1 whole numbers and arithmetic operations

The Basics

Whole numbers 1, 2, 3, 4,.. are called natural numbers and they are used to track whole items.

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 factor of x.

The Basics

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.

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:


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 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.



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.



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.


http://en.wikipedia.org/wiki/Euclid%27s_theorem

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

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

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

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

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

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

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

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

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


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



32 = 3*3 = 943 =


The Basics



32 = 3*3 = 9 (note that 3*2 = 6)43 =


The Basics



32 = 3*3 = 9 (note that 3*2 = 6)43 = 4*4*4


The Basics



32 = 3*3 = 9 (note that 3*2 = 6)43 = 4*4*4 = 64


The Basics



32 = 3*3 = 9 (note that 3*2 = 6)43 = 4*4*4 = 64 (note that 4*3 = 12)


The Basics

Numbers that are factored completely may be written using the exponential notation.

The Basics

Numbers that are factored completely may be written using the exponential notation.Hence, factored completely, 12 =

The Basics

Numbers that are factored completely may be written using the exponential notation.Hence, factored completely, 12 = 2*2*3

The Basics

Numbers that are factored completely may be written using the exponential notation.Hence, factored completely, 12 = 2*2*3 = 22*3,

The Basics

Numbers that are factored completely may be written using the exponential notation.Hence, factored completely, 12 = 2*2*3 = 22*3,200 =

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

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

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.

The Basics


The Basics

Each number has a unique form when its completely factored into prime factors and arranged in order.


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.


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.


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)


The Basics


14412 12


The Basics


14412 12

3 4 3 4


The Basics


14412 12

3 4 3 42 22 2


The Basics


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.


The Basics


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.

Basic LawsOf the four arithmetic operations +, – , *, and ÷, +, and * behave nicer than – or ÷.

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.

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 ÷.

Associative Law for Addition and Multiplication


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.


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)


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)


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.


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.


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.


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 )


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 )


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 LawsCommutative Law for Addition and Multiplication

Commutative Law for Addition and Multiplicationa + b = b + a a * b = b * a

Basic Laws

Commutative Law for Addition and Multiplicationa + b = b + a a * b = b * a For example, 3*4 = 4*3 = 12.

Basic Laws

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

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

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

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

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

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


14 + 3 + 16 + 8 + 35 + 15 = 30

Basic Laws


14 + 3 + 16 + 8 + 35 + 15 = 30 + 50

Basic Laws


14 + 3 + 16 + 8 + 35 + 15 = 30 + 11 + 50

Basic Laws


14 + 3 + 16 + 8 + 35 + 15 = 30 + 11 + 50 = 91

Basic Laws

Basic LawsWhen multiplying, the order of multiplication doesn’t matter.

Basic LawsWhen multiplying, the order of multiplication doesn’t matter.When multiplying many numbers, always multiply them in pairs first.

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.


Example E. 36

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



Example E. 36 = 3*3*3*3*3*3 = 9 * 9 * 9 = 81 * 9 = 729



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 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 )


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


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 = 35or by distribute law, 5*( 3 + 4 )


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 = 35or by distribute law, 5*( 3 + 4 ) = 5*3 + 5*4


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 = 35or by distribute law, 5*( 3 + 4 ) = 5*3 + 5*4 = 15 + 20 = 35

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

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

1 whole numbers and arithmetic operations

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