PROJECT IN MATH

CHAPTER 1

NUMBERS

REAL NUMBER

Real Numbers include all the rational and irrational numbers.

Real numbers are denoted by the letter R.

Real numbers consist of the natural numbers, whole numbers, integers, rational, and irrational numbers.

Examples of Real numbers

Natural numbers, whole numbers, integers, decimal numbers, rational numbers, and irrational numbers are the examples of real numbers.

Natural Numbers = {1, 2, 3,...} Whole Numbers = {0, 1, 2, 3,...} Integers = {..., -2, -1, 0, 1, 2,...} , 10.3, 0.6, , , 3.46466466646666..., , are

few more examples.

RATIONAL NUMBERS

A Rational Number is a real number written as a ratio of integers with a non-zero denominator.

Rational numbers are indicated by the symbol Q.

Rational number is written in p/q form, where p and q are integers and q is a non-zero denominator.

All the repeating or terminating decimal numbers are rational numbers.

Rational numbers are the subset of real numbers.

Examples of Rational Numbers

3/5, 10.3, 0.6 ,12/5, 3/4,  - All these are examples of rational numbers as they terminate.

IRRATIONAL NUMBERS

Irrational numbers are real numbers that cannot be expressed as fractions, terminating decimals, or repeating decimals.

Any real number that cannot be expressed as a ratio of integers.

Are real numbers that cannot be represented as terminating or repeating decimals.

Fractions.

Examples of Irrational numbers

℮,,√ 10 are few examples of irrational numbers.

INTEGERS

Commonly known as a "whole number", is a number that can be written without a fractional component. For example, 21, 4, and −2048 are integers, while 9.75, 5½, and √2 are not.

Positive or negative numbers or zero. Integers are the set of whole numbers and

their opposites.{. . . -3, -2, -1, 0, 1, 2, 3 . . . } is the set of integers.

Example of Integers

-12,315 733,751 -10 121 56 -89,100 26 1 -75

NON-INTEGERS

The non integer numbers are defined as the numbers as the decimal numbers, fractional number, mixed numbers, etc.

PROPERTIES OF REAL NUMBERS

DISTRIBUTIVE PROPERTY OF MULTIPLICATION OVER

ADDITION 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.

EXAMPLE: 2x(5 + y) = 10x + 2xy

COMMUTATIVE PROPERTY

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

ASSOCIATIVE PROPERTY

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)

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

CLOSURE PROPERTY

If you add real numbers the answer is also real.ADDITION OF CLOSURE PROPERTY

The problem 3 + 6 = 9 demonstrates the closure property of real number addition. Observe that the addends and the sum are real numbers.The closure property of real number addition states that when we add real numbers to other real numbers the result is also real.In the example above, 3, 6, and 9 are real numbers

If you multiply real numbers the answer is also real.

MULTIPLICATION OF CLOSURE PROPERTYThe problem 5 × 8 = 40 demonstrates the closure property of real number multiplication.Observe that the factors and the product are real numbers.The closure property of real number multiplication states that when we multiply real numbers with other real numbers the result is also real.In the example above, 5, 8, and 40 are real numbers.

INVERSE PROPERTY

Inverse Properties state that when a number is combined with its inverse, it is equal to its identity.

There are two types of inverses of a number: Additive Inverse and Multiplicative Inverse.

`- a` is said to be additive inverse of `a` if

a + (- a) = 0. `1\a` is said to be multiplicative Inverse

of `a` if ax 1\a=1 .

FACTORING

FACTORING

An integer that divides into another integer exactly is called a Factor, In other words, if any integer is divided by another integer without leaving any remainder, then the latter is the factor of the former.

For example: 2 is a factor of 8, because 2 divides evenly into 8. The quantities that are being multiplied

together to get a product are called factors. For example: 15 × 4 = 60 implies that 15 and 4 are the factors of 60.

