Download NUMBERS AND INEQUALITIES Introduction Sets

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.
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∈:
/ 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.
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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).
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- 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
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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.
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- 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
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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.
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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:
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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)
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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
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