The Set of Real Numbers and Its Properties
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The Set of Real Numbers and its Properties
Prepared by:Engr. Sandra Enn Bahinting
ALGEBRA
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OUTLINE
REAL NUMBERS
PROPERTIES and CATEGORIES of REAL NUMBERS
ALGEBRAIC EXPRESSIONS
POLYNOMIALS
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REAL NUMBERS
Real Numbers are every number
Any number that you can find on the number line
It has two categories.
Algebra
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The Real Number Line the line whose points are the real numbers
Order Property for Real Numbers - if a is to the left of b on the number line, a < b- if a is to the right of b on the number line, a > b
Algebra
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The Real Number System Tree Diagram
Algebra
REAL NUMBERS
RATIONAL NUMBERS
IRRATIONAL NUMBERS
Non-Terminating and
Non-repeating Decimals
INTEGERS
WHOLE NUMBERS
NATURAL NUMBERS
Terminating and Repeating Decimals
Negative Numbers
Zero
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Rational Numbers
A real number that can be written as a fraction
Rational Numbers written in decimal form are terminating or repeating
Algebra
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Example ½ -3/7 46 = 46/1 0.17 = 17/100 = 5
Repeating½ = 0.50002/3 = 0.6666669/7 = 1.285714285714
Algebra
Terminating3.56
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Integers
Integers are the whole numbers and their opposites
Consist of the numbers together with their negatives and 0
• Example:• 6, -12, 0, 143, -836
Algebra
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Types of Integers
Natural Numbers (N) = counting from 1,2,3,4,5,…………………
N = {1,2,3,4,5 …………………………}
Whole Numbers(W) = natural numbers including zero. They are 0,1,2,3,4,5………..
W = {0,1,2,3,4,5…..}W = 0 + N
Negative Numbers = {……..,-4, -3, -2, -1 }
Algebra
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Irrational Numbers
A number that cannot be written as a fraction of two integers
Irrational numbers written as decimals are non-terminating and non-repeating
• Example:
• = 1.414213562373095……• = 3.14159265….
Algebra
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Properties of REAL NUMBERS
Algebra
1. Closure PropertyLet a, b, and c represent real numbers
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Algebra
2. Commutative Property
3. Associative Property
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Algebra
4. Distributive Properties
5. Identity Property
6. Inverse Property
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Addition and Subtraction
Algebra
Subtraction = operation that undoes addition
a - b = a + (-b)
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Multiplication and Division
Algebra
Division = operation that undoes multiplication; to divide by a number, we multiply by the inverse of that number. If b 0, then, by definition
a b = a 1/b = a/b (quotient)
Example: 6 3 = 6 1/3
= 6/3 = 2
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Exponential Notation If a is any real number and n is a positive integer, then the nth
power of a is
The number a is called the base, and n is called the exponent.
Algebra
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Algebra
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Algebraic Expressions
Algebra
An algebraic expression is a constant, a variable or a combination of variables and constants involving a finite number of indicated operations on them. (operations such as addition, subtraction, multiplication, division, raising to a power and extraction of a root).
It is a collection of numerals, variables and operation symbols
Example:
5)a zxd 33)
xb 4) xy
xe 3)
xyzc) zyxf 33)
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Terms:Algebra
Variable = a letter that can represent any number from a given set of numbers.
Example: x, y, z
Constant = a number on its own, or sometimes a letter such as a, b or c to stand for a fixed number
Monomial = an expression of the form , where a is a real number and k is a nonnegative integer.
Example: 13, 3x, -57, x², 4y², -2xy, or 520x²y²
Binomial = a sum of two monomials
Example: 3x + 1, x² - 4x, 2x + y, or y - y²
Trinomial = a sum of three monomials
Example: x2 + 2x + 1, 3x² - 4x + 10, 2x + 3y + 2
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Polynomial
Algebra
A polynomial in the variable x is an expression of the form
where are real numbers, and n is a nonnegative integer. If 0, then the polynomial has degree n. The monomials that make up the polynomial are called the terms of the polynomial.
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Not a polynomial
Algebra
𝟑 𝒙−𝟒+𝟒 𝒙+𝟓
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Coefficients
Algebra
Coefficient is of two types.
Numbers form Numerical coefficientssymbols form literal coefficients.
Example:
2xy = 2 is the number or the Numerical coefficient xy, the symbol, is the Literal Coefficient. y = Numerical coefficient is 1 literal coefficient is y
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Multiplying Algebraic Expressions
Algebra
Multiplying Binomials
Example: 1. (2x+1) (3x-5)2. (3t + 2)(7t – 4)3. (2r - 5s)(3r - 2s)
Multiplying Polynomials
Example:
1. 2. 3.
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Special Products
Algebra
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Example:
Sum and Difference of same terms
Square of Sum
Square of Difference
Cube of Sum
Cube of Difference
Algebra
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Algebra
PASCAL TRIANGLE AND THE BINOMIAL THEOREM
The square of a binomial is a special case of the binomial theorem, which gives a pattern for finding any positive integer power of a binomial. The coefficient in the formula can be found from the following array of numbers, known as Pascal’s Triangle.
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1
10 yx
yxyx 1
222 2 yxyxyx
32233 33 yxyyxxyx
4322344 464 yxyyxyxxyx
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Algebra
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Algebra
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Algebra