2. The Basics Whole numbers 1, 2, 3, 4,.. are called natural numbers and they are used to track whole items.
3. The Basics Whole numbers 1, 2, 3, 4,.. are called natural numbers and they are used to track whole items. If x and y are two natural numbers and x can be divided by y, we say x is a multiple of y
4. The Basics Whole numbers 1, 2, 3, 4,.. are called natural numbers and they are used to track whole items. If x and y are two natural numbers and x can be divided by y, we say x is a multiple of y or y is a factorof x.
5. The Basics Whole numbers 1, 2, 3, 4,.. are called natural numbers and they are used to track whole items. If x and y are two natural numbers and x can be divided by y, we say x is a multiple of y or y is a factorof x. Hence 15 is a multiple of 3 (or 5),
6. The Basics Whole numbers 1, 2, 3, 4,.. are called natural numbers and they are used to track whole items. If x and y are two natural numbers and x can be divided by y, we say x is a multiple of y or y is a factorof x. Hence 15 is a multiple of 3 (or 5), and 3 (or 5) is a factor of 15.
7. The Basics Whole numbers 1, 2, 3, 4,.. are called natural numbers and they are used to track whole items. If x and y are two natural numbers and x can be divided by y, we say x is a multiple of y or y is a factorof x. Hence 15 is a multiple of 3 (or 5), and 3 (or 5) is a factor of 15. Note that a number x is always the multiple of itself and the number 1.
8. The Basics Whole numbers 1, 2, 3, 4,.. are called natural numbers and they are used to track whole items. If x and y are two natural numbers and x can be divided by y, we say x is a multiple of y or y is a factorof x. Hence 15 is a multiple of 3 (or 5), and 3 (or 5) is a factor of 15. Note that a number x is always the multiple of itself and the number 1. A prime number is a natural number that can only be divided by 1 and itself.
9. The Basics Whole numbers 1, 2, 3, 4,.. are called natural numbers and they are used to track whole items. If x and y are two natural numbers and x can be divided by y, we say x is a multiple of y or y is a factorof x. Hence 15 is a multiple of 3 (or 5), and 3 (or 5) is a factor of 15. Note that a number x is always the multiple of itself and the number 1. A prime number is a natural number that can only be divided by 1 and itself. Its factors consist of 1 and itself only.
10. The Basics Whole numbers 1, 2, 3, 4,.. are called natural numbers and they are used to track whole items. If x and y are two natural numbers and x can be divided by y, we say x is a multiple of y or y is a factorof x. Hence 15 is a multiple of 3 (or 5), and 3 (or 5) is a factor of 15. Note that a number x is always the multiple of itself and the number 1. A prime number is a natural number that can only be divided by 1 and itself. Its factors consist of 1 and itself only. Hence 2, 3, 5, 7, 11, 13,… are prime numbers.
11. The Basics Whole numbers 1, 2, 3, 4,.. are called natural numbers and they are used to track whole items. If x and y are two natural numbers and x can be divided by y, we say x is a multiple of y or y is a factorof x. Hence 15 is a multiple of 3 (or 5), and 3 (or 5) is a factor of 15. Note that a number x is always the multiple of itself and the number 1. A prime number is a natural number that can only be divided by 1 and itself. Its factors consist of 1 and itself only. Hence 2, 3, 5, 7, 11, 13,… are prime numbers. 15 is not a prime number because 15 can be divided by 3 or 5.
12. The Basics Whole numbers 1, 2, 3, 4,.. are called natural numbers and they are used to track whole items. If x and y are two natural numbers and x can be divided by y, we say x is a multiple of y or y is a factorof x. Hence 15 is a multiple of 3 (or 5), and 3 (or 5) is a factor of 15. Note that a number x is always the multiple of itself and the number 1. A prime number is a natural number that can only be divided by 1 and itself. Its factors consist of 1 and itself only. Hence 2, 3, 5, 7, 11, 13,… are prime numbers. 15 is not a prime number because 15 can be divided by 3 or 5. The number 1 is not a prime number.
13. The Basics Whole numbers 1, 2, 3, 4,.. are called natural numbers and they are used to track whole items. If x and y are two natural numbers and x can be divided by y, we say x is a multiple of y or y is a factorof x. Hence 15 is a multiple of 3 (or 5), and 3 (or 5) is a factor of 15. Note that a number x is always the multiple of itself and the number 1. A prime number is a natural number that can only be divided by 1 and itself. Its factors consist of 1 and itself only. Hence 2, 3, 5, 7, 11, 13,… are prime numbers. 15 is not a prime number because 15 can be divided by 3 or 5. The number 1 is not a prime number. Example A. List the factors and the multiples of 12.
