The two main pillars of mathematics are Deductive Reasoning and Abstraction.
Developed by the Greeks during their classical period of 600 B.C. to 300 B.C.
Is the type of reasoning at the heart of a ‘mathematical proof’.
The starting point for mathematics, in some sense, is a collection of assumptions (in your course there known as axioms!).
These axioms are accepted to be true.
Hence, one is not required to prove these axioms.
This area of mathematics is called ‘Foundations of mathematics’.
It is the foundation of which all mathematics is built.
So now how do we actually move on to obtain new mathematical conclusions and proofs?
ENTER Deductive Reasoning
So how does it work?
We start with premises(which are accepted facts) and then derive conclusions with certainty !
Here is an example..
Suppose my premises are that
Then, I conclude with certainty that
All students in the class have brown hair. Brian is a student in the class. Brian has brown hair.
Suppose my premise is that x is a number such that x - 4 = 6.
Then, I can conclude with certainty that x = 10.
If one does not employ deductive reasoning, then its conclusions are not necessarily certain.
Thales from Miletus (Turkey) regarded as first to require that deductive reasoning be used in order to prove something in mathematics.
After that Pythagoras, Plato and Euclid involved in developing the concept of deductive reasoning.
f(n) = n^2 + n + 41
Suppose my premise is that n is any positive number.
Can I use deductive reasoning to conclude with certainty (i.e. give a mathematical proof) that f(n) is a prime number?
Consider the function:
What is meant by mathematical proof?
A sequence of statements, each of which is either validly derived from those preceding it or is an axiom or assumption, and on the final member of which, the conclusion, is the statement of which the truth is thereby established.
Well f(n) is prime number for n = 1, 2, 3, 4, .., 30.
But f(n) is prime for 1 ≤ n ≤ 39, but
f(40) = 40(40) + 40 + 41 = 42^2
and so is NOT prime!
Conclusions remain certain forever.
E.g. Euclid’s Proof that there is infinitely many primes, is as valid today as it was 2,000 years ago.
Glossary of terms associated.
THEOREM: A statement deduced from axioms by logical arguments.
CONVERSE: Something that is opposite or contrary.
COROLLARY: A statement that follows, with little or no proof required, from a theorem that has already been proven.
IF AND ONLY IF: A logical statement connecting two conditions which depend on each other. Both statements must be true or both false.
IMPLIES: Suggests a logical consequence.
IS EQUIVALENT TO: Means ‘is equal to’.
To prove something by contradiction, we assume that what we want to prove is not true, and then show that the consequence of this are not possible.
That is, the consequences contradict either what we have just assumed, or something we already know to be true (or, indeed, both) - we call this a contradiction.
Proof by Contradiction
One well-known use of this method is in the proof that square root of 2 is irrational!
It is not always the best method for mathematical proof!
Draw CE perpendicular to BC
Construct CD equal to CA and join B to D
Applying Pythagoras’ Theorem to triangle BCD
BD 2 = BC 2 + DC 2 (I.47)
BD 2 = a 2 + b 2 (since BC = a and DC = b)
BD 2 = c 2 (since a 2 + b 2 = c 2 given )
BD = c
Triangles BCD and BCA are congruent by (SSS) ∴ angle α is a right angle QED
Euclid’s Proof of the Converse of Pythagoras’ Theorem (I.48) To prove that : If the square on the hypotenuse is equal to the sum of the squares on the other two sides then the triangle contains a right angle . α To prove that angle α is a right angle Given c 2 = a 2 + b 2 The Proof b c a C A B E D
Incommensurable Magnitudes (Irrational Numbers) 1 1 √ 2 The whole of Pythagorean mathematics and philosophy was based on the fact that any quantity or magnitude could always be expressed as a whole number or the ratio of whole numbers. Unit Square The discovery that the diagonal of a unit square could not be expressed in this way is reputed to have thrown the school into crisis, since it undermined some of their earlier theorems. Story has it that the member of the school who made the discovery was taken out to sea and drowned in an attempt to keep the bad news from other members of the school. He had discovered the first example of what we know today as irrational numbers .
1 1 √ 2 1 It is possible to draw a whole series of lengths that are irrational by following the pattern in the diagram below and using Pythagoras’ Theorem. Continue the diagram to produce lengths of √ 3, √ 5, √ 6, √ 7 , etc. See how many you can draw. You should get an interesting shape.