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MATH 1510
Lili Shen
ComplexNumbers
Inequalities
Modeling
Fundamentals of Mathematics(MATH 1510)
Instructor: Lili ShenEmail: [email protected]
Department of Mathematics and StatisticsYork University
September 28 - October 2, 2015
MATH 1510
Lili Shen
ComplexNumbers
Inequalities
Modeling
Outline
1 Complex Numbers
2 Inequalities
3 Modeling
MATH 1510
Lili Shen
ComplexNumbers
Inequalities
Modeling
Complex numbers
DefinitionA complex number is an expression of the form
a + bi ,
where a,b ∈ R and i2 = −1; a is called the real part and b iscalled the imaginary part. For two complex numbers a + bi ,c + di ,
a + bi = c + di ⇐⇒ a = c and b = d .
A complex number whose real part is zero is said to bepurely imaginary.
MATH 1510
Lili Shen
ComplexNumbers
Inequalities
Modeling
Complex numbers
If PJ stands for Poor Joke,then P+iJ is a Complex Poor Joke,and you did not laugh because the Joke part is imaginary.
MATH 1510
Lili Shen
ComplexNumbers
Inequalities
Modeling
Arithmetic operations on complex numbers
Proposition
(1) (a + bi)± (c + di) = (a± c) + (b ± d)i .(2) (a + bi)(c + di) = (ac − bd) + (ad + bc)i .
MATH 1510
Lili Shen
ComplexNumbers
Inequalities
Modeling
Arithmetic operations on complex numbers
ExampleCalculate:(1) (3 + 5i) + (4− 2i).(2) (3 + 5i)− (4− 2i).(3) (3 + 5i)(4− 2i).(4) i23.
MATH 1510
Lili Shen
ComplexNumbers
Inequalities
Modeling
Arithmetic operations on complex numbers
Solution.(1) (3 + 5i) + (4− 2i) = 7 + 3i .(2) (3 + 5i)− (4− 2i) = −1 + 7i .(3) (3 + 5i)(4− 2i) = 22 + 14i .(4) i23 = −i .
MATH 1510
Lili Shen
ComplexNumbers
Inequalities
Modeling
Complex conjugates
The complex conjugate of a complex number z = a + bi is
z = a− bi .
The product of a complex number and its conjugate isalways a nonnegative real number:
zz = (a + bi)(a− bi) = a2 + b2.
MATH 1510
Lili Shen
ComplexNumbers
Inequalities
Modeling
Arithmetic operations on complex numbers
PropositionThe division of two complex numbers is calculated as:a + bic + di
=ac + bdc2 + d2 +
bc − adc2 + d2 i .
MATH 1510
Lili Shen
ComplexNumbers
Inequalities
Modeling
Arithmetic operations on complex numbers
ExampleCalculate:
(1)3 + 5i1− 2i
.
(2)7 + 3i
4i.
MATH 1510
Lili Shen
ComplexNumbers
Inequalities
Modeling
Arithmetic operations on complex numbers
Solution.
(1)3 + 5i1− 2i
=(3 + 5i)(1 + 2i)(1− 2i)(1 + 2i)
= −75+
115
i .
(2)7 + 3i
4i=
(7 + 3i)i4i · i
=34− 7
4i .
MATH 1510
Lili Shen
ComplexNumbers
Inequalities
Modeling
Square roots of negative numbers
DefinitionFor any r ∈ R+, the principle square root of −r is
√−r = i
√r .
The two square roots of −r are ±i√
r .
We usually write i√
r instead of√
r i to avoid confusion with√ri .
MATH 1510
Lili Shen
ComplexNumbers
Inequalities
Modeling
Square roots of negative numbers
Although √a ·√
b =√
ab
when a,b ∈ R+, this is not true when a,b ∈ R−. Forexample:
√−2 ·
√−3 = i
√2 · i√
3 = −√
6,»(−2)(−3) =
√6.
MATH 1510
Lili Shen
ComplexNumbers
Inequalities
Modeling
Square roots of negative numbers
Example
Calculate (√
12−√−3)(3 +
√−4).
