in algebra, we work with the set of real numbers, which we can model using a number line. real...
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Real numbersIn 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
Properties of real numbersIn 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 .
1 .Commutative propertiesThe 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
multiplication3 x 8 x 5b = 5b x 3 x 8
2.Associative propertiesBoth 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)
3.Distributive propertyThe 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.
4 .Density propertyThe 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, 5.613 and so forth .Between 5.612 and 5.613, there is 5.6121, 5.6122 ...
and an endless list of other numbers !
5 .Identity propertyThe 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
Adding Real NumbersThis tutorial reviews adding real numbers as well as finding the additive inverse or opposite of a number
Adding Real Numbers with the same sign
Step 1: Add the absolute valuesStep 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.
Example 1: Add -6 + (-8). -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.
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Example 2: Add -5.5 + (-8.7). -5.5 + (-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 -14.2. 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.
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The Real NumbersSets of Numbers
*Natural Numbers {1, 2, 3, 4, . . .}
* Whole Numbers {0, 1, 2, 3, 4, . . .}
* Integers {. . . , -3, -2, -1, 0, 1, 2, 3, . . .}
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 :
, - , 1 ,- 5 = , 0 = , 0.125 = , 0.6666 = . . .
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 , , -
, -
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 .
Example
Given set A = { ,- , 0, 2.9, -5, 4, - , , -7 , 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 .
Order of Operations1 .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.
ExampleUse order of operations to evaluate:
a) 6(-5) – (-3)(2)
b )
c) -9 – {6 – 2[12 – (8 – 15)] – 4}
Solution:a) 6(-5) – (-3)(2) = 6(-5) – (-3)(16) No grouping symbols; power
calculated first
- =30-( – 48 )Multiplication performed- =30 + 48 Subtraction changed to addition
=18 Addition performed
Solution:b) Begin by simplifying the numerator and denominator of fraction.
= Calculate powers firs
=Perform multiplications
=Perform additions and subtractions
=Simplify
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
- =9 + 36 Change subtraction to addition =27 Add
Properties of the Real NumbersFor all real numbers a, b, and c:
1 .Commutative Property for Addition: a + b = b + a
2 .Commutative Property for Multiplication: ab = baThe commutative properties state that two numbers may be added or multiplied in any order.
Properties of the Real NumbersFor 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)cFor the associative properties, the order of the terms or factors remains the same; only the grouping is changed.
Properties of the Real NumbersFor 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 = aThe 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 = aThe identity property for multiplication tells us that multiplying any number by 1 will not change the number.
Properties of the Real NumbersFor 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 thata + (-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.
Properties of the Real NumbersFor all real numbers a, b, and c:
9 .Distributive Property: a(b + c) = ab + ac
Example:Identify the property illustrated in each statement:
a) (x + 7) + 8 = x + (7 + 8)b) 4x + 0 = 4xc) 10 · ( x) = (10 · )xd) (x + 1) · = 1e) 4(x + 5) = 4x + 20f) 3 · (5 · a) = 3 · (a · 5)g) -6x + 6x = 0h) (2 + y) + 5 = 5 + (2 + y)i) (y + 5)(y – 3) = (y – 3)(y + 5)j) 5 · 1 = 5
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
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.
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.
The Set of Real NumbersFirst, a few terms:
Terminating Decimal: A decimal that ends, having a finite number of digits after the decimal point. Sample: 3/4 = 0.75
The Set of Real NumbersFirst, 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 = 0.3333...
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 .
Exponents
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 .
Rules of 1
Exponents
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 !
Product Rule
Exponents
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 !
Product Rule
Exponents
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 .
Quotient Rule
Exponents
According to the "zero rule," any nonzero number raised to the power of zero equals 1 .
Zero Rule
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 .
Negative Exponents
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