2. Real numbers
In algebra, we work with the set of real numbers, which we can model using a number line.
Real numbers describe real-world quantities such as amounts, distances, age, temperature, and so on. A real number can be an integer, a fraction, or a decimal. They can also be either rational or irrational. Numbers that are not "real" are called imaginary. Imaginary numbers are used by mathematicians to describe numbers that cannot be found on the number line. They are a more complex subject than we will work with here

3. Properties of real numbers
In this lesson we look at some properties that apply to all real numbers. If you learn these properties, they will help you solve problems in algebra. Let's look at each property in detail, and apply it to an algebraic expression.

4. 1. Commutative properties
The commutative property of addition says that we can add numbers in any order. The commutative property of multiplication is very similar. It says that we can multiply numbers in any order we want without changing the result.
addition
5a + 4 = 4 + 5a
multiplication 3 x 8 x 5b = 5b x 3 x 8

5. 2.Associative properties
Both addition and multiplication can actually be done with two numbers at a time. So if there are more numbers in the expression, how do we decide which two to "associate" first? The associative property of addition tells us that we can group numbers in a sum in any way we want and still get the same answer. The associative property of multiplication tells us that we can group numbers in a product in any way we want and still get the same answer.
addition (4x + 2x) + 7x = 4x + (2x + 7x)
multiplication 2x2(3y) = 3y(2x2)

6. 3.Distributive property
The distributive property comes into play when an expression involves both addition and multiplication. A longer name for it is, "the distributive property of multiplication over addition." It tells us that if a term is multiplied by terms in parenthesis, we need to "distribute" the multiplication over all the terms inside.
2x(5 + y) = 10x + 2xy
Even though order of operations says that you must add the terms inside the parenthesis first, the distributive property allows you to simplify the expression by multiplying every term inside the parenthesis by the multiplier. This simplifies the expression.

7. 4. Density property
The density property tells us that we can always find another real number that lies between any two real numbers. For example, between 5.61 and 5.62, there is 5.611, 5.612, and so forth.
Between and 5.613, there is , and an endless list of other numbers!

8. 5. Identity property
The identity property for addition tells us that zero added to any number is the number itself. Zero is called the "additive identity." The identity property for multiplication tells us that the number 1 multiplied times any number gives the number itself. The number 1 is called the "multiplicative identity."
Addition 5y + 0 = 5y
Multiplication 2c × 1 = 2c

10. Adding Real Numbers
This tutorial reviews adding real numbers as well as finding the additive inverse or opposite of a number

11. Adding Real Numbers with the same sign
Step 1: Add the absolute values
Step 2: Attach their common sign to sum .
In other words:
If both numbers that you are adding are positive, then you will have a positive answer.
If both numbers that you are adding are negative then you
will have a negative answer.

12. Example 1: Add -6 + (-8). -6 + (-8) = -14
-6 + (-8) = -14
The sum of the absolute values would be 14 and their common sign is -. That is how we get the answer of -14.
You can also think of this as money. I know we can all relate to that. Think of the negative as a loss. In this
example, you can think of it as having
lost 6 dollars and then having lost
another 8 dollars for a total loss of 14
dollars. .

13. Example 2: Add -5.5 + (-8.7). -5.5 + (-8.7) = -14.2
(-8.7) = -14.2
The sum of the absolute values would be 14.2 and their common sign is -. That is how we get the answer of
You can also think of this as money - I know we can all
relate to that. Think of the negative as a loss. In this
example, you can think of it as having
lost 5.5 dollars and then having lost
another 8.7 dollars for a total loss of
14.2 dollars. .

14. The Real Numbers Sets of Numbers
* Natural Numbers {1, 2, 3, 4, . . .}
* Whole Numbers {0, 1, 2, 3, 4, . . .}
* Integers {. . . , -3, -2, -1, 0, 1, 2, 3, . . .}

15. Rational Numbers { | p and q are integers and q ¹ 0 }
The set of rational numbers contains all numbers that can be written as fractions, or quotients of integers. Integers are also rational numbers since they can be represented as fractions. All decimals that repeat or terminate belong to the set of rational numbers. The following are all rational numbers:

16. , , , -5 = , 0 = , = , =

17. {x | x is real but not rational }
, -
Irrational Numbers
{x | x is real but not rational }
The irrational numbers are nonrepeating, nonterminating decimals. They cannot be represented as the quotient of two integers. The following are all irrational numbers:
p , , -

18. Real Numbers {x | x corresponds to a point on the number line }
The set of real numbers consists of all the rational numbers together with all the irrational numbers.

19. Example
Given set A = { , , 0, 2.9, -5, 4, - , , , p}, list all the elements of A that belong to the set of :
a) natural numbers,
b) whole numbers,
c) integers,
d) rational numbers,
e) irrational numbers,
f) real numbers.

21. Order of Operations
1. Perform operations in grouping symbols (parentheses, brackets, braces, or fraction bars). Start with the innermost and work outward.
2. Calculate powers and roots, working from left to right.
3. Perform multiplication and division in order from left to right.
4. Perform addition and subtraction in order from left to right.