14. The Basics Whole numbers 1, 2, 3, 4,.. are called natural numbers and they are used to track whole items. If x and y are two natural numbers and x can be divided by y, we say x is a multiple of y or y is a factorof x. Hence 15 is a multiple of 3 (or 5), and 3 (or 5) is a factor of 15. Note that a number x is always the multiple of itself and the number 1. A prime number is a natural number that can only be divided by 1 and itself. Its factors consist of 1 and itself only. Hence 2, 3, 5, 7, 11, 13,… are prime numbers. 15 is not a prime number because 15 can be divided by 3 or 5. The number 1 is not a prime number. Example A. List the factors and the multiples of 12. The factors of 12: The multiples of 12:
15. The Basics Whole numbers 1, 2, 3, 4,.. are called natural numbers and they are used to track whole items. If x and y are two natural numbers and x can be divided by y, we say x is a multiple of y or y is a factorof x. Hence 15 is a multiple of 3 (or 5), and 3 (or 5) is a factor of 15. Note that a number x is always the multiple of itself and the number 1. A prime number is a natural number that can only be divided by 1 and itself. Its factors consist of 1 and itself only. Hence 2, 3, 5, 7, 11, 13,… are prime numbers. 15 is not a prime number because 15 can be divided by 3 or 5. The number 1 is not a prime number. Example A. List the factors and the multiples of 12. The factors of 12: 1, 2, 3, 4, 6, and 12. The multiples of 12:
16. The Basics Whole numbers 1, 2, 3, 4,.. are called natural numbers and they are used to track whole items. If x and y are two natural numbers and x can be divided by y, we say x is a multiple of y or y is a factorof x. Hence 15 is a multiple of 3 (or 5), and 3 (or 5) is a factor of 15. Note that a number x is always the multiple of itself and the number 1. A prime number is a natural number that can only be divided by 1 and itself. Its factors consist of 1 and itself only. Hence 2, 3, 5, 7, 11, 13,… are prime numbers. 15 is not a prime number because 15 can be divided by 3 or 5. The number 1 is not a prime number. Example A. List the factors and the multiples of 12. The factors of 12: 1, 2, 3, 4, 6, and 12. The multiples of 12: 12=1x12,
17. The Basics Whole numbers 1, 2, 3, 4,.. are called natural numbers and they are used to track whole items. If x and y are two natural numbers and x can be divided by y, we say x is a multiple of y or y is a factorof x. Hence 15 is a multiple of 3 (or 5), and 3 (or 5) is a factor of 15. Note that a number x is always the multiple of itself and the number 1. A prime number is a natural number that can only be divided by 1 and itself. Its factors consist of 1 and itself only. Hence 2, 3, 5, 7, 11, 13,… are prime numbers. 15 is not a prime number because 15 can be divided by 3 or 5. The number 1 is not a prime number. Example A. List the factors and the multiples of 12. The factors of 12: 1, 2, 3, 4, 6, and 12. The multiples of 12: 12=1x12, 24=2x12,
18. The Basics Whole numbers 1, 2, 3, 4,.. are called natural numbers and they are used to track whole items. If x and y are two natural numbers and x can be divided by y, we say x is a multiple of y or y is a factorof x. Hence 15 is a multiple of 3 (or 5), and 3 (or 5) is a factor of 15. Note that a number x is always the multiple of itself and the number 1. A prime number is a natural number that can only be divided by 1 and itself. Its factors consist of 1 and itself only. Hence 2, 3, 5, 7, 11, 13,… are prime numbers. 15 is not a prime number because 15 can be divided by 3 or 5. The number 1 is not a prime number. Example A. List the factors and the multiples of 12. The factors of 12: 1, 2, 3, 4, 6, and 12. The multiples of 12: 12=1x12, 24=2x12, 36=3x12, 48, 60, etc…
19. The Basics To factor a number x means to write x as a product, i.e. write x as a*b for some a and b.
20. The Basics To factor a number x means to write x as a product, i.e. write x as a*b for some a and b. For example 12 can be factored as 12 = 2*6 = 3*4 = 2*2*3.
