1 whole numbers and arithmetic operations
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Transcript of 1 whole numbers and arithmetic operations
![Page 1: 1 whole numbers and arithmetic operations](https://reader034.fdocuments.net/reader034/viewer/2022042611/58eed1ba1a28ab565e8b45e5/html5/thumbnails/1.jpg)
The Basics
![Page 2: 1 whole numbers and arithmetic operations](https://reader034.fdocuments.net/reader034/viewer/2022042611/58eed1ba1a28ab565e8b45e5/html5/thumbnails/2.jpg)
Whole numbers 1, 2, 3, 4,.. are called natural numbers and they are used to track whole items.
The Basics
![Page 3: 1 whole numbers and arithmetic operations](https://reader034.fdocuments.net/reader034/viewer/2022042611/58eed1ba1a28ab565e8b45e5/html5/thumbnails/3.jpg)
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 factor of x.
The Basics
![Page 4: 1 whole numbers and arithmetic operations](https://reader034.fdocuments.net/reader034/viewer/2022042611/58eed1ba1a28ab565e8b45e5/html5/thumbnails/4.jpg)
Example A. List the factors and the multiples of 12. The factors of 12:The multiples of 12:
The BasicsWhole 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 factor of x.
![Page 5: 1 whole numbers and arithmetic operations](https://reader034.fdocuments.net/reader034/viewer/2022042611/58eed1ba1a28ab565e8b45e5/html5/thumbnails/5.jpg)
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 BasicsWhole 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 factor of x.
![Page 6: 1 whole numbers and arithmetic operations](https://reader034.fdocuments.net/reader034/viewer/2022042611/58eed1ba1a28ab565e8b45e5/html5/thumbnails/6.jpg)
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 BasicsWhole 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 factor of x.
![Page 7: 1 whole numbers and arithmetic operations](https://reader034.fdocuments.net/reader034/viewer/2022042611/58eed1ba1a28ab565e8b45e5/html5/thumbnails/7.jpg)
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 x is a number that can only be divided by 1 and itself – so its factors consist of 1 and x only.
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 factor of x.
![Page 8: 1 whole numbers and arithmetic operations](https://reader034.fdocuments.net/reader034/viewer/2022042611/58eed1ba1a28ab565e8b45e5/html5/thumbnails/8.jpg)
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 x is a number that can only be divided by 1 and itself – so its factors consist of 1 and x only. The numbers 2, 3, 5, 7, 11, 13,… are prime numbers.
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 factor of x.
![Page 9: 1 whole numbers and arithmetic operations](https://reader034.fdocuments.net/reader034/viewer/2022042611/58eed1ba1a28ab565e8b45e5/html5/thumbnails/9.jpg)
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 x is a number that can only be divided by 1 and itself – so its factors consist of 1 and x only. The numbers 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 considered as a prime number.
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 factor of x.
![Page 10: 1 whole numbers and arithmetic operations](https://reader034.fdocuments.net/reader034/viewer/2022042611/58eed1ba1a28ab565e8b45e5/html5/thumbnails/10.jpg)
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 x is a number that can only be divided by 1 and itself – so its factors consist of 1 and x only. The numbers 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 considered as a prime number. Around 300 BC, Euclid gave an irrefutable argument that it’s impossible to have “only finitely many of prime numbers” so there are infinity many prime numbers.
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 factor of x.
