of the product is positive −12 ÷ −12 = 1 400 ÷ 50 = 8 Examples: 1. 45 ÷− 5 =− 2. −144 ÷− 12 = 3. −120 ÷ 4 =− 4. 40 ÷ 2 = 5. −5 ÷ 5 =− Recall: 1. In adding integers with like signs, just add the numbers and copy the sign. −4 + −5 =− 9 2. In adding integers with unlike sign, subtract each other then copy the sign of the number with a higher value 56 + −30 =− 26 3. In subtracting integers, change the sign of the subtrahend, then proceed to addition. 788 − −90 = 788 + 90 = 878 Recall: 4. In multiplying and dividing integers, take note that a. If the integers being multiplied or divided have the same sign, the answer is POSITIVE b. If the integers being multiplied or divided have unlike sign, the answer is NEGATIVE  TOPIC 2 PRIME AND COMPOSITE NUMBERS Recall: Prime Numbers A natural number that has only two factors (1 and itself) The numbers is only divisible by 1 and itself Examples: 3 5 7 Composite Numbers A natural number greater than 1 that has more than two factors (aside from 1 and itself) Examples: 35 = factors are 1, 5, 7, 35 42 = factors are 1, 2, 3, 6, 7, 14, 21, 42 Is 1 prime or composite? Neither. 1 is not a prime nor composite. 1’s factor is only 1 and it doesn’t qualify in any of the two definitions. Is 2 prime or composite? 2 is prime, it is the smallest prime number. 2’s factors are 1 and itself (2) only. Prime Factorization The process of finding the product of all its factors that are also prime numbers. Every number has one prime factorization. Theorem: Every integer greater than 1 can be represented uniquely as a product of prime numbers, up to the order of the factors Prime Factorization HOW? Divide the number by the smallest prime number 2 (if possible) Divide the number by the next prime number, which is 3 (if the number is not divisible by 2) Continuously divide the results by prime numbers (in increasing manner) Example: 1. 70 2 35 5 7 Therefore, 70 = 2 ∙ 5 ∙ 7 or 70 = 2 × 5 × 7 Example: 2. 190 2 95 5 19 Therefore, 190 = 2 ∙ 5 ∙ 19 or 190 = 2 5 19 or 190 = 2 × 5 × 19 Example: 3. 36 2 18 2 9 3 3 Therefore, 36 = 2 × 2 × 3 × 3 36 = 22 × 32 Example 4. 175 5 35 5 7 Therefore, 175 = 5 × 5 × 7 175 = 52 × 7 Example 5. 99 9 11 3 3 Therefore, 99 = 3 × 3 × 11 99 = 32 × 11 Example Find the prime factors of the ff: 400 Example Find the prime factors of the ff: 810 RECALL: Prime Numbers Composite Numbers Least Prime Number Prime Factorization EXERCISES: Find the prime factors of the following by showing their tree factorization. 1. 220 2. 420 3. 625