21. The Basics To factor a number x means to write x as a product, i.e. write x as a*b for some a and b. For example 12 can be factored as 12 = 2*6 = 3*4 = 2*2*3. We say the factorization is complete if all the factors are prime numbers.
22. The Basics To factor a number x means to write x as a product, i.e. write x as a*b for some a and b. For example 12 can be factored as 12 = 2*6 = 3*4 = 2*2*3. We say the factorization is complete if all the factors are prime numbers. Hence 12=2*2*3 is factoredcompletely,
23. The Basics To factor a number x means to write x as a product, i.e. write x as a*b for some a and b. For example 12 can be factored as 12 = 2*6 = 3*4 = 2*2*3. We say the factorization is complete if all the factors are prime numbers. Hence 12=2*2*3 is factoredcompletely, 2*6 is not complete because 6 is not a prime number.
24. The Basics To factor a number x means to write x as a product, i.e. write x as a*b for some a and b. For example 12 can be factored as 12 = 2*6 = 3*4 = 2*2*3. We say the factorization is complete if all the factors are prime numbers. Hence 12=2*2*3 is factoredcompletely, 2*6 is not complete because 6 is not a prime number. Exponents
25. The Basics To factor a number x means to write x as a product, i.e. write x as a*b for some a and b. For example 12 can be factored as 12 = 2*6 = 3*4 = 2*2*3. We say the factorization is complete if all the factors are prime numbers. Hence 12=2*2*3 is factoredcompletely, 2*6 is not complete because 6 is not a prime number. Exponents To simplify writing repetitive multiplication, we write 22 for 2*2, we write 23 for 2*2*2, we write 24 for 2*2*2*2 and so on.
26. The Basics To factor a number x means to write x as a product, i.e. write x as a*b for some a and b. For example 12 can be factored as 12 = 2*6 = 3*4 = 2*2*3. We say the factorization is complete if all the factors are prime numbers. Hence 12=2*2*3 is factoredcompletely, 2*6 is not complete because 6 is not a prime number. Exponents To simplify writing repetitive multiplication, we write 22 for 2*2, we write 23 for 2*2*2, we write 24 for 2*2*2*2 and so on. We write x*x*x…*x as xN where N is the number of x’s multiplied to itself.
27. The Basics To factor a number x means to write x as a product, i.e. write x as a*b for some a and b. For example 12 can be factored as 12 = 2*6 = 3*4 = 2*2*3. We say the factorization is complete if all the factors are prime numbers. Hence 12=2*2*3 is factoredcompletely, 2*6 is not complete because 6 is not a prime number. Exponents To simplify writing repetitive multiplication, we write 22 for 2*2, we write 23 for 2*2*2, we write 24 for 2*2*2*2 and so on. We write x*x*x…*x as xN where N is the number of x’s multiplied to itself. N is called the exponent, or the powerof x.
28. The Basics To factor a number x means to write x as a product, i.e. write x as a*b for some a and b. For example 12 can be factored as 12 = 2*6 = 3*4 = 2*2*3. We say the factorization is complete if all the factors are prime numbers. Hence 12=2*2*3 is factoredcompletely, 2*6 is not complete because 6 is not a prime number. Exponents To simplify writing repetitive multiplication, we write 22 for 2*2, we write 23 for 2*2*2, we write 24 for 2*2*2*2 and so on. We write x*x*x…*x as xN where N is the number of x’s multiplied to itself. N is called the exponent, or the powerof x. Example B. Calculate. 32 = 43 =
29. The Basics To factor a number x means to write x as a product, i.e. write x as a*b for some a and b. For example 12 can be factored as 12 = 2*6 = 3*4 = 2*2*3. We say the factorization is complete if all the factors are prime numbers. Hence 12=2*2*3 is factoredcompletely, 2*6 is not complete because 6 is not a prime number. Exponents To simplify writing repetitive multiplication, we write 22 for 2*2, we write 23 for 2*2*2, we write 24 for 2*2*2*2 and so on. We write x*x*x…*x as xN where N is the number of x’s multiplied to itself. N is called the exponent, or the powerof x. Example B. Calculate. 32 = 3*3 = 9 43 =