![Page 11: 1 whole numbers and arithmetic operations](https://reader034.fdocuments.net/reader034/viewer/2022042611/58eed1ba1a28ab565e8b45e5/html5/thumbnails/11.jpg)
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
![Page 12: 1 whole numbers and arithmetic operations](https://reader034.fdocuments.net/reader034/viewer/2022042611/58eed1ba1a28ab565e8b45e5/html5/thumbnails/12.jpg)
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
![Page 13: 1 whole numbers and arithmetic operations](https://reader034.fdocuments.net/reader034/viewer/2022042611/58eed1ba1a28ab565e8b45e5/html5/thumbnails/13.jpg)
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
![Page 14: 1 whole numbers and arithmetic operations](https://reader034.fdocuments.net/reader034/viewer/2022042611/58eed1ba1a28ab565e8b45e5/html5/thumbnails/14.jpg)
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
![Page 15: 1 whole numbers and arithmetic operations](https://reader034.fdocuments.net/reader034/viewer/2022042611/58eed1ba1a28ab565e8b45e5/html5/thumbnails/15.jpg)
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
![Page 16: 1 whole numbers and arithmetic operations](https://reader034.fdocuments.net/reader034/viewer/2022042611/58eed1ba1a28ab565e8b45e5/html5/thumbnails/16.jpg)
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
![Page 17: 1 whole numbers and arithmetic operations](https://reader034.fdocuments.net/reader034/viewer/2022042611/58eed1ba1a28ab565e8b45e5/html5/thumbnails/17.jpg)
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.ExponentsTo 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
![Page 18: 1 whole numbers and arithmetic operations](https://reader034.fdocuments.net/reader034/viewer/2022042611/58eed1ba1a28ab565e8b45e5/html5/thumbnails/18.jpg)
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.ExponentsTo 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
![Page 19: 1 whole numbers and arithmetic operations](https://reader034.fdocuments.net/reader034/viewer/2022042611/58eed1ba1a28ab565e8b45e5/html5/thumbnails/19.jpg)
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.ExponentsTo 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
![Page 20: 1 whole numbers and arithmetic operations](https://reader034.fdocuments.net/reader034/viewer/2022042611/58eed1ba1a28ab565e8b45e5/html5/thumbnails/20.jpg)
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.ExponentsTo 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
![Page 21: 1 whole numbers and arithmetic operations](https://reader034.fdocuments.net/reader034/viewer/2022042611/58eed1ba1a28ab565e8b45e5/html5/thumbnails/21.jpg)
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.ExponentsTo 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 = 943 =
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
![Page 22: 1 whole numbers and arithmetic operations](https://reader034.fdocuments.net/reader034/viewer/2022042611/58eed1ba1a28ab565e8b45e5/html5/thumbnails/22.jpg)
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.ExponentsTo 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
![Page 23: 1 whole numbers and arithmetic operations](https://reader034.fdocuments.net/reader034/viewer/2022042611/58eed1ba1a28ab565e8b45e5/html5/thumbnails/23.jpg)
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.ExponentsTo 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
![Page 24: 1 whole numbers and arithmetic operations](https://reader034.fdocuments.net/reader034/viewer/2022042611/58eed1ba1a28ab565e8b45e5/html5/thumbnails/24.jpg)
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.ExponentsTo 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
![Page 25: 1 whole numbers and arithmetic operations](https://reader034.fdocuments.net/reader034/viewer/2022042611/58eed1ba1a28ab565e8b45e5/html5/thumbnails/25.jpg)
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.ExponentsTo 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
![Page 26: 1 whole numbers and arithmetic operations](https://reader034.fdocuments.net/reader034/viewer/2022042611/58eed1ba1a28ab565e8b45e5/html5/thumbnails/26.jpg)
Numbers that are factored completely may be written using the exponential notation.
