Linear Algebra: Number Theory

Types of numbers: rational vs integer vs real. Prime numbers, Fundamental theorem of arithmetic.

32 flashcards · Mathematics · by lakesare

Define prime number.
Integer > 1 whose only factors (divisors) are itself and 1.
Name 3 first primes.
2, 3, 5. (not 1!)
x / m = n (without a remainder) m is a __________ of x?
factor or divisor. (Divides a multiple without remainders).
What's an opposite of prime number called?
A composite number.
State Fundamental theorem of arithmetic.
Every integer greater than 1 either: is prime itself, or is the product of primes, and this product is unique.
Name 2 facts Fundamental theorem of arithmetic states about 1200 = 24 × 31 × 52 = 5 × 2 × 5 × 2 × 3 × 2 × 2 = ...
1200 can be represented as a product of primes, no matter how this is done, there will always be four 2s, one 3, two 5s, and no other primes in the product.
Define rational numbers.
Number that can be expressed as a ratio p/q, where p and 1 are integers q is not 0 Denoted by Q, as in Quotient.
Express 0.153434... as fraction in its lowest terms.
x = 1.53434... 100x = 153.4343... (2 zeros because 34 is the repeating block) Solve for: 100x - x = 153.4343... - 1.53434... 99x = 151.9 x = 151.9/99 = 1519/990
Express 0.003 in floating-point notation VS scientific notation.
floating-point: 0.3 * 10-2 or 0.3E-2 scientific: 3 * 10-3 or 3E-3
What's meant by significand/mantissa?
Depending in notations: floating-point: 0.3E-2 scientific: 3E-3 Either 0.3 or 3 part of the number.
Define irrational numbers.
All numbers that are not rational! That is, numbers that can't be expressed as a ratio of 3 integers. Irrational numbers have decimal expansions that neither terminate nor become periodic.
Which of these numbers are irrational?
We know that Pi is irrational. Prove that Pi/2 is irrational too.
Suppose Pi/2 is rational. Then, by def, we could write that: Pi/2 = a/b, where a and b are integers. We could rewrite this as Pi = 2a/b, which would imply that Pi is rational. Contradiction!
Write: 2 first composite numbers 2 first prime numbers
composite: 4, 6 prime: 2, 3
Prove: Any product of four consecutive integers is one less than a perfect square.
n(n+1)(n+2)(n+3) => Links: https://math.stackexchange.com/a/738401/163615
How is this read: d | n?
The notation d | n is read “d divides n” Symbolically, if n and d are integers and d = 0: d | n ⇔ ∃ an integer k such that n = dk
Does 4 | 15?
No, 15 | 4 = 3.75, which is not an integer.
Why is 1 not a prime number?
Fundamental Theorem of Arithmetic: This theorem is the reason the number 1 is not allowed to be prime. If 1 were prime, then factorizations would not be unique. For example, 6 = 2 ·3 = 1 ·2 ·3, and so forth.
What's another name for Fundamental Theorem of Arithmetic?
Unique Factorization Theorem.
For each of the following values of n and d, find integers q and r such that n = dq + r and 0 ≤ r < d. a. n = 54, d = 4 b. n = −54, d = 4 c. n = 54, d = 70
a. 54 = 4·13 + 2; hence q = 13 and r = 2. b. −54 = 4·(−14) + 2; hence q = −14 and r = 2. c. 54 = 70·0 + 54; hence q = 0 and r = 54.
Prove the irrationality of 2\sqrt{2}2 .
Proof by contradiction: suppose 2\sqrt{2}2 is rational. Then: 2\sqrt{2}2 = a/b (where a and b don't share any denominators) 2 = a2/b2 a2 = 2b2 Now we'll show that both a and b are even: a is even because - a2 = 2k b is even because - a2 = 2b2 (2l)2 = 2b2 4l2 = 2b2 2l2 = b2 (and if square of the number is even - this number is even). Contradiction: we chose a and b to not have any common denominators, but since they are both even - they share 2!
Prove: if prime number p divides a, then p doesn't divide (a + 1).
Suppose p divides both a and (a+1): pk = a pl = a + 1 pk - pl = a - a - 1 p(k - l) = -1 p(l - k) = 1 Contradiction: only divisors of 1 are 1 and -1, neither of which are prime numbers.
Prove: set of all prime numbers is infinite.
Suppose set of all prime numbers is finite, then we can create a product of them + 1: N = (p1 * p2 * p3 * ... * pn) + 1 We know that N is not a prime number, because it's bigger than all the prime numbers we listed. Therefore N can be divided by one of these listed prime numbers. So: N = irrelevant_number * p (p1 * p2 * p3 * ... * pn) + 1 = irrelevant_number * p But since a product of primes involves all primes: (p1 * p2 * p3 * ... * pn) = irrelevant_number * p Contradiction: we know if prime n…
What is multiplicative inverse for a number x?
x−1 Number that when multiplied by x, will equal 1: x * multiplicative inverse of x = 1 Also called reciprocal of x.
Get the multiplicative inverse for 5.
5-1 = 1/5 Because 5 * 5-1 = 1. Also called reciprocal of 5.
When integers a and b are relatively prime?
When a and b have no common divisor (prime or not) greater than 1: gcd(a, b) = 1.
Define Euler's totient function.
φ(n) = amount of integers in [0, n), that are relatively prime to n. Euler's function is denoted as φ(n)Since 0 is not relatively prime to anything, φ(n) could equivalently be defined using the interval [1, n) rather than [0, n)
Find φ(7).
φ(7) = 6 Because 6 numbers in [0, 7) are relatively prime to 7: { 6, 5, 4, 3, 2, 1 }.
Find φ(12).
φ(12) = 4 Because 4 numbers in [0, 12) are relatively prime to 12: { 11, 7, 5, 1 }.
Solve:
Proof:
Most calculators only have keys: with base 10 (log), with base e (natural logarithms, ln) How to calculate log25 on such a calculator?
log25 = ln(5) / ln(2) = 2.321928095.
The simplest numbers are the natural numbers N, which are sufficient for all your math needs if all you are going to do is count things. How many goats? Five goats here and six goats there so the total is 11 goats. When do you start needing: integers Z rational numbers Q real numbers R complex numbers C? Numbers are the basic objects we use to calculate things. Mathematicians like to classify the different kinds of number-like objects into sets:Note also that each of the sets in the list contai…
integers Z - when you use the minus operator rational numbers Q - when you use the division operator real numbers R - when we start dealing with geometry complex numbers C - when we want to take the root of -1