30. The Basics To factor a number x means to write x as a product, i.e. write x as a*b for some a and b. For example 12 can be factored as 12 = 2*6 = 3*4 = 2*2*3. We say the factorization is complete if all the factors are prime numbers. Hence 12=2*2*3 is factoredcompletely, 2*6 is not complete because 6 is not a prime number. Exponents To simplify writing repetitive multiplication, we write 22 for 2*2, we write 23 for 2*2*2, we write 24 for 2*2*2*2 and so on. We write x*x*x…*x as xN where N is the number of x’s multiplied to itself. N is called the exponent, or the powerof x. Example B. Calculate. 32 = 3*3 = 9 (note that 3*2 = 6) 43 =
31. The Basics To factor a number x means to write x as a product, i.e. write x as a*b for some a and b. For example 12 can be factored as 12 = 2*6 = 3*4 = 2*2*3. We say the factorization is complete if all the factors are prime numbers. Hence 12=2*2*3 is factoredcompletely, 2*6 is not complete because 6 is not a prime number. Exponents To simplify writing repetitive multiplication, we write 22 for 2*2, we write 23 for 2*2*2, we write 24 for 2*2*2*2 and so on. We write x*x*x…*x as xN where N is the number of x’s multiplied to itself. N is called the exponent, or the powerof x. Example B. Calculate. 32 = 3*3 = 9 (note that 3*2 = 6) 43 = 4*4*4
32. The Basics To factor a number x means to write x as a product, i.e. write x as a*b for some a and b. For example 12 can be factored as 12 = 2*6 = 3*4 = 2*2*3. We say the factorization is complete if all the factors are prime numbers. Hence 12=2*2*3 is factoredcompletely, 2*6 is not complete because 6 is not a prime number. Exponents To simplify writing repetitive multiplication, we write 22 for 2*2, we write 23 for 2*2*2, we write 24 for 2*2*2*2 and so on. We write x*x*x…*x as xN where N is the number of x’s multiplied to itself. N is called the exponent, or the powerof x. Example B. Calculate. 32 = 3*3 = 9 (note that 3*2 = 6) 43 = 4*4*4 = 64
33. The Basics To factor a number x means to write x as a product, i.e. write x as a*b for some a and b. For example 12 can be factored as 12 = 2*6 = 3*4 = 2*2*3. We say the factorization is complete if all the factors are prime numbers. Hence 12=2*2*3 is factoredcompletely, 2*6 is not complete because 6 is not a prime number. Exponents To simplify writing repetitive multiplication, we write 22 for 2*2, we write 23 for 2*2*2, we write 24 for 2*2*2*2 and so on. We write x*x*x…*x as xN where N is the number of x’s multiplied to itself. N is called the exponent, or the powerof x. Example B. Calculate. 32 = 3*3 = 9 (note that 3*2 = 6) 43 = 4*4*4 = 64 (note that 4*3 = 12)
34. The Basics Numbers that are factored completely may be written using the exponential notation.
35. The Basics Numbers that are factored completely may be written using the exponential notation. Hence, factored completely, 12 =
36. The Basics Numbers that are factored completely may be written using the exponential notation. Hence, factored completely, 12 = 2*2*3
37. The Basics Numbers that are factored completely may be written using the exponential notation. Hence, factored completely, 12 = 2*2*3 = 22*3,
38. The Basics Numbers that are factored completely may be written using the exponential notation. Hence, factored completely, 12 = 2*2*3 = 22*3, 200 =
39. The Basics Numbers that are factored completely may be written using the exponential notation. Hence, factored completely, 12 = 2*2*3 = 22*3, 200 = 8*25
40. The Basics Numbers that are factored completely may be written using the exponential notation. Hence, factored completely, 12 = 2*2*3 = 22*3, 200 = 8*25 = 2*2*2*5*5
41. The Basics Numbers that are factored completely may be written using the exponential notation. Hence, factored completely, 12 = 2*2*3 = 22*3, 200 = 8*25 = 2*2*2*5*5 = 23*52.
42. The Basics Numbers that are factored completely may be written using the exponential notation. Hence, factored completely, 12 = 2*2*3 = 22*3, 200 = 8*25 = 2*2*2*5*5 = 23*52. Each number has a unique form when its completely factored into prime factors and arranged in order.
43. The Basics Numbers that are factored completely may be written using the exponential notation. Hence, factored completely, 12 = 2*2*3 = 22*3, 200 = 8*25 = 2*2*2*5*5 = 23*52. Each number has a unique form when its completely factored into prime factors and arranged in order. This is analogous to chemistry where each chemical has a unique chemical composition of basic elements such as H20 for water.
