The Set of Real Numbers and its Properties

Prepared by:Engr. Sandra Enn Bahinting

ALGEBRA

OUTLINE

REAL NUMBERS

PROPERTIES and CATEGORIES of REAL NUMBERS

ALGEBRAIC EXPRESSIONS

POLYNOMIALS

REAL NUMBERS

Real Numbers are every number

Any number that you can find on the number line

It has two categories.

Algebra

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

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

Rational Numbers

A real number that can be written as a fraction

Rational Numbers written in decimal form are terminating or repeating

Algebra

Example ½ -3/7 46 = 46/1 0.17 = 17/100 = 5

Repeating½ = 0.50002/3 = 0.6666669/7 = 1.285714285714

Algebra

Terminating3.56

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

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

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

Properties of REAL NUMBERS

Algebra

1. Closure PropertyLet a, b, and c represent real numbers

Algebra

2. Commutative Property

3. Associative Property

Algebra

4. Distributive Properties

5. Identity Property

6. Inverse Property

Addition and Subtraction

Algebra

Subtraction = operation that undoes addition

a - b = a + (-b)

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

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

Algebra

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)

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

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.

Not a polynomial

Algebra

𝟑 𝒙−𝟒+𝟒 𝒙+𝟓

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

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.

Special Products

Algebra

Example:

Sum and Difference of same terms

Square of Sum

Square of Difference

Cube of Sum

Cube of Difference

Algebra

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

Algebra

Algebra

Algebra