The Basics
![Page 27: 1 whole numbers and arithmetic operations](https://reader034.fdocuments.net/reader034/viewer/2022042611/58eed1ba1a28ab565e8b45e5/html5/thumbnails/27.jpg)
Numbers that are factored completely may be written using the exponential notation.Hence, factored completely, 12 =
The Basics
![Page 28: 1 whole numbers and arithmetic operations](https://reader034.fdocuments.net/reader034/viewer/2022042611/58eed1ba1a28ab565e8b45e5/html5/thumbnails/28.jpg)
Numbers that are factored completely may be written using the exponential notation.Hence, factored completely, 12 = 2*2*3
The Basics
![Page 29: 1 whole numbers and arithmetic operations](https://reader034.fdocuments.net/reader034/viewer/2022042611/58eed1ba1a28ab565e8b45e5/html5/thumbnails/29.jpg)
Numbers that are factored completely may be written using the exponential notation.Hence, factored completely, 12 = 2*2*3 = 22*3,
The Basics
![Page 30: 1 whole numbers and arithmetic operations](https://reader034.fdocuments.net/reader034/viewer/2022042611/58eed1ba1a28ab565e8b45e5/html5/thumbnails/30.jpg)
Numbers that are factored completely may be written using the exponential notation.Hence, factored completely, 12 = 2*2*3 = 22*3,200 =
The Basics
![Page 31: 1 whole numbers and arithmetic operations](https://reader034.fdocuments.net/reader034/viewer/2022042611/58eed1ba1a28ab565e8b45e5/html5/thumbnails/31.jpg)
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
![Page 32: 1 whole numbers and arithmetic operations](https://reader034.fdocuments.net/reader034/viewer/2022042611/58eed1ba1a28ab565e8b45e5/html5/thumbnails/32.jpg)
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
![Page 33: 1 whole numbers and arithmetic operations](https://reader034.fdocuments.net/reader034/viewer/2022042611/58eed1ba1a28ab565e8b45e5/html5/thumbnails/33.jpg)
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
![Page 34: 1 whole numbers and arithmetic operations](https://reader034.fdocuments.net/reader034/viewer/2022042611/58eed1ba1a28ab565e8b45e5/html5/thumbnails/34.jpg)
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.
![Page 35: 1 whole numbers and arithmetic operations](https://reader034.fdocuments.net/reader034/viewer/2022042611/58eed1ba1a28ab565e8b45e5/html5/thumbnails/35.jpg)
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.
![Page 36: 1 whole numbers and arithmetic operations](https://reader034.fdocuments.net/reader034/viewer/2022042611/58eed1ba1a28ab565e8b45e5/html5/thumbnails/36.jpg)
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.
![Page 37: 1 whole numbers and arithmetic operations](https://reader034.fdocuments.net/reader034/viewer/2022042611/58eed1ba1a28ab565e8b45e5/html5/thumbnails/37.jpg)
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)
![Page 38: 1 whole numbers and arithmetic operations](https://reader034.fdocuments.net/reader034/viewer/2022042611/58eed1ba1a28ab565e8b45e5/html5/thumbnails/38.jpg)
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)
14412 12
![Page 39: 1 whole numbers and arithmetic operations](https://reader034.fdocuments.net/reader034/viewer/2022042611/58eed1ba1a28ab565e8b45e5/html5/thumbnails/39.jpg)
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)
14412 12
3 4 3 4
![Page 40: 1 whole numbers and arithmetic operations](https://reader034.fdocuments.net/reader034/viewer/2022042611/58eed1ba1a28ab565e8b45e5/html5/thumbnails/40.jpg)
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)
14412 12
3 4 3 42 22 2
![Page 41: 1 whole numbers and arithmetic operations](https://reader034.fdocuments.net/reader034/viewer/2022042611/58eed1ba1a28ab565e8b45e5/html5/thumbnails/41.jpg)
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)
14412 12
3 4 3 42 22 2
Gather all the prime numbers at the end of the branches we have 144 = 3*3*2*2*2*2 = 3224.
![Page 42: 1 whole numbers and arithmetic operations](https://reader034.fdocuments.net/reader034/viewer/2022042611/58eed1ba1a28ab565e8b45e5/html5/thumbnails/42.jpg)
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)
14412 12
3 4 3 42 22 2
Gather all the prime numbers at the end of the branches we have 144 = 3*3*2*2*2*2 = 3224.Note that we obtain the same answer regardless how we factor at each step.
![Page 43: 1 whole numbers and arithmetic operations](https://reader034.fdocuments.net/reader034/viewer/2022042611/58eed1ba1a28ab565e8b45e5/html5/thumbnails/43.jpg)
Basic LawsOf the four arithmetic operations +, – , *, and ÷, +, and * behave nicer than – or ÷.
![Page 44: 1 whole numbers and arithmetic operations](https://reader034.fdocuments.net/reader034/viewer/2022042611/58eed1ba1a28ab565e8b45e5/html5/thumbnails/44.jpg)
Basic LawsOf 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.
