MATH 1510 Lili Shen Numbers Fundamentals of...

MATH 1510

Lili Shen

ComplexNumbers

Inequalities

Modeling

Fundamentals of Mathematics(MATH 1510)

Instructor: Lili ShenEmail: shenlili@yorku.ca

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.