MATH107 Basic Mathematics Lecture 1 PDF

# MATH107 BASIC MATHEMATICS ## THE REAL NUMBER SYSTEM: - **Real Numbers** - **Rational Numbers** - **Integers** - **Negative Integers** - **Zero** - **Positive Integers** (also called natural numbers, IN) - **Nonintegers** - **Irrational Numbers** ### INTEGERS - Z= {..., -3, -2, -1, 0, 1, 2, 3,...} → Integers - Z+ = {1, 2, 3,...} → Positive integers - Z- = {-3, -2, -1} → Negative integers The set of positive and negative integers, together with 0 (zero) is called the set of "INTEGERS". Note that zero is an integer number but it is neither positive nor negative integer (e.g., $\frac{a}{0}$ is undefined where "a" is a real number) ## RATIONAL NUMBERS: The numbers which can be expressed as the ratio of 2 integers are called "RATIONAL NUMBERS" and the set of rational numbers denoted by "Q". $Q = \{ \frac{m}{n}$: m and n are elements of Z and n≠0 } - m is numerator - n is denominator ## IRRATIONAL NUMBERS: The number which cannot be expressed as the ratio of 2 integers are called "IRRATIONAL NUMBERS" and the irrational numbers set is denoted by $Z_I$. - $Z_I= \{ e, \pi, \sqrt{2}, \sqrt{3}, .... \}$ - e = 2.718..... - $\pi$ = 3.14...... ## REAL NUMBERS: The combination of integers, rational, irrational numbers are called REAL NUMBERS and it is denoted by R. - e.g., 2, 3, 2.1, 0, $\sqrt{2}$ - all of them are elements of the set of real numbers The set of real numbers can be represented by using a "number line". - The number line is a line with a point marked at 0, and points to the right of it marked 1,2,3... and points to the left marked -1, -2, -3... - The end point of this line on the right is positive infinity and the end point on the left is negative infinity. ## The important symbols: - >`:` (is greater than) e.g. a>b a is greater than b, 3>2 - `<:` (is less than) e.g. a<b a is less than b, -1<1 - `≥:` (greater or equal to) e.g. a≥b a is greater or equal to b, -1≥-3 - `≤:` (less or equal to) e.g. a≤b a is less or equal to b, 0≤4 ## PROPERTIES OF INEQUALITIES; 1. If a b, then a + c < b + d. e.g., 2 < 3 and 4 < 5 then 2 + 4 < 3 + 5 = 6 < 8 2. If a < b and c is any number then a + c b c, c is negative integer. e.g., 2 < 3 - 2 . (-2) < 3 . (-2) = -4 < -6 5. If a < b and c is positive number then $\frac{a}{c}$ < $\frac{b}{c}$. e.g., 2 < 3 - $\frac{2}{1}$ < $\frac{3}{1}$ ⇒ 2 < 3 - $\frac{2}{2}$ < $\frac{3}{2}$ ⇒ 1 < 1.5 6. If a $\frac{b}{c}$. e.g., 2 < 3 - $\frac{2}{-1}$ < $\frac{3}{-1}$ ⇒ -2 > -3 - $\frac{2}{-2}$ < $\frac{3}{-2}$ ⇒ -1 > -1.5 7. If a and b are positive numbers and a $\frac{1}{b}$. e.g., 2 < 3 - $\frac{1}{2}$ > $\frac{1}{3}$ ⇒ 0.5 > 0.3 ## DEFINITION: (OPEN INTERVAL) Let a and b are real numbers with a<b. The open internal (a,b) consists of all numbers x such that a< x < b. - a and b are not elements of this set. ## DEFINITION: (CLOSED INTERVAL) Let a and b be real numbers with a<b. The closed interval [a,b] consists of all numbers x such that a≤x≤b. - a and b are included in the set. ## DEFINITION: (HALF OPEN INTERVALS) Let a and b be real numbers with a <b. The half open interval (a, b) consists of all numbers x such that a ≤ x a - (-∞, a]; x ≤ a - (-∞, a); x < a - (-∞, ∞); all x ## ABSOLUTE VALUE! The absolute value of a number "a" is the distance from that number to zero and written |a|. - $|3|$ = 3, because 3 is 3 units away from