![Page 45: 1 whole numbers and arithmetic operations](https://reader034.fdocuments.net/reader034/viewer/2022042611/58eed1ba1a28ab565e8b45e5/html5/thumbnails/45.jpg)
Basic LawsOf 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 ÷.
![Page 46: 1 whole numbers and arithmetic operations](https://reader034.fdocuments.net/reader034/viewer/2022042611/58eed1ba1a28ab565e8b45e5/html5/thumbnails/46.jpg)
Associative Law for Addition and Multiplication
Basic LawsOf 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 ÷.
![Page 47: 1 whole numbers and arithmetic operations](https://reader034.fdocuments.net/reader034/viewer/2022042611/58eed1ba1a28ab565e8b45e5/html5/thumbnails/47.jpg)
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 LawsOf 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 ÷.
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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) + 31 + (2 + 3)
Basic LawsOf 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 ÷.
![Page 49: 1 whole numbers and arithmetic operations](https://reader034.fdocuments.net/reader034/viewer/2022042611/58eed1ba1a28ab565e8b45e5/html5/thumbnails/49.jpg)
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 LawsOf 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 ÷.
![Page 50: 1 whole numbers and arithmetic operations](https://reader034.fdocuments.net/reader034/viewer/2022042611/58eed1ba1a28ab565e8b45e5/html5/thumbnails/50.jpg)
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 LawsOf 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 ÷.
![Page 51: 1 whole numbers and arithmetic operations](https://reader034.fdocuments.net/reader034/viewer/2022042611/58eed1ba1a28ab565e8b45e5/html5/thumbnails/51.jpg)
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 LawsOf 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 ÷.
![Page 52: 1 whole numbers and arithmetic operations](https://reader034.fdocuments.net/reader034/viewer/2022042611/58eed1ba1a28ab565e8b45e5/html5/thumbnails/52.jpg)
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 LawsOf 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 ÷.
![Page 53: 1 whole numbers and arithmetic operations](https://reader034.fdocuments.net/reader034/viewer/2022042611/58eed1ba1a28ab565e8b45e5/html5/thumbnails/53.jpg)
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 LawsOf 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 ÷.
![Page 54: 1 whole numbers and arithmetic operations](https://reader034.fdocuments.net/reader034/viewer/2022042611/58eed1ba1a28ab565e8b45e5/html5/thumbnails/54.jpg)
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 LawsOf 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 ÷.
![Page 55: 1 whole numbers and arithmetic operations](https://reader034.fdocuments.net/reader034/viewer/2022042611/58eed1ba1a28ab565e8b45e5/html5/thumbnails/55.jpg)
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 LawsOf 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 ÷.
![Page 56: 1 whole numbers and arithmetic operations](https://reader034.fdocuments.net/reader034/viewer/2022042611/58eed1ba1a28ab565e8b45e5/html5/thumbnails/56.jpg)
Basic LawsCommutative Law for Addition and Multiplication
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Commutative Law for Addition and Multiplicationa + b = b + a a * b = b * a
Basic Laws
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Commutative Law for Addition and Multiplicationa + b = b + a a * b = b * a For example, 3*4 = 4*3 = 12.
Basic Laws
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Commutative Law for Addition and Multiplicationa + b = b + a a * b = b * a For example, 3*4 = 4*3 = 12. Subtraction and division don't satisfy commutative laws.