44. The Basics Numbers that are factored completely may be written using the exponential notation. Hence, factored completely, 12 = 2*2*3 = 22*3, 200 = 8*25 = 2*2*2*5*5 = 23*52. Each number has a unique form when its completely factored into prime factors and arranged in order. This is analogous to chemistry where each chemical has a unique chemical composition of basic elements such as H20 for water. Larger numbers may be factored using a vertical format.
45. The Basics Numbers that are factored completely may be written using the exponential notation. Hence, factored completely, 12 = 2*2*3 = 22*3, 200 = 8*25 = 2*2*2*5*5 = 23*52. Each number has a unique form when its completely factored into prime factors and arranged in order. This is analogous to chemistry where each chemical has a unique chemical composition of basic elements such as H20 for water. Larger numbers may be factored using a vertical format. Example C. Factor 144 completely. (Vertical format)
46. The Basics Numbers that are factored completely may be written using the exponential notation. Hence, factored completely, 12 = 2*2*3 = 22*3, 200 = 8*25 = 2*2*2*5*5 = 23*52. Each number has a unique form when its completely factored into prime factors and arranged in order. This is analogous to chemistry where each chemical has a unique chemical composition of basic elements such as H20 for water. Larger numbers may be factored using a vertical format. Example C. Factor 144 completely. (Vertical format) 144 12 12
47. The Basics Numbers that are factored completely may be written using the exponential notation. Hence, factored completely, 12 = 2*2*3 = 22*3, 200 = 8*25 = 2*2*2*5*5 = 23*52. Each number has a unique form when its completely factored into prime factors and arranged in order. This is analogous to chemistry where each chemical has a unique chemical composition of basic elements such as H20 for water. Larger numbers may be factored using a vertical format. Example C. Factor 144 completely. (Vertical format) 144 12 12 3 4 3 4
48. The Basics Numbers that are factored completely may be written using the exponential notation. Hence, factored completely, 12 = 2*2*3 = 22*3, 200 = 8*25 = 2*2*2*5*5 = 23*52. Each number has a unique form when its completely factored into prime factors and arranged in order. This is analogous to chemistry where each chemical has a unique chemical composition of basic elements such as H20 for water. Larger numbers may be factored using a vertical format. Example C. Factor 144 completely. (Vertical format) 144 12 12 3 4 3 4 2 2 2 2
49. The Basics Numbers that are factored completely may be written using the exponential notation. Hence, factored completely, 12 = 2*2*3 = 22*3, 200 = 8*25 = 2*2*2*5*5 = 23*52. Each number has a unique form when its completely factored into prime factors and arranged in order. This is analogous to chemistry where each chemical has a unique chemical composition of basic elements such as H20 for water. Larger numbers may be factored using a vertical format. Example C. Factor 144 completely. (Vertical format) 144 Gather all the prime numbers at the end of the branches we have 144 = 3*3*2*2*2*2 = 3224. 12 12 3 4 3 4 2 2 2 2
50. The Basics Numbers that are factored completely may be written using the exponential notation. Hence, factored completely, 12 = 2*2*3 = 22*3, 200 = 8*25 = 2*2*2*5*5 = 23*52. Each number has a unique form when its completely factored into prime factors and arranged in order. This is analogous to chemistry where each chemical has a unique chemical composition of basic elements such as H20 for water. Larger numbers may be factored using a vertical format. Example C. Factor 144 completely. (Vertical format) 144 Gather all the prime numbers at the end of the branches we have 144 = 3*3*2*2*2*2 = 3224. 12 12 3 4 3 4 Note that we obtain the same answer regardless how we factor at each step. 2 2 2 2
51. Basic Laws Of the four arithmetic operations +, – , *, and ÷, +, and * behave nicer than – or ÷.
52. Basic Laws Of the four arithmetic operations +, – , *, and ÷, +, and * behave nicer than – or ÷. We often take advantage of the this nice “behavior” of the addition and multiplication operations to get short cuts.
53. Basic Laws Of the four arithmetic operations +, – , *, and ÷, +, and * behave nicer than – or ÷. We often take advantage of the this nice “behavior” of the addition and multiplication operations to get short cuts. However, be sure you don’t mistaken apply these short cuts to – and ÷.
