2. Whole numbers 1, 2, 3, 4,.. are called natural numbers because
they are used to count whole items like chickens and cows.
The Basics
3. Whole numbers 1, 2, 3, 4,.. are called natural numbers because
they are used to count whole items like chickens and cows.
If x and y are two natural numbers and x can be divided by y,
we say y is a factor of x.
The Basics
4. Whole numbers 1, 2, 3, 4,.. are called natural numbers because
they are used to count whole items like chickens and cows.
If x and y are two natural numbers and x can be divided by y,
we say y is a factor of x.
The Basics
15 can be divided by 3
5. Whole numbers 1, 2, 3, 4,.. are called natural numbers because
they are used to count whole items like chickens and cows.
If x and y are two natural numbers and x can be divided by y,
we say y is a factor of x.
Hence 3 is a factor of 15
The Basics
15 can be divided by 3
6. Whole numbers 1, 2, 3, 4,.. are called natural numbers because
they are used to count whole items like chickens and cows.
If x and y are two natural numbers and x can be divided by y,
we say y is a factor of x.
Hence 3 is a factor of 15
Example A. List the factors and the multiples of 12.
The factors of 12:
The Basics
15 can be divided by 3
7. Whole numbers 1, 2, 3, 4,.. are called natural numbers because
they are used to count whole items like chickens and cows.
If x and y are two natural numbers and x can be divided by y,
we say y is a factor of x.
Hence 3 is a factor of 15
Example A. List the factors and the multiples of 12.
The factors of 12: 1,
The Basics
15 can be divided by 3
8. Whole numbers 1, 2, 3, 4,.. are called natural numbers because
they are used to count whole items like chickens and cows.
If x and y are two natural numbers and x can be divided by y,
we say y is a factor of x.
Hence 3 is a factor of 15
Example A. List the factors and the multiples of 12.
The factors of 12: 1, 2,
The Basics
15 can be divided by 3
9. Whole numbers 1, 2, 3, 4,.. are called natural numbers because
they are used to count whole items like chickens and cows.
If x and y are two natural numbers and x can be divided by y,
we say y is a factor of x.
Hence 3 is a factor of 15
Example A. List the factors and the multiples of 12.
The factors of 12: 1, 2, 3, 4, 6, and 12.
The Basics
15 can be divided by 3
10. Whole numbers 1, 2, 3, 4,.. are called natural numbers because
they are used to count whole items like chickens and cows.
If x and y are two natural numbers and x can be divided by y,
we say y is a factor of x. We also say that x is a multiple of y.
Hence 3 is a factor of 15
Example A. List the factors and the multiples of 12.
The factors of 12: 1, 2, 3, 4, 6, and 12.
The Basics
15 can be divided by 3
11. Whole numbers 1, 2, 3, 4,.. are called natural numbers because
they are used to count whole items like chickens and cows.
If x and y are two natural numbers and x can be divided by y,
we say y is a factor of x. We also say that x is a multiple of y.
Hence 3 is a factor of 15 and that 15 is a multiple of 3.
Example A. List the factors and the multiples of 12.
The factors of 12: 1, 2, 3, 4, 6, and 12.
The Basics
12. Whole numbers 1, 2, 3, 4,.. are called natural numbers because
they are used to count whole items like chickens and cows.
If x and y are two natural numbers and x can be divided by y,
we say y is a factor of x. We also say that x is a multiple of y.
Hence 3 is a factor of 15 and that 15 is a multiple of 3.
A number x is always the multiple and a factor of itself.
Example A. List the factors and the multiples of 12.
The factors of 12: 1, 2, 3, 4, 6, and 12.
The Basics
13. Whole numbers 1, 2, 3, 4,.. are called natural numbers because
they are used to count whole items like chickens and cows.
If x and y are two natural numbers and x can be divided by y,
we say y is a factor of x. We also say that x is a multiple of y.
Hence 3 is a factor of 15 and that 15 is a multiple of 3.
A number x is always the multiple and a factor of itself.
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:
The Basics
14. Whole numbers 1, 2, 3, 4,.. are called natural numbers because
they are used to count whole items like chickens and cows.
If x and y are two natural numbers and x can be divided by y,
we say y is a factor of x. We also say that x is a multiple of y.
Hence 3 is a factor of 15 and that 15 is a multiple of 3.
A number x is always the multiple and a factor of itself.
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,
The Basics
15. Whole numbers 1, 2, 3, 4,.. are called natural numbers because
they are used to count whole items like chickens and cows.
If x and y are two natural numbers and x can be divided by y,
we say y is a factor of x. We also say that x is a multiple of y.
Hence 3 is a factor of 15 and that 15 is a multiple of 3.
A number x is always the multiple and a factor of itself.
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…
The Basics
16. Whole numbers 1, 2, 3, 4,.. are called natural numbers because
they are used to count whole items like chickens and cows.
If x and y are two natural numbers and x can be divided by y,
we say y is a factor of x. We also say that x is a multiple of y.
Hence 3 is a factor of 15 and that 15 is a multiple of 3.
A number x is always the multiple and a factor of itself.
