1. The Mysterious World of Number Identity…
What’s In A Number?
The Mysterious World of
Number Identity…

2. Categories of Numbers in the REAL Number System
Natural Numbers
Whole Numbers
Integers
Rational Numbers
Irrational Numbers

3. Natural Numbers
Are the counting numbers, or all positive numbers, that mean we exclude negative numbers and zero
{1, 2, 3, 4, 5, 6, 7, 8, …}

4. Whole Numbers All of the counting numbers and zero.
{0, 1, 2, 3, 4, 5, 6, 7, …}

5. Integers
Are all of the natural numbers, their opposites (negative numbers) and zero.
{…, -4, -3, -2, -1, 0, 1, 2, 3, 4, …}

6. Rational Numbers Numbers that can be expressed as a fraction (a/b).
This set includes the integers, terminating decimals, and repeating decimals.
Some examples:
2 = 2/1
3 ¼ = 13/4
-0.25 = -25/100
1/3 =

7. Irrational Numbers
Numbers that CANNOT be expressed as a fraction of integers.
In decimal form, they are the numbers that go on forever without a repeating pattern.
Some examples:
√2 = …
π = … …

8. Venn Diagram of REAL Number System
Rational Numbers
Integers
Whole
Natural
Irrational Numbers

9. Tree Diagram of Real Number System
0.5, 1.4, 0.256
0.3, 0.45 …
Π, √2
1/5, 4/11, 12/3
-5, -80
1, 2, 3…

10. Classify each number as natural, whole, integer, rational, or irrational. Write as many as apply.…
-5 ¾
-79 3
√16

11. The Real Number System zero
The natural numbers are the counting numbers, without _______.
Whole numbers include the natural numbers and ____________.
Integers include all whole numbers and their ________________.
Rational numbers are real numbers that can be written as a _____________ where a and b are integers and b ≠ 0. Any rational number can be represented as a terminating or a repeating ___________.
Irrational numbers are any real numbers that are not _________.
zero
opposites
decimal
rational

12. Set
A collection of objects.
Set Notation
{ }
Natural
numbers
Counting numbers {1,2,3, …}
Whole
Numbers
Natural numbers and 0.
{0,1,2,3, …}
Integers
Positive and negative natural numbers and zero {… -2, -1, 0, 1, 2, 3, …}
Vocabulary
Rational
Number
A real number that can be expressed as a ratio of integers (fraction)
Irrational
Number
Any real number that is not rational.
Real Numbers
All numbers associated with the number line.

13. Translating English to Maths
sum of two numbers
difference between two numbers
The product of two numbers
the quotient of two numbers is
a + b
a - b ab b a =

14. B I M D A S ORDER OF OPERATIONS ultiplication rackets ubtraction
When there is more than one symbol of operation in an expression, it is agreed to complete the operations in a certain order. A mnemonic to help you remember this order is below.
Complete multiplication and division from left to right
Apply Indices
Complete addition and subtraction from left to right
Do any simplifying possible inside of brackets starting with innermost brackets and working out
rackets
ultiplication
ndices
ivision
ddition
ubtraction
B I M D A S

15. BIMDAS BIMDAS BIMDAS BIMDAS BIMDAS
complete addition and subtraction, left to right
complete multiplication and division, left to right
indices – apply the indice now
brackets – combine these first
BIMDAS
BIMDAS
BIMDAS
BIMDAS
BIMDAS

17. Properities of real numbers:

18. COMMUTATIVE PROPERTY
The operations of both addition and multiplication are commutative, that not applied for division or subtraction.
When adding, you can “commute” or trade the terms places
When multiplying, you can “commute” or trade the factors places

19. ASSOCIATIVE PROPERTY
The operations of both addition and multiplication are associative
When adding, you can “associate” and add any terms first and then add the other term.
When multiplying, you can “associate” and multiply any factors first and then multiply the other factor.

20. DISTRIBUTIVE PROPERTY
The operation of multiplication distributes over addition
The distributive property also holds for a factor that is multiplied on the left.

21. CAUTION: Remember that the value for a and/or b could also be positive or negative.
Rules of Signs
A positive times a negative is
A negative times a positive is
NEGATIVE
NEGATIVE
The negative of a negative
POSITIVE
A negative divided by a positive is
NEGATIVE
A positive divided by a negative or
A negative divided by a negative is
POSITIVE

24. You are familiar with the following type of numerical expressions:
3 (12)
6 ( )
(6)

25. What is a variable?
In the expression 12 + B, the letter “B” is a variable. A variable is a letter or symbol that represents an unknown value.

26. Algebraic Expressions
When variables are used with other numbers, parentheses, or operations, they create an algebraic expression. a + 2 (a) (b) 3m + 6n - 6

27. What are coefficients?
A coefficient is the number multiplied by the variable in an algebraic expression. Algebraic Expression Coefficient 6m r + 7m + 4 8, 7 14b

28. What is a term?
A term is the name given to a number, a variable, or a number and a variable combined by multiplication or division. Algebraic Expressions Terms a + 2 a, 2 3m + 6n - 6 3m, 6n, - 6

29. What are constants?
A constant is a number that cannot change its value.
In the expression: 5x + 7y - 2
the constant is - 2.

30. Figure it out!
Identify the terms, coefficients, and constants a - 6b x - 2y 3. c x + 2

31. Writing Algebraic Expressions
You can translate word phrases into variable expressions.
Examples:
Three more than a number = x + 3
The quotient of a number and 8 = y/8
Six times a number = 6 x n or 6n
15 less than a number = z – 15
The quotient of 30 and a number plus 10 = 30/x + 10.

32. The Laws of Exponents

33. Exponents
exponent
Power
base
53 means 3 factors of 5 or 5 x 5 x 5

34. #1: Exponential form: The exponent of a power indicates
The Laws of Exponents:
#1: Exponential form: The exponent of a power indicates
how many times the base multiplies itself.
n factors of x

35. #2: Multiplying Powers: If you are multiplying Powers with the same base, KEEP the BASE & ADD the EXPONENTS!
So, I get it! When you multiply Powers, you add the exponents!

36. #3: Dividing Powers: When dividing Powers with the same base, KEEP the BASE & SUBTRACT the EXPONENTS!
So, I get it!
When you divide Powers, you subtract the exponents!

37. Try these:

38. SOLUTIONS

39. SOLUTIONS

40. #4: Power of a Power: If you are raising a Power to an exponent, you multiply the exponents!
So, when I take a Power to a power, I multiply the exponents

41. #5: Product Law of Exponents: If the product of the bases is powered by the same exponent, then the result is a multiplication of individual factors of the product, each powered by the given exponent.
So, when I take a Power of a Product, I apply the exponent to all factors of the product.

42. #6: Quotient Law of Exponents: If the quotient of the
bases is powered by the same exponent, then the result is both
numerator and denominator , each powered by the given exponent.
So, when I take a Power of a Quotient, I apply the exponent to all parts of the quotient.

43. Try these:

44. SOLUTIONS

45. SOLUTIONS

46. #7: Negative Law of Exponents: If the base is powered
by the negative exponent, then the base becomes reciprocal with the
positive exponent.
So, when I have a Negative Exponent, I switch the base to its reciprocal with a Positive Exponent.
Ha Ha!
If the base with the negative exponent is in the denominator, it moves to the numerator to lose its negative sign!

47. #8: Zero Law of Exponents: Any base powered by zero exponent equals one.
So zero factors of a base equals 1. That makes sense! Every power has a coefficient of 1.

48. Try these:

49. SOLUTIONS

50. SOLUTIONS

52. Recognize Monomials