Basic Laws
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Commutative Law for Addition and Multiplicationa + 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
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Commutative Law for Addition and Multiplicationa + 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
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Commutative Law for Addition and Multiplicationa + 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
![Page 63: 1 whole numbers and arithmetic operations](https://reader034.fdocuments.net/reader034/viewer/2022042611/58eed1ba1a28ab565e8b45e5/html5/thumbnails/63.jpg)
Commutative Law for Addition and Multiplicationa + 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
![Page 64: 1 whole numbers and arithmetic operations](https://reader034.fdocuments.net/reader034/viewer/2022042611/58eed1ba1a28ab565e8b45e5/html5/thumbnails/64.jpg)
Commutative Law for Addition and Multiplicationa + 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
![Page 65: 1 whole numbers and arithmetic operations](https://reader034.fdocuments.net/reader034/viewer/2022042611/58eed1ba1a28ab565e8b45e5/html5/thumbnails/65.jpg)
Commutative Law for Addition and Multiplicationa + 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
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Commutative Law for Addition and Multiplicationa + 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
![Page 67: 1 whole numbers and arithmetic operations](https://reader034.fdocuments.net/reader034/viewer/2022042611/58eed1ba1a28ab565e8b45e5/html5/thumbnails/67.jpg)
Commutative Law for Addition and Multiplicationa + 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
![Page 68: 1 whole numbers and arithmetic operations](https://reader034.fdocuments.net/reader034/viewer/2022042611/58eed1ba1a28ab565e8b45e5/html5/thumbnails/68.jpg)
Commutative Law for Addition and Multiplicationa + 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
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Basic LawsWhen multiplying, the order of multiplication doesn’t matter.
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Basic LawsWhen multiplying, the order of multiplication doesn’t matter.When multiplying many numbers, always multiply them in pairs first.
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Basic LawsWhen multiplying, the order of multiplication doesn’t matter.When multiplying many numbers, always multiply them in pairs first. This is useful for raising exponents.
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Basic LawsWhen 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.
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Basic LawsWhen 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.
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Basic LawsWhen 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.
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Basic LawsWhen 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
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Basic LawsWhen 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 )
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Basic LawsWhen 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
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Basic LawsWhen 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 = 35or by distribute law, 5*( 3 + 4 )
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Basic LawsWhen 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 = 35or by distribute law, 5*( 3 + 4 ) = 5*3 + 5*4
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Basic LawsWhen 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 = 35or by distribute law, 5*( 3 + 4 ) = 5*3 + 5*4 = 15 + 20 = 35
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Exercise A. Do the following problems two ways. * Add the following by summing the multiples of 10 first. * Add by adding in the order.to find the correct answer. 1. 3 + 5 + 7 2. 8 + 6 + 2 3. 1 + 8 + 9 4. 3 + 5 + 15 5. 9 + 14 + 6 6. 22 + 5 + 8 7. 16 + 5 + 4 + 3 8. 4 + 13 + 5 + 79. 19 + 7 + 1 + 3 10. 4 + 5 + 17 + 311. 23 + 5 + 17 + 3 12. 22 + 5 + 13 + 2813. 35 + 6 + 15 + 7 + 14 14. 42 + 5 + 18 + 1215. 21 + 16 + 19 + 7 + 44 16. 53 + 5 + 18 + 27 + 2217. 155 + 16 + 25 + 7 + 344 18. 428 + 3 + 32 + 227 + 22
Basic Laws
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Basic Laws
19. List the all the factors and the first 4 multiples of the following numbers. 6, 9, 10, 15, 16, 24, 30, 36, 42, 56, 60.
B. Calculate.1. 33 2. 42 3. 52 4. 53 5. 62 6. 63 7. 72
8. 82 9. 92 10. 102 11. 103 12. 104 13. 105
14. 1002 15. 1003 16. 1004 17. 112 18. 122 20. Factor completely and arrange the factors from smallest to the largest in the exponential notation: 4, 8, 12, 16, 18, 24, 27, 32, 36, 45, 48, 56, 60, 63, 72, 75, 81, 120. 21. 3 * 5 * 4 * 2 22. 6 * 5 * 4 * 3 23. 6 * 15 * 3 *
224. 7 * 5 * 5 * 4 25. 6 * 7 * 4 * 3 26. 9 * 3 * 4 * 427. 2 * 25 * 3 * 4 * 2 28. 3 * 2 * 3 * 3 * 2 * 429. 3 * 5 * 2 * 5 * 2 * 4 30. 4 * 2 * 3 * 15 * 8 * 431. 24 32. 25 33. 26 34. 27 35. 28
36. 29 37. 210 38. 34 39. 35 40. 36
C. Multiply in two ways to find the correct answer.
© F. Ma