54. Basic Laws Of the four arithmetic operations +, – , *, and ÷, +, and * behave nicer than – or ÷. We often take advantage of the this nice “behavior” of the addition and multiplication operations to get short cuts. However, be sure you don’t mistaken apply these short cuts to – and ÷. Associative Law forAddition and Multiplication
55. Basic Laws Of the four arithmetic operations +, – , *, and ÷, +, and * behave nicer than – or ÷. We often take advantage of the this nice “behavior” of the addition and multiplication operations to get short cuts. However, be sure you don’t mistaken apply these short cuts to – and ÷. Associative Law forAddition and Multiplication ( a + b ) + c = a + ( b + c ) ( a * b ) * c = a * ( b * c ) Note: We are to perform the operations inside the “( )” first.
56. Basic Laws Of the four arithmetic operations +, – , *, and ÷, +, and * behave nicer than – or ÷. We often take advantage of the this nice “behavior” of the addition and multiplication operations to get short cuts. However, be sure you don’t mistaken apply these short cuts to – and ÷. Associative Law forAddition and Multiplication ( a + b ) + c = a + ( b + c ) ( a * b ) * c = a * ( b * c ) Note: We are to perform the operations inside the “( )” first. For example, (1 + 2) + 3 1 + (2 + 3)
57. Basic Laws Of the four arithmetic operations +, – , *, and ÷, +, and * behave nicer than – or ÷. We often take advantage of the this nice “behavior” of the addition and multiplication operations to get short cuts. However, be sure you don’t mistaken apply these short cuts to – and ÷. Associative Law forAddition and Multiplication ( a + b ) + c = a + ( b + c ) ( a * b ) * c = a * ( b * c ) Note: We are to perform the operations inside the “( )” first. For example, (1 + 2) + 3 = 3 + 3 = 6, 1 + (2 + 3)
58. Basic Laws Of the four arithmetic operations +, – , *, and ÷, +, and * behave nicer than – or ÷. We often take advantage of the this nice “behavior” of the addition and multiplication operations to get short cuts. However, be sure you don’t mistaken apply these short cuts to – and ÷. Associative Law forAddition and Multiplication ( a + b ) + c = a + ( b + c ) ( a * b ) * c = a * ( b * c ) Note: We are to perform the operations inside the “( )” first. For example, (1 + 2) + 3 = 3 + 3 = 6, 1 + (2 + 3) = 1 + 5 = 6.
59. Basic Laws Of the four arithmetic operations +, – , *, and ÷, +, and * behave nicer than – or ÷. We often take advantage of the this nice “behavior” of the addition and multiplication operations to get short cuts. However, be sure you don’t mistaken apply these short cuts to – and ÷. Associative Law forAddition and Multiplication ( a + b ) + c = a + ( b + c ) ( a * b ) * c = a * ( b * c ) Note: We are to perform the operations inside the “( )” first. For example, (1 + 2) + 3 = 3 + 3 = 6, which is the same as 1 + (2 + 3) = 1 + 5 = 6.
60. Basic Laws Of the four arithmetic operations +, – , *, and ÷, +, and * behave nicer than – or ÷. We often take advantage of the this nice “behavior” of the addition and multiplication operations to get short cuts. However, be sure you don’t mistaken apply these short cuts to – and ÷. Associative Law forAddition and Multiplication ( a + b ) + c = a + ( b + c ) ( a * b ) * c = a * ( b * c ) Note: We are to perform the operations inside the “( )” first. For example, (1 + 2) + 3 = 3 + 3 = 6, which is the same as 1 + (2 + 3) = 1 + 5 = 6. Subtraction and division are not associative.