MATH 1510
Lili Shen
ComplexNumbers
Inequalities
Modeling
Square roots of negative numbers
Solution.
(√
12−√−3)(3 +
√−4)
= (2√
3− i√
3)(3 + 2i)
= (6√
3 + 2√
3) + i(4√
3− 3√
3)
= 8√
3 + i√
3.
MATH 1510
Lili Shen
ComplexNumbers
Inequalities
Modeling
Complex solutions of quadratic equations
We already know that the solutions of a quadratic equationax2 + bx + c = 0 (a 6= 0) are
x =−b ±
√b2 − 4ac
2a.
If b2 − 4ac < 0, the equation has no real solution.
However, in the complex number system, the equationalways have solutions even when b2 − 4ac < 0:
x =−b ± i
√4ac − b2
2a(if b2 − 4ac < 0).
MATH 1510
Lili Shen
ComplexNumbers
Inequalities
Modeling
Complex solutions of quadratic equations
ExampleSolve the following equations:(1) x2 + 9 = 0.(2) x2 + 4x + 5 = 0.
(3)13
x2 − 2x + 4 = 0.
MATH 1510
Lili Shen
ComplexNumbers
Inequalities
Modeling
Complex solutions of quadratic equations
Solution.(1) x = ±3i .
(2) x =−4±
√42 − 4 · 52
= −2± i .
MATH 1510
Lili Shen
ComplexNumbers
Inequalities
Modeling
Complex solutions of quadratic equations
(3)
13
x2 − 2x + 4 = 0,
x2 − 6x + 12 = 0,
(x − 3)2 = −3,
x − 3 = ±i√
3,
x = 3± i√
3.
MATH 1510
Lili Shen
ComplexNumbers
Inequalities
Modeling
Complex conjugate root theorem
TheoremFor any equation equivalent to the form
anxn + an−1xn−1 + · · ·+ a1x + a0 = 0,
where a0,a1, . . . ,an ∈ R, if c + di (c,d ∈ R) is a solution ofthis equation, then so is its conjugate c − di.
MATH 1510
Lili Shen
ComplexNumbers
Inequalities
Modeling
Outline
1 Complex Numbers
2 Inequalities
3 Modeling
MATH 1510
Lili Shen
ComplexNumbers
Inequalities
Modeling
Inequalities
Inequalities look like equations but the equal sign isreplaced by <, >, ≤ or ≥. For example:
4x + 7 ≤ 19.
To solve an inequality that contains a variable means to findall values of the variable that make the inequality true.Unlike an equation, an inequality generally has infinitelymany solutions which form the solution set of the inequality.
MATH 1510
Lili Shen
ComplexNumbers
Inequalities
Modeling
Rules for inequalities
Proposition(1) A ≤ B ⇐⇒ A + C ≤ B + C.(2) A ≤ B ⇐⇒ A− C ≤ B − C.(3) If C > 0, then A ≤ B ⇐⇒ CA ≤ CB.(4) If C < 0, then A ≤ B ⇐⇒ CA ≥ CB.
(5) If A > 0 and B > 0, then A ≤ B ⇐⇒ 1A≥ 1
B.
(6) If A ≤ B and C ≤ D, then A + C ≤ B + D.(7) If A ≤ B and B ≤ C, then A ≤ C.
MATH 1510
Lili Shen
ComplexNumbers
Inequalities
Modeling
Solving linear inequalities
ExampleSolve the inequality
3x < 9x + 4.
MATH 1510
Lili Shen
ComplexNumbers
Inequalities
Modeling
Solving linear inequalities
Solution.
3x < 9x + 4,−6x < 4,
6x > −4,
x > −23.
So the solution set is(− 2
3,∞).
MATH 1510
Lili Shen
ComplexNumbers
Inequalities
Modeling
Solving nonlinear inequalities
ExampleSolve the following inequalities:(1) x2 ≤ 5x − 6.(2) x(x − 1)2(x − 3) < 0.
(3)1 + x1− x
≥ 1.
MATH 1510
Lili Shen
ComplexNumbers
Inequalities
Modeling
Solving nonlinear inequalities
Solution.(1)
x2 ≤ 5x − 6,
x2 − 5x + 6 ≤ 0,(x − 2)(x − 3) ≤ 0.
