Math1 1

Fundamental concepts of

Algebra

Fujairah Collage Department of Information Technology

Fundamental concepts of Algebra Asmaa Abdullah

Real Numbers


Different types of Numbers are:1.Positive integers or natural numbers. N={1,2,3…..}

2.Whole numbers or non-negative numbers. W={0} + N …., -4,-3,-2,-1,0,1 ,2,3,4,…

3.Rational numbers.A rational number is a number that can be expressed as a fraction or ratio )rational). The numerator and the denominator of the fraction are both integers.When the fraction is divided out, it becomes a terminating or repeating decimal. Rational numbers can be ordered on a number line.

Examples of rational numbers are :

6 or can also be written as 6.0

can also be written as -2 or -2.0

can also be written as .5

can also be written as -1.25

can also be written as

Rational numbers are "nice" numbers.They are easy to write on paper

This means that the rational numbers are :

* It’s a real number can be expressed in the for a/b, b=0 * every integer is a rational number. * every real number can be expressed as decimal either: - terminated(5/4) - non-terminated (177/55)


Examples: Write each rational number as a fraction:

1- 0.3

2- 0.007

3- -5.9


4- An irrational number can not be expressed as a fraction. In decimal form, irrational numbers do not repeat in a pattern or terminate. They "go on forever" (infinity (

Examples of irrational numbers are:

3.141592654 =……

1.414213562 =……

Note: Many students think is a terminating decimal, 3.14, but "we" have rounded it to do math calculations.

is actually a non-ending decimal and is an irrational number.Irrational numbers are "not nice " numbers. The decimal is impossible

to write on paper because it goes on and on and on…..

Rational and irrational numbers are real numbers

Complex Numbers

Real Numbers

Rational Numbers Irrational Numbers

Integers

0Positive integers Negative integers

The family tree For our numbers


PROPERTIES OF EQUALITY

If a , b are real numbers and a=b , c is any real number, then

• 1. a + c=b + c• 2. ac=b c


PRODUCTS INVOLVING ZERO

• 1 a.0 = 0 for every real number a.• 2. If a b=0,then either a=0 or b=0.


PROPERTIES OF NEGATIVES.

• 1.-(-a)=a.• 2. (-a) b=-(a b)=a(-b)• 3.(-a)(-b)=a b• 4.(-1)a=-a


Subtraction and Division.

• a-b=a+(-b); To subtract one number from another, add the negative.

• a/b=a.1/b, where b is not equal to 0.• To divide one number by a non zero number,

multiply by the reciprocal.


Properties of Quotients.

1. a/b=c/d if ad=bc.2. ad/bd=a/b3. a/-b=-a/b=-(a/b)4. a/b+c/b=(a+c)/b5. a/b+c/d=(ad+bc)/bd6. a/b.c/d=ac/bd7. (a/b)/(c/d)=a/b.d/c=ad/bc


Inequality signs

• The symbols < and > are inequality signs., and the expressions a >b and a<b are called inequalities.


Laws of signs.

• If a and b have the same sign, then, ab and a/b are positive.

• If a and b have opposite signs, then ab and a/b are negative.


The absolute value notation

• Absolute value of any real number “a” is |a|=a and also |-a|=a. i.e., absolute value of any number is always positive.



Basic Number Properties: Associative, Commutative, and Distributive

There are three basic properties of numbers, and you'll probably have just a little section on these properties, maybe at the beginning of the course, and then you'll probably never see them again (until the beginning of the next course). Covering these properties is a holdover from the "New Math" fiasco of the 1960s. While these properties will start to become relevant in matrix algebra and calculus (and become amazingly important in advanced math, a couple years after calculus), they really don't matter a whole lot now.Why not? Because every math system you've ever worked with has obeyed these properties. You have never dealt with a system where a×b didn't equal b×a, for instance, or where (a×b)×c didn't equal a×(b×c). Which is why the properties probably seem somewhat pointless to you. Don't worry about their "relevance" for now; just make sure you can keep the properties straight so you can pass the next test. The lesson below explains how I kept track of the properties.

Distributive Property

The Distributive Property is easy to remember, if you recall that "multiplication distributes over addition". Formally, they write this property as "a(b + c) = ab + ac". In numbers, this means, for example, that 2(3 + 4) = 2×3 + 2×4. Any time they refer in a problem to using the Distributive Property, they want you to take something through the parentheses (or factor something out); any time a computation depends on multiplying through a parentheses (or factoring something out), they want you to say that the computation uses the Distributive Property. So, for instance:Why is the following true? 2(x + y) = 2x + 2y Since they distributed through the parentheses, this is true by the Distributive Property


Associative Property

"Associative" comes from "associate" or "group", so the Associative Property is the rule that refers to grouping. For addition, the rule is "a + (b + c) = (a + b) + c"; in numbers, this means 2 + (3 + 4) = (2 + 3) + 4. For multiplication, the rule is "a(bc) = (ab)c"; in numbers, this means 2(3×4) = (2×3)4. Any time they refer to the Associative Property, they want you to regroup things; any time a computation depends on things being regrouped, they want you to say that the computation uses the Associative Property.