61. Basic Laws Of the four arithmetic operations +, – , *, and ÷, +, and * behave nicer than – or ÷. We often take advantage of the this nice “behavior” of the addition and multiplication operations to get short cuts. However, be sure you don’t mistaken apply these short cuts to – and ÷. Associative Law forAddition and Multiplication ( a + b ) + c = a + ( b + c ) ( a * b ) * c = a * ( b * c ) Note: We are to perform the operations inside the “( )” first. For example, (1 + 2) + 3 = 3 + 3 = 6, which is the same as 1 + (2 + 3) = 1 + 5 = 6. Subtraction and division are not associative. For example, ( 3 – 2 ) – 1 3 – ( 2 – 1 )
62. Basic Laws Of the four arithmetic operations +, – , *, and ÷, +, and * behave nicer than – or ÷. We often take advantage of the this nice “behavior” of the addition and multiplication operations to get short cuts. However, be sure you don’t mistaken apply these short cuts to – and ÷. Associative Law forAddition and Multiplication ( a + b ) + c = a + ( b + c ) ( a * b ) * c = a * ( b * c ) Note: We are to perform the operations inside the “( )” first. For example, (1 + 2) + 3 = 3 + 3 = 6, which is the same as 1 + (2 + 3) = 1 + 5 = 6. Subtraction and division are not associative. For example, ( 3 – 2 ) – 1 = 0 3 – ( 2 – 1 )
63. Basic Laws Of the four arithmetic operations +, – , *, and ÷, +, and * behave nicer than – or ÷. We often take advantage of the this nice “behavior” of the addition and multiplication operations to get short cuts. However, be sure you don’t mistaken apply these short cuts to – and ÷. Associative Law forAddition and Multiplication ( a + b ) + c = a + ( b + c ) ( a * b ) * c = a * ( b * c ) Note: We are to perform the operations inside the “( )” first. For example, (1 + 2) + 3 = 3 + 3 = 6, which is the same as 1 + (2 + 3) = 1 + 5 = 6. Subtraction and division are not associative. For example, ( 3 – 2 ) – 1 = 0 is different from 3 – ( 2 – 1 ) = 2.
66. Basic Laws Commutative Law for Addition and Multiplication a + b = b + a a * b = b * a For example, 3*4 = 4*3 = 12.
67. Basic Laws Commutative Law for Addition and Multiplication a + b = b + a a * b = b * a For example, 3*4 = 4*3 = 12. Subtraction and division don't satisfy commutative laws.
68. Basic Laws Commutative Law for Addition and Multiplication a + b = b + a a * b = b * a For example, 3*4 = 4*3 = 12. Subtraction and division don't satisfy commutative laws. For example: 2 1 1 2.
69. Basic Laws Commutative Law for Addition and Multiplication a + b = b + a a * b = b * a For example, 3*4 = 4*3 = 12. Subtraction and division don't satisfy commutative laws. For example: 2 1 1 2. From the above laws, we get the following important facts.
70. Basic Laws Commutative Law for Addition and Multiplication a + b = b + a a * b = b * a For example, 3*4 = 4*3 = 12. Subtraction and division don't satisfy commutative laws. For example: 2 1 1 2. From the above laws, we get the following important facts. When adding, the order of addition doesn’t matter.
71. Basic Laws Commutative Law for Addition and Multiplication a + b = b + a a * b = b * a For example, 3*4 = 4*3 = 12. Subtraction and division don't satisfy commutative laws. For example: 2 1 1 2. From the above laws, we get the following important facts. When adding, the order of addition doesn’t matter. Hence to add a list numbers, it's easier to add the ones that add to multiples of 10 first.
72. Basic Laws Commutative Law for Addition and Multiplication a + b = b + a a * b = b * a For example, 3*4 = 4*3 = 12. Subtraction and division don't satisfy commutative laws. For example: 2 1 1 2. From the above laws, we get the following important facts. When adding, the order of addition doesn’t matter. Hence to add a list numbers, it's easier to add the ones that add to multiples of 10 first. Example D. 14 + 3 + 16 + 8 + 35 + 15
73. Basic Laws Commutative Law for Addition and Multiplication a + b = b + a a * b = b * a For example, 3*4 = 4*3 = 12. Subtraction and division don't satisfy commutative laws. For example: 2 1 1 2. From the above laws, we get the following important facts. When adding, the order of addition doesn’t matter. Hence to add a list numbers, it's easier to add the ones that add to multiples of 10 first. Example D. 14 + 3 + 16 + 8 + 35 + 15 = 30
74. Basic Laws Commutative Law for Addition and Multiplication a + b = b + a a * b = b * a For example, 3*4 = 4*3 = 12. Subtraction and division don't satisfy commutative laws. For example: 2 1 1 2. From the above laws, we get the following important facts. When adding, the order of addition doesn’t matter. Hence to add a list numbers, it's easier to add the ones that add to multiples of 10 first. Example D. 14 + 3 + 16 + 8 + 35 + 15 = 30 + 50
75. Basic Laws Commutative Law for Addition and Multiplication a + b = b + a a * b = b * a For example, 3*4 = 4*3 = 12. Subtraction and division don't satisfy commutative laws. For example: 2 1 1 2. From the above laws, we get the following important facts. When adding, the order of addition doesn’t matter. Hence to add a list numbers, it's easier to add the ones that add to multiples of 10 first. Example D. 14 + 3 + 16 + 8 + 35 + 15 = 30 + 11 + 50
76. Basic Laws Commutative Law for Addition and Multiplication a + b = b + a a * b = b * a For example, 3*4 = 4*3 = 12. Subtraction and division don't satisfy commutative laws. For example: 2 1 1 2. From the above laws, we get the following important facts. When adding, the order of addition doesn’t matter. Hence to add a list numbers, it's easier to add the ones that add to multiples of 10 first. ExampleD. 14 + 3 + 16 + 8 + 35 + 15 = 30 + 11 + 50 = 91
