Why Are Mathematical Proofs Important

WHAT ARE MATHEMATICAL PROOFS AND WHY

THEY ARE IMPORTANT?

introduction

Many students seem to have trouble with the notion of a mathemat-

ical proof. People that come up to a form likdue east Math 216, who certainly

know a dandy deal of mathematics - Calculus, Trigonometry, Geometry

and Algebra, all of the sudden come up to run across a new kind of mathemat-

ics, an abstract mathematics that requires proofs. In this document

westeast will endeavor to explicate the importance of proofs in mathematics, and

to give a you an idea what are mathematical proofs. This will requite

you some reference to checg if your proofs are correct. Weastward begin by

describing the role of proofs in mathematics, and so we deﬁne the logical

language which serves equally the basis for proofs and logical deductions.

Adjacent we talk over brieﬂy the role of axioms in mathematics. Finally due weste

givdue east several examples of mathematical proofs using various techniques.

At that place is also an first-class practicecument on proofs written past Prof. Jim

Hurley (liknked to the course homepage), and I recommend to all of

you to read his exercisecument as well. You will ﬁnd in that location more examples

and additional explanations. It is my hope that reading these docu-

ments will make your ﬁrst steps in writing proofs easier.

1. The importance of Proofs in mathematics

It is diﬃcult to overestimate the importance of proofs in mathemat-

ics. If you hafivee a theorize, the only way that you can safely exist

sure that it is true, is by presenting a valid mathematical proof. For

example, consider the follofly well known mathematical theorem:

Theorem i (Euclid). There are inﬁnitely many prime numbers.

For those of you who don't remember, a prime number is a posi-

tive integer p > i that cannot be written equally a product of two strictly

smaller positive integers a and b. For case, the number 91 is non

prime since information technology can be written every bit 91 = 13 · seven, merely 67 is a prime number. Although

about people ha5e a hunch that there are inﬁnitely humany primes, it is

non obvious at all. Even today with the powerful computers, all we

tin can exercise is to verify that there are very big prime numbers, but westwarde

Why Are Mathematical Proofs Important,

Source: https://docslib.org/doc/8763246/mathematical-proofs-and-why-they-are-important