22. Example a) 6(-5) – (-3)(2) b) c) -9 – {6 – 2[12 – (8 – 15)] – 4}
Use order of operations to evaluate:
a) 6(-5) – (-3)(2) b)
c) -9 – {6 – 2[12 – (8 – 15)] – 4}

23. Solution:
a) 6(-5) – (-3)(2) = 6(-5) – (-3)(16) No grouping symbols; power
calculated first
= -30 – (-48) Multiplication performed
= Subtraction changed to addition
= 18 Addition performed

24. Solution:
b) Begin by simplifying the numerator and denominator of fraction.
= Calculate powers firs
= Perform multiplications
= Perform additions and subtractions
= Simplify

25. c) -9 – {6 – 2[12 – (8 – 15)] – 4} = -9 – {6 – 2[12 – (-7)] – 4} Start with innermost grouping symbol, parentheses, and subtract
= -9 – {6 – 2[19] – 4} Working outward, perform subtraction in brackets
= -9 – {6 – 38 – 4} Within braces, multiply
= -9 – {-36} Within braces, subtract
= Change subtraction to addition
= 27 Add

26. Properties of the Real Numbers For all real numbers a, b, and c:
1. Commutative Property for Addition: a + b = b + a
2. Commutative Property for Multiplication: ab = ba
The commutative properties state that two numbers may be added or multiplied in any order.

27. Properties of the Real Numbers For all real numbers a, b, and c:
3. Associative Property for Addition: a + (b + c) = (a + b) + c
4. Association Property for Multiplication: a(bc) = (ab)c
For the associative properties, the order of the terms or factors remains the same; only the grouping is changed.

28. Properties of the Real Numbers For all real numbers a, b, and c:
5. Identity Property for Addition: There is a unique real number, 0, such that a + 0 = a and 0 + a = a
The identity property for addition tells us that adding 0 to any number will not change the number.
6. Identity Property for Multiplication: There is a unique real number, 1, such that a·1 = a and 1·a = a
The identity property for multiplication tells us that multiplying any number by 1 will not change the number.

29. Properties of the Real Numbers For all real numbers a, b, and c:
7. Inverse Property for Addition: Each nonzero real number a has a unique additive inverse, represented by –a, such that
a + (-a) = 0 and –a + a = 0
Additive inverses are called opposites.
8. Inverse Property for Multiplication: Each nonzero real number a has
unique multiplicative inverse, represented by , such that and
Multiplicative inverses are called
reciprocals.

30. Properties of the Real Numbers For all real numbers a, b, and c:
9. Distributive Property: a(b + c) = ab + ac

31. Example: Identify the property illustrated in each statement:
a) (x + 7) + 8 = x + (7 + 8)
b) 4x + 0 = 4x
c) 10 · ( x) = (10 · )x
d) (x + 1) · = 1
e) 4(x + 5) = 4x + 20
f) 3 · (5 · a) = 3 · (a · 5)
g) -6x + 6x = 0
h) (2 + y) + 5 = 5 + (2 + y)
i) (y + 5)(y – 3) = (y – 3)(y + 5)
j) 5 · 1 = 5

32. Solution:
a) Associative Property for Addition. Order of terms remains the same. Only the grouping changes.
b) Identity Property for Addition. Adding zero to something does not change it.
c) Associative Property for Multiplication. Order of factors is the same. Only the grouping changes

33. Solution:
d) Inverse Property for Multiplication. The product of reciprocals is 1.
e) Distributive Property.
f) Commutative Property for Multiplication. Order of the factors is changed.

34. Solution: g) Inverse Property for Addition. The sum of opposites is 0.
h) Commutative Property for Addition. The order of the terms is changed.
i) Commutative Property for Multiplication. The order of the factors is changed.
j) Identity Property for Multiplication. Multiplying
a number by 1 does not change it.

35. The Set of Real Numbers
First, a few terms:
Terminating Decimal: A decimal that ends, having a finite number of digits after the decimal point. Sample: 3/4 = 0.75

36. The Set of Real Numbers
First, a few terms:
Repeating Decimal: A decimal that doesn't end; it shows a repeating pattern of digits after the decimal point. Sample: 1/3 =

37. Exponents
Exponents are used in many algebra problems, so it's important that you understand the rules for working with exponents. Let's go over each rule in detail, and see some examples.

38. Exponents Rules of 1 There are two simple "rules of 1" to remember.
First, any number raised to the power of "one" equals itself. This makes sense, because the power shows how many times the base is multiplied by itself. If it's only multiplied one time, then it's logical that it equals itself.
Secondly, one raised to any power is one. This, too, is logical, because one times one times one, as many times as you multiply it, is always equal to one.

39. Exponents Product Rule
The exponent "product rule" tells us that, when multiplying two powers that have the same base, you can add the exponents. In this example, you can see how it works. Adding the exponents is just a short cut!

40. Exponents Product Rule
The exponent "product rule" tells us that, when multiplying two powers that have the same base, you can add the exponents. In this example, you can see how it works. Adding the exponents is just a short cut!

41. Exponents Quotient Rule
The quotient rule tells us that we can divide two powers with the same base by subtracting the exponents. You can see why this works if you study the example shown.

42. Exponents
Zero Rule
According to the "zero rule," any nonzero number raised to the power of zero equals 1.

43. Exponents Negative Exponents
The last rule in this lesson tells us that any nonzero number raised to a negative power equals its reciprocal raised to the opposite positive power.

44. ​Sαяα Al-Qαsim
Shαhdα Al-ariny
S3 ​