Numbers And Arithmatic Flashcards
Define complex numbers?
What are the main things about the properties of complex numbers?
Additive, multiplicative identity and inverse
How does he do the complex congregate?
Using a line on top of the Z
What is the definition of a complex congregate?
What does i.e mean?
In other words
What are some useful lemmas to do with complex numbers?
Define to polar form of a complex number?
What is the polar form of 0?
0
How should i rethink about trig form?
It is the exponential form first then made into the trig form
What can we say about the unit circle in the complex plane?
It is a group and it is closed under complex multiplication
What are the 2 ways to prove something by induction for all negative integers?
Use =k-1 or use -(k+1)
How do you define the roots of unity?
The solutions to the equation z^n=1
Where:
n is a natural number
And z a complex number
How can you write the condensed factorised form of the roots of unity?
What do you need to do when you see a proof?
Firstly, wrestle with the mathematics
Then understand the point
What does it mean to get rid of the complex coefficients in an factorised form of a polynomial?
Yu get rid of the complex root by expanding them. (Just A level factorisation ignoring complex numbers)
Define prime numbers
What statements follow from the definition of a prime number?
What si the idea behind the proof of the following statements?
Prove directly following the definitions and substitutions
What is the idea behind the proof of the following result?
Use induction with a contradiction for m=a+1
Eventually x is a divisor at a+1 which bounds n,x when inductive hypothesis is used
What is the general idea behind the proof of there being infinitely many primes?
Define lcm?
Define gcd
What is some insights we can make about lcm and gcd definitions
What is the Bezout identity
How does Euclid’s algorithm work?
Provide a worked example
Why did we bother to prove Bezout’s identity?
It tells us that for all numbers we could use Euclid’s algorithm and then work backwards to express the gcd(a,b) as ax+by
What is Euclid’s Lemma?
What is the fundamental theorem of arithmetic?
