Algebra

1. Fundamental concepts of Algebra Fujairah Collage Department of Information Technology Fundamental concepts of Algebra Asmaa Abdullah

2. Real Numbers Fundamental concepts of Algebra Asmaa Abdullah

3. 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 .

4. Examples of rational numbers are : 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) Fundamental concepts of Algebra Asmaa Abdullah can also be written as -1.25 can also be written as .5 can also be written as -2.0 can also be written as -2 or 6.0 6 or can also be written as

5. Examples : Write each rational number as a fraction : 1- 0.3 2- 0.007 3- -5.9 Fundamental concepts of Algebra Asmaa Abdullah

6. 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 = ……

7. 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

8. The family tree For our numbers Fundamental concepts of Algebra Asmaa Abdullah Complex Numbers Real Numbers Rational Numbers Irrational Numbers Integers 0 Positive integers Negative integers

17. Fundamental concepts of Algebra Asmaa Abdullah 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.

18. 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 ) = 2 x + 2 y Since they distributed through the parentheses, this is true by the Distributive Property Fundamental concepts of Algebra Asmaa Abdullah

19. 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. Fundamental concepts of Algebra Asmaa Abdullah

20. 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. Fundamental concepts of Algebra Asmaa Abdullah

21. Let a , b , and c be real numbers, variables, or algebraic expressions . Multiplicative Inverse Property Note : a can not = 0 9. 3 + (-3) = 0 Additive Inverse Property a + ( - a)=0 8. 3 • 1 = 3 Multiplicative Identity Property a • 1 = a 7. 3 + 0 = 3 Additive Identity Property a + 0 = a 6. 2 • ( 3 + 4 ) = 2 • 3 + 2 • 4 Distributive Property a • (b + c) ) = a • b ) + ( a • c) 5. 2 • ( 3 • 4 ) = ( 2 • 3 ) • 4 Associative Property of Multiplication a • (b • c ) = ( a • b • ) c 4. 2 + ( 3 + 4 ) = ( 2 + 3 ) + 4 Associative Property of Addition a + (b + c ) = ( a + b ) + c 3. 2 • ( 3 ) = 3 • ( 2 ) Commutative Property of Multiplication a • b = b • a 2. 2 + 3 = 3 + 2 Commutative Property of Addition a + b = b + a 1. Example Property

23. 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“. "5 3 " is "five, raised to the third power". When we deal with numbers, we usually just simplify; we'd rather deal with "27" than with "3 3 ". But with variables, we need the exponents, because we'd rather deal with " x 6 " than with " xxxxxx ". Fundamental concepts of Algebra Asmaa Abdullah

24. 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 ( x 3 )( x 4 ) Think in terms of what the exponents mean: ( x 3 )( x 4 ) = ( xxx )( xxxx ) = xxxxxxx = x 7 Exponents: Basic Rules ...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 (x 4 )(y 3 ), because the bases are different: (x 4 )(y 3 ) = xxxxyyy = (x 4 )(y 3 ). Fundamental concepts of Algebra Asmaa Abdullah

25. Simplify ( x 2 ) 4 Again, think in terms of what the exponents mean: ( x 2 ) 4 = ( x 2 )( x 2 )( x 2 )( x 2 ) = ( xx )( xx )( xx )( xx ) = xxxxxxxx = x 8 ...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: ( x m ) 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, ( xy 2 ) 3 = ( xy 2 )( xy 2 )( xy 2 ) = ( xxx )( y 2 y 2 y 2 ) = ( xxx )( yyyyyy ) = x 3 y 6 = ( x ) 3 ( y 2 ) 3

26. Fundamental concepts of Algebra Asmaa Abdullah Exponents: Basic Rules 5 -3 =1/5 3 a -n = 1 /a n 3 0 =1 a 0 =1 Illustration Definition ) a=0)

27. Fundamental concepts of Algebra Asmaa Abdullah Exponents: Basic Rules (2 3 /2 5 ) =1/(2 5 - 3 ) =1/(2 2 ) a n /a m = 1/ a n-m (2 5 /2 3 ) =(2 5 - 3 ) =(2 2 ) a m /a n = a m-n (2/5) 3 =(2 5 )/ (5 3 ) (a/b) n = a n /b n (2.10 ) 3 = (2) 3 .(10 ) 3 (ab) n =a n b n (2 3 ) 4 = (2 3.4 ) = (2 12 ) (a m ) n =a mn 3 2 3 4 =3 2+4 =3 6 a m a n =a m+n Illustration Law

28. Fundamental concepts of Algebra Asmaa Abdullah Mathematical Terms The set of numbers beginning with one {1, 2, 3, ...} used for most counting Natural Numbers or Counting Numbers A set contained in another set Subset Set A set in which it is not possible to name all members Infinite A set in which all members can be listed Finite Set A set with no members Empty Set/Null Set A group or collection of objects Set

29. Mathematical Terms Find the numerical value of an expression Evaluate A combination of numbers and mathematical operations Expression The set of integers and all fractions and their decimal equivalents Rational Numbers The set of whole numbers and their opposites Integers The set of natural numbers that includes zero as an element {0, 1, 2, ...} Whole Numbers