zero. - $|-2|$ = 2, because -2 is 2 units away from zero. - We have; $|a|$ = { a if a ≥ 0 - a if a < 0 - e.g., |5| = 5 - |-4| = -(-4) = 4 ## SOME PROPERTIES OF ABSOLUTE VALUES: 1. |a| ≥ 0 where "a" is any real number. - e.g., |-5| = 5 ≥ 0 - |5| = 5 ≥ 0 - |0| = 0 ≥ 0 2. |-a| = |a| where "a" is any real number - e.g., |-5| = |5| - 5 = 5 - |3| = |-3| - 3 = 3 3. |a-b| = |b - a| where "a" and "b" are real numbers - e.g.,|4-3| = |3-4| - 1 = 1 - |1-1| = |1-1| - 1 = 1 4. |ab|= |a||b| - e.g., |4.3| = |4||3| - 12 = 12 - |-2.3| = |-2||3| ⇒ 6 = 6 - |-6| = 2.3 ⇒ 6 = 6 ## POLYNOMIALS ### POSITIVE INTEGER EXPONENTS. **DEFINITION:** If n is a positive integer, and "a" is any real number then; - a^n = a . a . a . .... a - n factors. **Ex:** 2^3 = 2 . 2 . 2 = 8 - (-5)^2 = (-5) (-5) = 25 - -3^3 = -(3 . 3 . 3) = -27 - aaa bbbb ccc c = a^3 b^4 c^4 **DEFINITION:** If n is a positive integer and a≠0 - a^(-n) = 1/a^n **Ex:** 2^(-3) = 1/2^3 = 1/8 - 5^(-2) = 1/5^2 = 1/25 **Example** - a^(-10). b^(-4). c^2 = c^2/a^(10) b^(4) **DEFINITION:** If "a" is a real number and a≠0, then - a^0 = 1 **Ex:** 10^0 = 1 - (-4)^0 = 1 - 8^0 = 9^2.1^0 = 81 - -6b^0 = -6.1 = -6 if b≠0 ## LAWS OF EXPONENTS: 1. a^m.a^n = a^(m+n) - m ∈ IR, n ∈ IR 2. (a^m)^n = a^(m.n) 3. (ab)^m = a^m b^m 4. a^m/a^n = a^(m - n), a≠0 5. (a/b)^m = a^m/b^m **Ex:** 5^2 . 5^3 = 5^(2+3) = 5^5 - (2 . 3)^2 = 2^2 . 3^2 = 4 . 9 = 36 - (4/3)^3 = 4^3/3^3 = 64/27 - (4^2) ^4 = 4^(2 . 4) = 4^8 - (2/3)^(-2) = 2^(-2)/ 3^(-2) = 3^2/2^2 = 9/4 ## EXERCISES: 1. (2a/5b^2)^3 2. [(-1)^3]^5 3. x^5/x^5 ## ADDITION AND SUBTRACTION OF POLYNOMIALS: In adding and subtracting polynomials we combine the terms which are involving the same variables. **Ex:** - (5y^3 - 2y^2 + y) + (4y^2 - 5y) - = 5y^3 + (-2y^2 + 4y^2) + (y - 5y) - = 5y^3 + 2y^2 - 4y **Ex:** - (3x^2 + 5x - 2) - (x^2 + 2x - 8) - = (3x^2 - x^2) + (5x - 2x) + (-2 + 8) - = 2x^2 + 3x + 6 ## MULTIPLICATION OF POLYNOMIALS: All the rules and properties of multiplication for real numbers apply when polynomials are multiplied. **Ex:** x(4x + 4) = 4x^2 + 4x **Ex:** (x - 2)(x^2 - 4x + 4) = x^3 - 4x^2 + 4x - 2x^2 + 8x - 8 = x^3 - 6x^2 + 12x - 8 **Ex:** (x - y)(x - 3y) = x^2 - 3xy - xy + 3y^2 = x^2 - 4xy + 3y^2 ## DIVISION OF POLYNOMIALS: **Ex:** - x^3 +2x^2 + 4x - 1 - x - 2 - = x^2 + 4x + 12 + 23/x-2 **Ex:** - 24a^4b^5 + 18a^2b^3 / -3a^2b^4 - = ## ACTORING Factoring a polynomial means that expressing it as the product of two or more other polynomials. 1. **Distributive Law** - ab + ac = a (b + c) - **Ex:** x^3 - x^2 + x = x (x^2 - x + 1) 2. **Factoring a Trinomial** - x^2+ (a+b) x + ab = (x+a) (x+b) - **Ex:** x^2+x-6 = (x+3)(x-2) - x^2- 5x+6= (x-3)(x-2) 3. **Factoring perfect squares** - x^2-a^2 = (x-a)(x+a) - **Ex:** x^2 - 2^2 = (x - 2)(x + 2) - a^2 - b^2 = (a - b) (a + b) - y^2 - 100 = (y - 10) (y + 10) 4. **Difference between two cubes** - a^3 - b^3 = (a - b) (a^2 + ab + b^2) - **Ex:** x^3 - 27 = (x - 3)(x^2 + 3x + 9) - x^3 - 1 = (x - 1)(x^2 + x + 1) - 8x^3 - 1 = (2x - 1) (4x^2 + 2x + 1) 5. **Sum of two cubes** - a^3 + b^3 = (a + b) (a^2 - ab + b^2) - **Ex:** x^3 + 8 = (x + 2) (x^2 - 2x + 4) - x^3 + 27 = (x + 3) (x^2 - 3x + 1) 6. **Square of a binomial** - (x + a)^2 = x^2 + 2ax + a^2 - **Ex:** (x + 5)^2 = x^2 + 2 . 5x + 5^2 - = x^2 + 10x + 25

MATH107 Basic Mathematics Lecture 1 PDF

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