Commutative Property

"Commutative" comes from "commute" or "move around", so the Commutative Property is the one that refers to moving stuff around. For addition, the rule is "a + b = b + a"; in numbers, this means 2 + 3 = 3 + 2. For multiplication, the rule is "ab = ba"; in numbers, this means 2×3 = 3×2. Any time they refer to the Commutative Property, they want you to move stuff around; any time a computation depends on moving stuff around, they want you to say that the computation uses the Commutative Property.


Let a, b, and c be real numbers, variables, or algebraic expressions.

Property Example

1. Commutative Property of Addition a + b = b + a 2 + 3 = 3 + 2

2. Commutative Property of Multiplication a • b = b • a 2 • ( 3 ) = 3 • ( 2 )

3. Associative Property of Addition a +(b + c ) = ( a + b) + c 2 + ( 3 + 4 ) = ( 2 + 3 ) + 4

4. Associative Property of Multiplication a • (b • c ) = ( a • b •) c 2 • ( 3 • 4 ) = ( 2 • 3 ) • 4

5. Distributive Property a • (b + c) ) = a • b )+( a • c) 2 • ( 3 + 4 ) = 2 • 3 + 2 • 4

6. Additive Identity Propertya + 0 = a 3 + 0 = 3

7. Multiplicative Identity Property a • 1 = a 3 • 1 = 3

8. Additive Inverse Property a + ( -a)=0 3 + (-3) = 0

9. Multiplicative Inverse Property Note: a can not = 0

Distance formula

• Let a and b be the coordinates of two points A and B respectively. Then the distance between A and B is d (A ,B)=|b-a|


Exponents: Basic Rules

Exponents are shorthand for multiplication: (5)(5) = 25, (5)(5)(5) = 135. The "exponent" stands for however many times the thing is being multiplied.The thing that's being multiplied is called the "base". This process of using exponents is called "raising to a power", where the exponent is the "power“. "53" is "five, raised to the third power". When we deal with numbers, we usually just simplify; we'd rather deal with "27" than with "33". But with variables, we need the exponents, because we'd rather deal with "x6" than with "xxxxxx".


There are a few rules that simplify our dealings with exponents. Given the same base, there are ways that we can simplify various expressions. For instance:Simplify (x3)(x4)Think in terms of what the exponents mean:(x3)(x4) = (xxx)(xxxx) = xxxxxxx = x7


...which also equals x(3+4). This demonstrates a basic exponent rule: Whenever you multiply two terms with the same base, you can add the exponents:

( x m ) ( x n ) = x( m + n )

Note that we cannot simplify (x4)(y3), because the bases are different:

(x4)(y3) = xxxxyyy = (x4)(y3).


Simplify (x2)4 Again, think in terms of what the exponents mean:(x2)4 = (x2)(x2)(x2)(x2) = (xx)(xx)(xx)(xx) = xxxxxxxx = x8 ...which also equals x( 2×4 ). This demonstrates another rule: Whenever you have an exponent expression that is raised to a power, you can multiply the exponent and power:( xm ) n = x m n

If you have a product inside parentheses, and a power on the parentheses, then the power goes on each element inside. For instance,(xy2)3 = (xy2)(xy2)(xy2) = (xxx)(y2y2y2) = (xxx)(yyyyyy) = x3y6 = (x)3(y2)3



Definition)a=0) Illustration

a0=1 30=1

a-n=1/an 5-3=1/53

a

acbb

2

42



Law Illustration

aman=am+n 3234=32+4=36

(am)n=amn (23) 4= (23.4) = (212)

(ab)n=anbn

(2.10 ) 3= (2) 3.(10 ) 3

(a/b) n= an/bn (2/5)3=(25)/ (53)

am/an=am-n (25/23) =(25-3) =(22)

an/am= 1/an-m (23/25) =1/(25-3) =1/(22)


Mathematical Terms

Set A group or collection of objectsEmpty Set/Null Set A set with no members

Finite Set A set in which all members can be listed

Infinite Set A set in which it is not possible to name all members

Subset A set contained in another set

Natural Numbers or Counting Numbers

The set of numbers beginning with one {1, 2, 3, ...} used for most counting

Whole NumbersThe set of natural numbers that includes zero as an element {0, 1, 2, ...}

Integers The set of whole numbers and their opposites

Rational Numbers The set of integers and all fractions and their decimal equivalents

ExpressionA combination of numbers and mathematical operations

Evaluate Find the numerical value of an expression

Mathematical Terms

Date post:	26-May-2015
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