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…
The Basics
A prime number is a natural number that can not be evenly
divided into smaller numbers (except as 1’s),
17. Whole numbers 1, 2, 3, 4,.. are called natural numbers because
they are used to count whole items like chickens and cows.
If x and y are two natural numbers and x can be divided by y,
we say y is a factor of x. We also say that x is a multiple of y.
Hence 3 is a factor of 15 and that 15 is a multiple of 3.
A number x is always the multiple and a factor of itself.
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…
The Basics
A prime number is a natural number that can not be evenly
divided into smaller numbers (except as 1’s), that is to say,
a prime number only has 1 and itself as factors.
18. Whole numbers 1, 2, 3, 4,.. are called natural numbers because
they are used to count whole items like chickens and cows.
If x and y are two natural numbers and x can be divided by y,
we say y is a factor of x. We also say that x is a multiple of y.
Hence 3 is a factor of 15 and that 15 is a multiple of 3.
A number x is always the multiple and a factor of itself.
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…
The Basics
A prime number is a natural number that can not be evenly
divided into smaller numbers (except as 1’s), that is to say,
a prime number only has 1 and itself as factors.
The numbers 2, 3, 5, 7, 11, 13,… are prime numbers.
19. Whole numbers 1, 2, 3, 4,.. are called natural numbers because
they are used to count whole items like chickens and cows.
If x and y are two natural numbers and x can be divided by y,
we say y is a factor of x. We also say that x is a multiple of y.
Hence 3 is a factor of 15 and that 15 is a multiple of 3.
A number x is always the multiple and a factor of itself.
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…
The Basics
A prime number is a natural number that can not be evenly
divided into smaller numbers (except as 1’s), that is to say,
a prime number only has 1 and itself as factors.
The numbers 2, 3, 5, 7, 11, 13,… are prime numbers.
Even numbers are not prime so prime numbers must be odd
(except for 2).
20. Whole numbers 1, 2, 3, 4,.. are called natural numbers because
they are used to count whole items like chickens and cows.
If x and y are two natural numbers and x can be divided by y,
we say y is a factor of x. We also say that x is a multiple of y.
Hence 3 is a factor of 15 and that 15 is a multiple of 3.
A number x is always the multiple and a factor of itself.
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…
The Basics
A prime number is a natural number that can not be evenly
divided into smaller numbers (except as 1’s), that is to say,
a prime number only has 1 and itself as factors.
The numbers 2, 3, 5, 7, 11, 13,… are prime numbers.
Even numbers are not prime so prime numbers must be odd
(except for 2). The number 1 is not a prime number.
21. To factor a number x means to write x as a product, i.e. write
x as a*b for some a and b.
The Basics
22. 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.
The Basics
23. 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.
The Basics
24. 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 factored completely,
The Basics
25. 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 factored completely,
2*6 is not complete because 6 is not a prime number.
The Basics
26. 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 factored completely,
2*6 is not complete because 6 is not a prime number.
Exponents
The Basics
27. 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 factored completely,
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.
The Basics
28. 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 factored completely,
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.
The Basics
29. 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 factored completely,
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 power of x.
The Basics
30. 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 factored completely,
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.
Example B. Calculate.
32 =
43 =
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 power of x.
The Basics
31. 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 factored completely,
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.
Example B. Calculate.
32 = 3*3 = 9
43 =
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 power of x.
The Basics
32. 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 factored completely,
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.
Example B. Calculate.
32 = 3*3 = 9 (note that 3*2 = 6)
43 =
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 power of x.
The Basics
33. 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 factored completely,
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.
Example B. Calculate.
32 = 3*3 = 9 (note that 3*2 = 6)
43 = 4*4*4
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 power of x.
The Basics
34. 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 factored completely,
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.
Example B. Calculate.
32 = 3*3 = 9 (note that 3*2 = 6)
43 = 4*4*4 = 64
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 power of x.
The Basics
35. 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 factored completely,
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.
Example B. Calculate.
32 = 3*3 = 9 (note that 3*2 = 6)
43 = 4*4*4 = 64 (note that 4*3 = 12)
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 power of x.
The Basics
36. Numbers that are factored completely may be written
using the exponential notation.
The Basics
37. Numbers that are factored completely may be written
using the exponential notation.
Hence, factored completely, 12 =
The Basics
38. Numbers that are factored completely may be written
using the exponential notation.
Hence, factored completely, 12 = 2*2*3
The Basics
39. Numbers that are factored completely may be written
using the exponential notation.
Hence, factored completely, 12 = 2*2*3 = 22*3,
The Basics
40. Numbers that are factored completely may be written
using the exponential notation.
Hence, factored completely, 12 = 2*2*3 = 22*3,
200 =
The Basics
41. Numbers that are factored completely may be written
using the exponential notation.
Hence, factored completely, 12 = 2*2*3 = 22*3,
200 = 8*25
The Basics
42. 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
The Basics
43. 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.
The Basics
44. 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.
The Basics
Each number has a unique form when its completely factored
into prime factors and arranged in order.
45. 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.
The Basics
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.
46. 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.
The Basics
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.
47. 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.
The Basics
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)
48. 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.
The Basics
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
49. 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.
The Basics
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
50. 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.
The Basics
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 22 2
51. 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.
The Basics
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 22 2
Gather all the prime numbers at
the end of the branches we have
144 = 3*3*2*2*2*2 = 3224.
52. 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.
The Basics
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 22 2
Gather all the prime numbers at
the end of the branches we have
144 = 3*3*2*2*2*2 = 3224.
Your Turn: Starting with:
show that we obtain the
same prime factors.
144
9 16
53. Basic Laws
Of the four arithmetic operations +, – , *, and ÷,
+, and * behave nicer than – or ÷.
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.
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 ÷.
56. Associative Law for Addition and Multiplication
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 ÷.
57. Associative Law for Addition and Multiplication
( a + b ) + c = a + ( b + c )
( a * b ) * c = a * ( b * c )
Note: We are to perform the operations inside the “( )” first.
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 ÷.
58. Associative Law for Addition 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)
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 ÷.
59. Associative Law for Addition 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)
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 ÷.
60. Associative Law for Addition 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.
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 ÷.
61. Associative Law for Addition 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.
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 ÷.
62. Associative Law for Addition 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.
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 ÷.
63. Associative Law for Addition 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 )
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 ÷.
64. Associative Law for Addition 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 )
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 ÷.
65. Associative Law for Addition 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.
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 ÷.
67. Commutative Law for Addition and Multiplication
a + b = b + a
a * b = b * a
Basic Laws
68. Commutative Law for Addition and Multiplication
a + b = b + a
a * b = b * a
For example, 3*4 = 4*3 = 12.
Basic Laws
69. 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.
Basic Laws
70. 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.
Basic Laws
71. 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.
Basic Laws
72. 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.
Basic Laws
73. 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.
Basic Laws
74. 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
Basic Laws
75. 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
Basic Laws
76. 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
Basic Laws
77. 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
Basic Laws
78. 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 = 91
Basic Laws
80. Basic Laws
When multiplying, the order of multiplication doesn’t
matter.
When multiplying many numbers, always multiply them in
pairs first.
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.
82. Basic Laws
When multiplying, the order of multiplication doesn’t
matter.
Example E.
36
When multiplying many numbers, always multiply them in
pairs first. This is useful for raising exponents.
83. Basic Laws
When multiplying, the order of multiplication doesn’t
matter.
Example E.
36 = 3*3*3*3*3*3
= 9 * 9 * 9
When multiplying many numbers, always multiply them in
pairs first. This is useful for raising exponents.
84. Basic Laws
When multiplying, the order of multiplication doesn’t
matter.
Example E.
36 = 3*3*3*3*3*3
When multiplying many numbers, always multiply them in
pairs first. This is useful for raising exponents.
85. Basic Laws
When multiplying, the order of multiplication doesn’t
matter.
Example E.
36 = 3*3*3*3*3*3
= 9 * 9 * 9
= 81 * 9 = 729
When multiplying many numbers, always multiply them in
pairs first. This is useful for raising exponents.
86. Basic Laws
When multiplying, the order of multiplication doesn’t
matter.
Example E.
36 = 3*3*3*3*3*3
= 9 * 9 * 9
= 81 * 9 = 729
When multiplying many numbers, always multiply them in
pairs first. This is useful for raising exponents.
Distributive Law : a*(b ± c) = a*b ± a*c
87. Basic Laws
When multiplying, the order of multiplication doesn’t
matter.
Example E.
36 = 3*3*3*3*3*3
= 9 * 9 * 9
= 81 * 9 = 729
When multiplying many numbers, always multiply them in
pairs first. This is useful for raising exponents.
Distributive Law : a*(b ± c) = a*b ± a*c
Example F.
5*( 3 + 4 )
88. Basic Laws
When multiplying, the order of multiplication doesn’t
matter.
Example E.
36 = 3*3*3*3*3*3
= 9 * 9 * 9
= 81 * 9 = 729
When multiplying many numbers, always multiply them in
pairs first. This is useful for raising exponents.
Distributive Law : a*(b ± c) = a*b ± a*c
Example F.
5*( 3 + 4 ) = 5*7 = 35
89. Basic Laws
When multiplying, the order of multiplication doesn’t
matter.
Example E.
36 = 3*3*3*3*3*3
= 9 * 9 * 9
= 81 * 9 = 729
When multiplying many numbers, always multiply them in
pairs first. This is useful for raising exponents.
Distributive Law : a*(b ± c) = a*b ± a*c
Example F.
5*( 3 + 4 ) = 5*7 = 35
or by distribute law,
5*( 3 + 4 )
90. Basic Laws
When multiplying, the order of multiplication doesn’t
matter.
Example E.
36 = 3*3*3*3*3*3
= 9 * 9 * 9
= 81 * 9 = 729
When multiplying many numbers, always multiply them in
pairs first. This is useful for raising exponents.
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
91. Basic Laws
When multiplying, the order of multiplication doesn’t
matter.
Example E.
36 = 3*3*3*3*3*3
= 9 * 9 * 9
= 81 * 9 = 729
When multiplying many numbers, always multiply them in
pairs first. This is useful for raising exponents.
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
