Prime numbers – a number that can Flashcards
What is a prime number?
A number that can only be divided by itself and 1 without remainders. It is not a product of two smaller natural numbers.
What is a composite number?
Numbers that have more than two factors and can be formed by multiplying two smaller positive integers.
What is a rule of inference?
A logical form or guide consisting of premises (or hypotheses) that draws a conclusion.
What is a conjecture?
A mathematical statement which appears to be true, but has not yet been rigorously proved.
Example: The Goldbach Conjecture: Every even integer greater than 2 can be expressed as the sum of two primes.
What is an axiom (or postulate)?
A statement that is assumed to be true without proof. A postulate is true based on geometry principles.
Example: In geometry, ‘Through any two points, there is exactly one line.’
What is a theorem?
A statement that has been or can be proven to be true based on known facts and mathematical operations.
Example: Pythagorean Theorem a^2 + b^2 = c^2 for a right triangle.
What is a lemma (or pre-theorem)?
A minor result whose sole purpose is to help in proving a theorem. It is a stepping stone on the path to proving a theorem.
What is a corollary (or post-theorem)?
A result in which the (usually short) proof relies heavily on a given theorem.
What is a proof?
A logical argument that shows a mathematical statement is true.
What does divisibility mean?
This means a number can be divided without leaving a remainder.
What is the greatest common divisor?
The greatest value that can divide two integers evenly.
What is the least common multiple?
The smallest positive multiple that two or more numbers share.
What are relatively prime integers?
There are no common factors other than 1, meaning no other integer could divide both numbers evenly.
What is a direct proof?
A method that involves proving a statement by assuming the hypothesis is true and demonstrating that the conclusion logically follows.
What is a proof of contrapositive?
Uses the contrapositive of a conditional statement to prove the statement itself.
Example: p → q is equivalent to ~q → ~p.
What is a proof of contradiction?
It is when we assume a proposition is not true and show that it leads to a contradiction, which proves that the proposition is true.
What is a proof of exhaustion?
To prove a statement, you must break the statement into a finite amount of cases.
What is a proof of counterexamples?
Providing a single example where a statement is false to disprove it.
Example: Disprove ‘All prime numbers are odd.’ Counterexample: 2 is a prime number but is not odd.
What is an appeal to authority?
Occurs when we accept a claim merely because someone tells us that an authority figure supports that claim.
What is an appeal to force?
Argumentation using force or the threat of force to convince others to accept an argument’s conclusion.
What is an appeal to ignorance?
Occurs when you argue that your conclusion must be true, because there is no evidence against it.
What is an appeal to pity?
An attempt to persuade others by exploiting and provoking feelings of guilt or pity instead of presenting factual evidence.
What is an appeal to people?
Argues that a claim is true simply because that’s what a large number of people believe.
What is argumentum ad hominem?
Occurs when, instead of addressing someone’s argument or position, you irrelevantly attack the person or some aspect of the person who is making the argument.
What is a circular argument?
A logical fallacy wherein it tries to prove itself using its conclusion as evidence.
What is equivocation?
Happens when a word or phrase in an argument has more than one meaning, and the person switches between those meanings to make their point seem more logical or convincing.
What is the fallacy of division?
This happens when a person believes that what is true for the entire group or thing also holds true for each of its component pieces.
What is a false dilemma?
A logical fallacy that presents only two options or sides to an issue when there are actually more complexities.
What is a hasty generalization?
Happens when someone makes a decision or forms a conclusion without having enough evidence or by only looking at one side of the story.
What is a red herring?
Occurs when someone introduces an irrelevant point or topic to divert attention from the original issue.
What is a slippery slope?
A logical fallacy that assumes a cause-and-effect relationship between events without evidence.
What is a strawman fallacy?
The distortion of someone else’s argument to make it easier to attack or refute.