77. Basic Laws When multiplying, the order of multiplication doesn’t matter.
78. Basic Laws When multiplying, the order of multiplication doesn’t matter. When multiplying many numbers, always multiply them in pairs first.
79. Basic Laws When multiplying, the order of multiplication doesn’t matter. When multiplying many numbers, always multiply them in pairs first. This is useful for raising exponents.
80. Basic Laws When multiplying, the order of multiplication doesn’t matter. When multiplying many numbers, always multiply them in pairs first. This is useful for raising exponents. Example E. 36
81. Basic Laws When multiplying, the order of multiplication doesn’t matter. When multiplying many numbers, always multiply them in pairs first. This is useful for raising exponents. Example E. 36 = 3*3*3*3*3*3 = 9 * 9 * 9
82. Basic Laws When multiplying, the order of multiplication doesn’t matter. When multiplying many numbers, always multiply them in pairs first. This is useful for raising exponents. Example E. 36 = 3*3*3*3*3*3 = 9 * 9 * 9 = 81 * 9 = 729
83. Basic Laws When multiplying, the order of multiplication doesn’t matter. When multiplying many numbers, always multiply them in pairs first. This is useful for raising exponents. Example E. 36 = 3*3*3*3*3*3 = 9 * 9 * 9 = 81 * 9 = 729 Distributive Law :a*(b ± c) = a*b ± a*c
84. Basic Laws When multiplying, the order of multiplication doesn’t matter. When multiplying many numbers, always multiply them in pairs first. This is useful for raising exponents. Example E. 36 = 3*3*3*3*3*3 = 9 * 9 * 9 = 81 * 9 = 729 Distributive Law :a*(b ± c) = a*b ± a*c Example F. 5*( 3 + 4 )
85. Basic Laws When multiplying, the order of multiplication doesn’t matter. When multiplying many numbers, always multiply them in pairs first. This is useful for raising exponents. Example E. 36 = 3*3*3*3*3*3 = 9 * 9 * 9 = 81 * 9 = 729 Distributive Law :a*(b ± c) = a*b ± a*c Example F. 5*( 3 + 4 ) = 5*7 = 35
86. Basic Laws When multiplying, the order of multiplication doesn’t matter. When multiplying many numbers, always multiply them in pairs first. This is useful for raising exponents. Example E. 36 = 3*3*3*3*3*3 = 9 * 9 * 9 = 81 * 9 = 729 Distributive Law :a*(b ± c) = a*b ± a*c Example F. 5*( 3 + 4 ) = 5*7 = 35 or by distribute law, 5*( 3 + 4 )
87. Basic Laws When multiplying, the order of multiplication doesn’t matter. When multiplying many numbers, always multiply them in pairs first. This is useful for raising exponents. Example E. 36 = 3*3*3*3*3*3 = 9 * 9 * 9 = 81 * 9 = 729 Distributive Law :a*(b ± c) = a*b ± a*c Example F. 5*( 3 + 4 ) = 5*7 = 35 or by distribute law, 5*( 3 + 4 ) = 5*3 + 5*4
88. Basic Laws When multiplying, the order of multiplication doesn’t matter. When multiplying many numbers, always multiply them in pairs first. This is useful for raising exponents. Example E. 36 = 3*3*3*3*3*3 = 9 * 9 * 9 = 81 * 9 = 729 Distributive Law :a*(b ± c) = a*b ± a*c Example F. 5*( 3 + 4 ) = 5*7 = 35 or by distribute law, 5*( 3 + 4 ) = 5*3 + 5*4 = 15 + 20 = 35
