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essential concepts of algebra
- 1. Essential Concepts of AlgebraEssential Concepts of Algebra Business Mathematics Lecture : 1 By: Lamya Bint-al Islam Eastern University Faculty of Business Administration
- 2. Numbers & Integers • Numbers: A number is a digit or a collection of digits. Numbers can be positive, negative, odd, even, fractions, decimals and even weird numbers such as √2. • Integers: All whole numbers are integers, they can be positive, negative and zero, thus, the set of integers is {……-3,-2,-1,0,1,2,3,…....}.
- 3. Numbers & Integers • The difference between ‘number’ and ‘integer’ is that number can mean fractions or whole number, 3 is not the only number between 2 & 4, there are many numbers in between such as 2.5, 2.9, and 3.9. While integer only means whole number, so 3 is the only integer between 2 & 4. • Only integers can be even or odd. Fractions, decimals and other non-integers can never be even or odd.
- 5. Real Number • The set of all rational and irrational numbers is called the set of real numbers.
- 6. Rational Numbers • The integers combined with the fractions form the set of rational numbers. Thus a rational number is a number that can be expressed in the form of a fraction that has integers as numerator and denominator, such as p/q where p & q are integers and q ≠0. Example: 5/4, 9/10, 6/1. Here 5/4= 1.25, 1/3 = 0.33333, 1/22 = 0.045454545, 15/14 = 1.0714285714285 • So every rational number can be expressed as a terminating or repeating decimal.
- 7. Irrational Number • Irrational numbers cannot be expressed as a simple fraction, because the decimals do not terminate or repeat, such as √2, Π, e, and √15. • √2= 1.414213…. Π= 3.14159265…. √7= 2.645751….
- 8. Complex Numbers • Square root of a negative number is called an imaginary number such as √-1=i, numbers with an imaginary component are called complex numbers such as a+ib.
- 9. Properties of Zero Zero is a special number with some unique properties: • O is even • It is an integer but it is neither positive nor negative. • O + any other number is equal to that number. • O multiplied by any other number is equal to 0. • Any number divided by 0 will be infinite or undefined. Any number/ 0 = undefined or ∞ • 0 divided by any number equals to 0. 0/any number = 0. • 00 is undefined.
- 10. Rules of Sign Addition & Subtraction - (+2) = -2 + ( -2) = -2 - (-2) = +2 + (+9) = +9 + ( -2) = - (+2) (+7) + ( -3) = +4 (-7) + (+3) = - 4 Multiplication (4) (2) = 8 2(- 4) = - 8 (- 4)(2) = - 8 (- 4)(- 2) = 8 -4(3) + (-6) (2) = -24 Division 8/4 = 2 8/ -4 = -2 -8/4 = -2 -8/ -4 = 2 -4(2) (-3) 24 -2 (-1) (4) 8 = 3
- 11. Order of Operations • 4 (1-3) + 5x 6/2 = 4 (-2) + 5 x 6/2 = - 8 + 5 x 3 = - 8 + 15 = 7
- 12. Properties of Algebra Property Addition Multiplication Commutitative If a and b are real, then a + b = b + a If a and b are real, then a.b = b. a Associative If a, b and c are real, then (a+b) + c = a + ( b+c) If a, b and c are real, then (ab) c = a (bc) Distributive If a, b and c are real, then a (b+c) = a.b + a.c
- 13. Factoring • Common factor: 2xy + axy = xy (2 + a) • Middle term : 6x2 + 5x - 4
- 14. Fractions • A fraction is a number of the form a/b where a and b are both integers and b ≠0. • The integer a is called the numerator and b is called the denominator of the fraction. For example, -7/ 5 is a fraction where -7 is the numerator and 5 is the denominator. • If both the numerator a and denominator b are multiplied by the same nonzero integer then the resulting fraction will be equal to a/b. For example, (-7)4 / (5)(4) = -28/ 20 = -7/5
- 15. Rules of Fractions • A fraction with a negative sign in either the numerator or denominator can be written with the negative sign in front of the fraction, for example, -7 /5 = 7/ -5 = - (7/5) • If both the numerator and denominator have a common factor, then the numerator and denominator can be factored and reduced to an equivalent fraction, for example, 40/ 72 = (8) (5) / (8) (9) = 5/9
- 16. Addition & Subtraction of Fractions • To add two fractions with the same denominator, we add the numerator and keep the same denominator, for example, -8/11 + 5/ 11 = -8 + 5/11 = -3/11 • To add two fractions with different denominators, we first find the LCM of the denominators, then add the numerators . For example, 1/3 + -2/5 = 5+ (-6) / 15 = 1/15 • The same method applies to subtraction of fractions.
- 17. Multiplication & Division of Fractions • To multiply two fractions, multiply the two numerators and multiply the two denominators. For example, (10/7) ( -1/ 3) = -10/ 21 • To divide one fraction by another, first invert the second fraction then multiply the first fraction by the inverted fraction. For example, • 17/8 ÷ 3/4 = (17/8) (4/3) = (17/2) (1/3) = 17/6
- 18. Mixed Number • An expression such as 4⅜ is called a mixed number. It consists of an integer part and a fraction part, the mixed number means 4⅜ = 4 + 3/8 = 35/8