Checking the sign of x − 2, x − 3 in the intervals
(−∞,2), (2,3), (3,∞),
we obtain the solution set [2,3].
MATH 1510
Lili Shen
ComplexNumbers
Inequalities
Modeling
Solving nonlinear inequalities
(2) Checking the sign of x , (x − 1)2 and x − 3 in theintervals
(−∞,0), (0,1), (1,3), (3,∞),
we obtain the solution set
(0,1) ∪ (1,3).
MATH 1510
Lili Shen
ComplexNumbers
Inequalities
Modeling
Solving nonlinear inequalities
(3)
1 + x1− x
≥ 1,
1 + x1− x
− 1 ≥ 0,
2x1− x
≥ 0.
Checking the sign of 2x , 1− x in the intervals
(−∞,0), (0,1), (1,∞),
we obtain the solution set [0,1).
MATH 1510
Lili Shen
ComplexNumbers
Inequalities
Modeling
Absolute value inequalities
Proposition
For any c ∈ R+:(1) |x | ≤ c ⇐⇒ −c ≤ x ≤ c.(2) |x | ≥ c ⇐⇒ x ≤ −c or x ≥ c.
MATH 1510
Lili Shen
ComplexNumbers
Inequalities
Modeling
Solving absolute value inequalities
ExampleSolve the following inequalities:(1) |x − 5| < 2.(2) |3x + 2| ≥ 4.
MATH 1510
Lili Shen
ComplexNumbers
Inequalities
Modeling
Solving absolute value inequalities
Solution.(1)
|x − 5| < 2,−2 < x − 5 < 2,
3 < x < 7.
So the solution set is (3,7).
MATH 1510
Lili Shen
ComplexNumbers
Inequalities
Modeling
Solving absolute value inequalities
(2)
|3x + 2| ≥ 4,3x + 2 ≥ 4 or 3x + 2 ≤ −4,
3x ≥ 2 or 3x ≤ −6,
x ≥ 23
or x ≤ −2,
So the solution set is (−∞,−2] ∪[23,∞).
MATH 1510
Lili Shen
ComplexNumbers
Inequalities
Modeling
Outline
1 Complex Numbers
2 Inequalities
3 Modeling
MATH 1510
Lili Shen
ComplexNumbers
Inequalities
Modeling
Modeling with equations
ExampleA car rental company charges 30 dollars a day and 15 centsa mile for renting a car. Helen rents a car for two days, andher bill comes to 108 dollars. How many miles did shedrive?
MATH 1510
Lili Shen
ComplexNumbers
Inequalities
Modeling
Modeling with equations
Solution.Let x be the number of miles driven. Then
30 · 2 + 0.15x = 108,0.15x = 48,
x = 320.
So, Helen drove her rental car 320 miles.
MATH 1510
Lili Shen
ComplexNumbers
Inequalities
Modeling
Modeling with equations
ExampleA rectangular building lot is 8 ft longer than it is wide andhas an area of 2900 ft2. Find the dimensions of the lot.
MATH 1510
Lili Shen
ComplexNumbers
Inequalities
Modeling
Modeling with equations
Solution.Let x be the width of the lot. Then
x(x + 8) = 2900,
x2 + 8x − 2900 = 0,(x − 50)(x + 58) = 0.
Since the width of the lot must be a positive number, weconclude that x = 50 ft.So, the width and the length of the lot are respectively 50 ftand 58 ft.
MATH 1510
Lili Shen
ComplexNumbers
Inequalities
Modeling
Modeling with inequalities
ExampleA carnival has two plans for tickets:
Plan A: 5 dollars as entrance fee and 25 cents for eachride.Plan B: 2 dollars as entrance fee and 50 cents for eachride.
How many rides would you have to take for Plan A to beless expensive than Plan B?
MATH 1510
Lili Shen
ComplexNumbers
Inequalities
Modeling
Modeling with inequalities
Solution.Let x be the number of rides. Then
5 + 0.25x < 2 + 0.50x ,0.25x > 3,
x > 12.
So if you plan to take more than 12 rides, Plan A is lessexpensive.