TYPES OF FACTORING

COMMON MONOMIAL FACTOR (CMF)

Examples: 1. 12x2 – 9x3 = 3x2 (4 – 3x) 2. -10a6b5 – 15a4b6 – 20a3b4

Solution: The common factor of -10, -15 and -20 is -5; For a6, a4, a3, the common factor is a3

For b5, b6 and b4, the common factor is b4

Therefore, the common monomial factor of the given polynomial is -5a3b4

After getting the common monomial factor, divide the given polynomial by this to get the other factor. -10a6b5 – 15a4b6 – 20a3b4 = -5a3b4(2a3b + 3ab6 + 4)

DIFFERENCE OF TWO SQUARES

The difference of two squares is equal to the product of the sum and difference of the square roots of the terms. 1. = (x+y) (x-y)2. 4x2-16y2= (2x+4y)(2x-4y)

PERFECT SQUARE TRINOMIAL

The square of any binomial is a perfect square trinomial where the first and the last terms are the square of the first and square of last term of the binomial and the other term is a plus (or minus) the product of the first and the product of the first and last term of the binomial. x2+ 2xy + y2= (x + y)2

x2 – 2xy + y2= (x – y)2 To check if the trinomial is a perfect square trinomial, • two terms should be perfect squares • the other term should be a plus or minus twice the product

of the square roots of the other terms.

SUM AND DIFFERENCE OF TWO CUBES

1. x3+y3=(x+y)(x2-xy+y2)2. x3-y3=(x-y)(x2+xy+y2)3. 27x3 – 8y3 = (3x – 2y) (9x2 + 6xy +

4y2) 4. a3 + 64 = (a + 4)(a2 – 4a + 16) 5. 125b4c – bc4 = bc(125b3 – c3) = bc(5b – c)(25b2 + 5bc + c2)

FACTORING BY GROUPINGSometimes proper grouping of terms is necessary to make the given polynomial factorable. After terms are grouped, a complicated expression may be factored easily by applying Types 1 to 5 formulas. This type of factoring is usually applied to algebraic expressions consisting of at least four terms.Examples: 1. 3x(a – b) + 4y(a – b) =(a – b)(3x + 4y) 2. bx + by + 2hx + 2hy = (bx + by) + (2hx + 2hy) grouping = b(x + y) + 2h(x + y) removal of common factor from each group = (x + y)(b+ 2h) Another solution: = (bx + 2hx) + (by + 2hy) = x(b + 2h) + y(b + 2h) = (b + 2h)(x + y) 3. 3x(2a – b) + 4y(b – 2a) = 3x(2a – b) – 4y(2a – b) alteration of sign = (2a – b)(3x – 4y) 4. xz – kx + kw – wz = (xz – kx) – (wz – kw) = x(z – k) – w(z – k) = (z – k)(x – w) 5. ab3 – 3b2 – 4a + 12 = (ab3 – 3b2) – (4a – 12) = b2(a – 3) – 4(a – 3) = (a – 3) (b2 – 4) = (a – 3) (b – 2) (b+2)

QUADRATIC FACTORING

A "quadratic" is a polynomial that looks like "ax2 + bx + c", where "a", "b", and "c" are just numbers.

For the easy case of factoring, you will find two numbers that will not only multiply to equal the constant term "c", but also add up to equal "b", the coefficient on the x-term. For instance:

Factor x2 + 5x + 6.I need to find factors of 6 that add up to 5. Since 6 can be written as the product of 2 and 3, and since 2 + 3 = 5, then I'll use 2 and 3. I know from multiplying polynomials that this quadratic is formed from multiplying two factors of the form "(x + m)(x + n)", for some numbers m and n. So I'll draw my parentheses, with an "x" in the front of each:

(x )(x )

Then I'll write in the two numbers that I found above:

(x + 2)(x + 3)

This is the answer: x2 + 5x + 6 = (x + 2)(x + 3)

This is how all of the "easy" quadratics will work: you will find factors of the constant term that add up to the middle term, and use these factors to fill in your parentheses.

Note that you can always check your work by multiplying back to get the original answer. In this case:

(x + 2)(x + 3) = x2 + 5x + 6

RATIONAL EXPRESSION

Can be written , b ≠ 0 Numerator and Denominator are Polynomials

Simplify the following expression:

To simplify a numerical fraction, I would cancel off any common numerical factors. For this rational expression (this polynomial fraction), I can similarly cancel off any common numerical or variable factors.

The numerator factors as (2)(x); the denominator factors as (x)(x). Anything divided by itself is just "1", so I can cross out any factors common to both the numerator and the denominator. Considering the factors in this particular fraction, I get:

Then the simplified form of the expression is:

=

Simplify the following rational expression: How nice! This one is already factored for me! However (warning!), you will usually need to do the factorization yourself, so make sure you are comfortable with the process!The only common factor here is "x + 3", so I'll cancel that off and get:

Then the simplified form is: