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NUMBERS AND INEQUALITIES Introduction Sets - A set is a collection of objects. The objects of a set are called elements or members. - The set of natural numbers: N = {1, 2, 3, . . .} I are needed for counting -The set of integers: Z = {. . . − 2, −1, 0, 1, 2, . . .} I are needed for describing below-zero temperature or debt - The set of rational numbers: m Q = { | m, n 6= 0 are integers} n I are needed for concepts like half an apple ITheir corresponding decimals are repeating - The set of irrational numbers: The set of numbers that can not be represented as quotients of integers. I are needed to measure distances like the diagonal of a square ITheir decimal √ representations are non- repeating IExamples: π, 3. - The set of real numbers denoted as R is the set of all rational and irrational numbers. Set Notation and Set Operations ∈: The symbol used to denote that an element is a member of a given set. For example, 2 is an element of the set A = {1, 2, 3}. This is written as 2 ∈ A. 1 ∈: / The symbol used to denote that an element is not a member of a given set. For example, 5 is not an element of the set A. This is written as 5 ∈ / A. - If a set A concludes all elements of another set B, it is said that A is a subset of B. This is written as A ⊂ B. - The union of two sets A and B consists of all elements that are in one or the other of the sets. The union is denoted as A ∪ B. - The intersection of two sets A and B is the set of all elements that are members of both sets. The intersection is denoted as A ∩ B. - The empty set, denoted by ∅, is the set that contains no element. Remark: ∅ ⊂ N ⊂ Z ⊂ Q ⊂ R. - Let A ⊂ B. The complement of A, denoted by Ac , is the set of all elements in B which are not in A. Set of Real Numbers -The set of real numbers denoted as R. I The real numbers are ordered. That is, given a, b ∈ R we have either a < b or a > b or a = b. I We can add, subtract and multiply any two real numbers. We can also divide any number by any non-zero number. Number Line The set of real numbers can be represented visually as a line. - Each point on the line represents a unique real number. - We choose an arbitrary point to be zero. - Positive numbers are to the right of zero and negative numbers are to the left. - In general, a < b means that a lies to the left of b on the number line. 2 Intervals Let a, b ∈ R with a < b. - An open interval (a, b) consists of all real numbers between a and b, but not including a and not including b. This is signified by the use of round brackets. - (a, b) is a subset of R. So, we write (a, b) ⊂ R. - The notation (a, b) is interval notation. In set notation: (a, b) = {x ∈ R | a < x < b}. - If we include the endpoints in the interval, we use square brackets. That is, [a, b] is the set of all real numbers between a and b, including a and b. This is called a closed interval. - We can also have half-open or half-closed intervals where we include one endpoint but not the other. For example, [a, b) is the set of all real numbers between a and b, including a but not including b. In summary, (a, b) [a, b] [a, b) (a, b] = = = = {x ∈ R | a < x < b} {x ∈ R | a ≤ x ≤ b} {x ∈ R | a ≤ x < b} {x ∈ R | a < x ≤ b} - Given a ∈ R, the set of {x ∈ R | x > a} in the interval notation can be written as (a, ∞). - Similarly, the set of all numbers less than a, {x ∈ R | x < a} can be written as (−∞, a). 3 - We use a square bracket for the a if we want to include it in the interval. The symbols ∞ and −∞ always have a round bracket since infinity is not a number and so cannot be included in the interval. - Since intervals are sets we can perform set operations on them. Given intervals (a, b) and (c, d) we have (a, b) ∪ (c, d) = {x ∈ R | a < x < b or c < x < d} (a, b) ∩ (c, d) = {x ∈ R | a < x < b and c < x < d} For example, (1, 3) ∪ (0, 2) = (0, 3) and (1, 3) ∩ (0, 2) = (1, 2). Exercises: 1. Find (−∞, 4) ∪ (1, 6]. Answer: (−∞, 6] 2. Find (2, 8] ∪ (−4, 10]. Answer: (−4, 10] 3. Find [−8, 4) ∩ (−2, 12). Answer: (-2,4) 4. Find [5, ∞)c . Answer: (−∞, 5) Natural and Rational Exponents Natural Exponents For a real number a and n = 1, 2, 3, . . . we define an = a | · a · a{z· · · · · a} n times - By definition, a0 = 1, for a 6= 0. - The letter n is called the exponent, and a is called the base. - If a 6= 0 and n = 1, 2, 3, . . ., we define 1 . an For natural numbers m and n, we have the following rules for working with natural exponents. a−n = 4 1. am · an = am+n 2. (ab)n = an bn 3. (am )n = amn am = am−n an an a n 5. n = b b 4. Rational Exponents √ The expression an = b, where n is any positive integer, can also be written as n b = a. Thus, √ n an = b ⇐⇒ b = a. √ - The symbol n b is called a radical. √ √ - For n = 2, we usually drop the 2 on the top of the radical 2 b and write b. √ √ Example: Calculate 3 125. From the definition above 3 125 is the number when raised to the power 3 gives 125. That number is 5 since 53 = 125. Therefore, √ 3 125 = 5. √ n - If n is even, there are two possibilities for b. For example, if n = 2 and b = 36, √ then from what we have said so far 36 could be either 6 or −6 since 62 = 36 and (−6)2 = 36. To remove this √ ambiguity we define b. Therefore, 36 = 6. - √ n √ n b for n even to be the positive nth root of 0 = 0 for all n = 2, 3, 4 . . .. √ - For n even and b < 0, n b is not defined since any real number raised to an √ even power is positive. For example, suppose n = 2 and b = −36. If a = −36, then a2 = −36. But, the square of a real number is always positive, therefore, no such a exists. We have the following rules for working with n-th roots: √ 1. n am = am/n 5 2. 3. 4. 5. r √ n a a √ = n n b b √ √ √ n a n b = n ab p √ √ m n a = mn ab √ a if n is odd n n a = −a if n is even Algebraic Expressions - A variable is a letter that can represent any number in a given set of numbers. - If variables are combined by any or all of the operations of addition, subtraction, multiplication, division, exponentiation, and extraction of roots, then the resulting expression is called an algebraic expression. Example:The following expression is an algebraic expression in the variable x: r 3 3 3x − 5x − 2 10 − x Polynomials - A polynomial of degree n in one variable is an algebraic expression of the form: an xn + an−1 xn−1 + · · · + a1 x + a0 where n is a non-negative integer, an 6= 0, x is a variable, and a0 , a1 , . . . , an are fixed constants. - Each part ai xi for i = 0, 1, . . . , n of a polynomial is called a term. - Polynomials of the form ai xi are called monomials. - A binomial is a polynomial of the form ai xi + aj xj . - A trinomial is a polynomial of the form ai xi + aj xj + ak xk . - A polynomial in one variable of degree one is called linear, for example 2x − 1. 6 - A polynomial in one variable of degree two is called quadratic, for example 2x2 − x + 1. - A polynomial in one variable of degree three is called cubic, for example x3 − 1. Adding or Subtracting Polynomials To add or subtract polynomials we add or subtract like terms, i.e., terms with the same variables raised to the same power. For example, (x3 − 6x2 + 2x + 4) + (x3 + 5x2 − 7x) = 2x3 − x2 − 5x + 4 Multiplying Polynomials To multiply polynomials, we apply distributive property: a(b + c) = ab + ac (b + c)a = ab + ac. - To find the product of a + b and c + d, we multiply each term of the first bracket with each term of the second bracket. (a + b) · (c + d) = ac + ad + bc + bd Some special cases: (a + b)2 (a − b)2 (a + b)(a − b) (a + b)3 (a − b)3 = = = = = (a + b)(a + b) = a2 + 2ab + b2 (a − b)(a − b) = a2 − 2ab + b2 a2 − b2 a3 + 3a2 b + 3ab2 + b3 a3 − 3a2 b + 3ab2 − b3 Example: (6x3 − 2x)(6x3 + 2x) = (6x3 )2 − (2x)2 = 36x6 − 4x4 (x2 − 3)(x3 + 2x + 1) = x5 − x3 + x2 − 6x − 3 7 Factoring Sometimes it will be useful to write a polynomial as a product of two or more polynomials. Some of the useful formulas are listed below: IDifference of squares: a2 − b2 = (a + b)(a − b) IDifference of cubes: a3 − b3 = (a − b)(a2 + ab + b2 ) ISum of cubes: a3 + b3 = (a + b)(a2 − ab + b2 ) Example: x6 + 8 = (x2 )3 + 23 = (x2 + 2)((x2 )2 − (x2 )(2) + 22 ) = (x2 + 2)(x4 − 2x2 + 4) - Another method of factoring polynomials is to factor out common factors: Example: 3x2 − 6x = 3x(x − 2) - We can also use the technique of factoring by grouping to help factor polynomials. That is, we group some terms in the polynomial and factor out their common factor. We use this method if the polynomial has at least 4 terms. Example: x3 + x2 + 4x + 4 = (x3 + x2 ) + (4x + 4) = x2 (x + 1) + 4(x + 1) = (x2 + 4)(x + 1) Factoring by Trial and Error Given a quadratic polynomial x2 + bx + c, we want to factor this as: (x + m)(x + n) for some numbers m and n. Note that (x + m)(x + n) = x2 + (m + n)x + mn. Since this is equal to the original polynomial then the coefficients must be the same. So, m + n = b and mn = c. If the numbers we are dealing with aren’t too large and if integer solutions exist, then these equations can often be solved by trial and error. 8 Example: If x2 + 7x + 12 = (x + m)(x + n) then m + n = 7 and mn = 12. But, mn = 12 for the following integer values of m and n. m 1 -1 2 -2 3 -3 n 12 -12 6 -6 4 -4 But, the only values in this list which also satisfy m + n = 7 are m = 3 and n = 4. Therefore, x2 + 7x + 12 = (x + 3)(x + 4). Factoring Expressions with Fractional Exponents For expressions with fractional exponents, sometimes it is possible to factor out the fractional exponent as a common factor such that the remaining elements are a polynomial. Example: 3x3/2 − 9x1/2 + 6x−1/2 = 3x−1/2 (x2 − 3x + 2) = 3x−1/2 (x − 1)(x − 2) Exercise: Factor (1 + x)−2/3 x + (1 + x)1/3 . Answer: (1 + x)−2/3 (2x + 1) Fractions I A rational expression is a quotient of two polynomials. I The fractions that we consider are assumed to have nonzero denominators. I Some rules for manipulating fractions are as follows: 9 A·C A = , C 6= 0 B·C B A C A·C · = B D B·D A/B A D = · C/D B C Example: Rewrite 3 x−1 + x x+2 A B A+B + = C C C as a single fraction. Solution: 3 x 3(x + 2) x(x − 1) + = + x−1 x+2 (x − 1)(x + 2) (x − 1)(x + 2) 3x + 6 + x2 − x = (x − 1)(x + 2) x2 + 2x + 6 = (x − 1)(x + 2) Compound Fractions In a compound fraction the numerator or the denominator or both are a rational fraction. Example : 1 − 3/x . 1 − 6/x + 9/x2 Example: Simplify 1 x+h − x1 . h Solution: 1 x+h − h 1 x = = x x(x+h) − x+h x(x+h) h −h x(x+h) h 1 = = x−(x+h) x(x+h) h −h 1 =− x(x + h) x(x + h) 10 Exercise:If two electrical resistors with resistances R1 and R2 are connected in parallel, then the total resistance R is given by R= 1 R1 1 + 1 R2 . Simplify the expression for R. Polynomial Division To divide a polynomial by a polynomial, we use long division when the degree of the divisor is less than or equal to the degree of dividend, as the next example shows. Example: Divide 2x3 − 14x − 5 by x − 3. Solution: Here, 2x3 − 14x − 5 is the dividend, and x − 3 is the divisor. Note that the powers of x are in decreasing order. 2x2 + 6x + 4 ← quotient x − 3 2x3 + 0x2 − 14x − 5 2x3 − 6x2 6x2 − 14x 6x2 − 18x 4x − 5 4x − 12 7 ← remainder - We always stop when the remainder is 0 or is a polynomial whose degree is less than the degree of the divisor. - Our answer may be written as 2x2 + 6x + 4 + 7 . x−3 - We can always write the answer in the form quotient + remainder . divisor 11