MAT01A1: Complex Roots & Intro to Logic

Dr Craig

18 February 2020

From powers to roots in CSuppose we want to find the n-th root of

z = r(cos θ + i sin θ). That is, we want the

complex number w = s(cosϕ + i sinϕ) such

that wn = z. From De Moivre’s Theorem we

want

sn(cosnϕ + i sinnϕ) = r(cos θ + i sin θ)

To get this, we need

sn = r and cosnϕ = cos θ and sinnϕ = sin θ

Thus nϕ = θ + 2kπ.

Roots of a complex number: n-th roots

Let z = r(cos θ + i sin θ) and let n be any

positive integer.

Then z has n distinct n-th roots. That is,

for k = 0, 1, 2, . . . , n− 1 the roots are

wk = r1/n

[cos

(θ + 2kπ

n

)+ i sin

(θ + 2kπ

n

)]All of the roots of z lie on the circle of radius

r1/n in the complex plane.

It often helps to think of the argument of a

complex root in the following way:

θ + 2kπ

n=θ

n+ k

(2π

n

)The argument of the first root will simply beθn (because k = 0). Each root after that has

the same modulus (r1/n) but is rotated

anti-clockwise by 2πn . You only need to

calculate solutions up to k = n− 1. If you let

k = n then you will have the same complex

number as w0 but with a different argument.

Example of roots of a complex number

Find the cube roots of z = i.

Note that a = 0 and b = 1. Thus we get

r = 1 and arg(z) = θ =π

2

Solutions to 3√i

w0 =3√1(cos(π6

)+ i sin

(π6

))w1 =

3√1

(cos

(π

6+

2π

3

)+ i sin

(π

6+

2π

3

))w2 =

3√1

(cos

(π

6+

4π

3

)+ i sin

(π

6+

4π

3

))Exercise: convert each solution to the form

a+ bi. You will see that w2 is a solution that

you might have found by inspection.

Another exercise:

Take w0, w1 and w2 and cube each of them

using the rectangular form (a + bi). Check

that in each case you get i as the solution.

Also, look at Example 7 on page A62 of the

textbook.

LOGIC

Notes for this section have been uploaded to

Blackboard as a pdf under “Course content”.

Please take the time to read these printed

notes.

Introduction

Mathematics is about reasoning. In this course weprove certain facts about real numbers, aboutfunctions, about differentiation, etc.

In order to prove mathematical statements we needa system of reasoning. The next few lectures onlogic provide a mathematical foundation for thesystem of reasoning that we will use.

Instead of using numbers, our newelements/variables are propositions. Instead ofoperations like +,−,×,÷, we use logical operationswhich operate on our propositions.

Introduction continued...

Language is very important in logic. We are

translating statements from English into a

mathematical language.

Our mathematical language will be very

precise, therefore we also have to be very

precise about our use of the English

language.

Please ask if there is something that doesn’t

make sense.

Logic at UJ:

Dr Wilmari Morton, Dr Claudette Robinson

& Dr Craig.

We conduct research into non-classical and

multivalued logics.

We collaborate with researchers from the

Netherlands, Sweden, Italy, Slovakia.

In 2016 we hosted LATD (an international

logic conference) in Phalaborwa.

Some famous logicians:

I George Boole (Boolean logics/circuits)

I Kurt Godel (completeness of first-order

logic and incompleteness theorem)

I Bertrand Russell (Russell’s paradox)

I Alfred Tarski (pea and the sun paradox)

Propositions

Definition: A proposition is a statement that iseither true or false, but not both.

Examples:

I UJ is in Johannesburg.

I Durban is the capital of South Africa.

I 3 plus 3 equals 6.

I 3 plus 3 equals 5.

Note: a proposition does not have to be true. Butit can’t be both false and true, and it can’t beneither true nor false.

Definition: A proposition is a statement that iseither true or false, but not both.

The following are not propositions:

I Are you tired? (Not a statement.)

I She is a college student.(The truth value depends on who ‘she’ is.)

I x is a rational number.(The truth value depends on what x is.)

I This statement is false.(If it is true, then it must also be false, acontradiction. If it is false, then it must be true.Again, a contradiction.)

Logical connectives

not negation ¬and conjunction ∧or disjunction ∨if ... then ... implication →if and only if biconditional ↔

Suppose p is the primitive proposition “2 is a primenumber” and q is the proposition “The sun is hot”.Then¬p “2 is not a prime number”

p ∧ q “2 is a prime number and the sun is hot”

q → p “If the sun is hot, then 2 is a prime number”

Truth values for logical connectives

For negation:

p ¬pT FF T

For conjunction:

p q p ∧ qT T T

T F FF T FF F F

For disjunction:

p q p ∨ qT T TT F TF T TF F F

In ordinary language ‘or’ is often used in anexclusive sense, e.g. in ‘I will win or I will lose’. Informal logic ‘or’ is used in an inclusive sense, thus‘It will rain or it will be cold’ will still be true if it isboth rainy and cold.

In CompSci you might have seen this difference withthe logical command XOR (exclusive OR).

For implication (here we call p the “antecedent”and q the “consequent”):

p q p→ q

T T TT F FF T TF F T

For the biconditional:

p q p↔ q

T T TT F FF T FF F T

Computing truth values for propositionsThe truth value of a compound proposition dependson the truth values of its primitive propositions.Suppose we have:

I “Dineo is clever” is true

I “Dineo is lazy” is false

I “Dineo likes mathematics” is true

We can now calculate whether the followingsentence is true or false:

“Dineo is not clever, or, if she likes mathematics,then she is clever and is not lazy”

¬p ∨ (r → (p ∧ ¬q))

BODMAS for logical connectives

The order of priority is as follows:

1. ¬2. ∧ and ∨3. →4. ↔

For example:

¬p→ q = (¬p)→ q

p ∧ q → r = (p ∧ q)→ r

p ∨ q ↔ r = (p ∨ q)↔ r

p→ q ↔ r = (p→ q)↔ r