How do we do proofs? Part I - Dr Joel Feinstein

This is the first of two sessions on how to do proofs. See also Dr Feinstein's blog at http://explainingmaths.wordpress.com/
The aim of these sessions on how we do proofs is to help students with some of the relatively routine aspects of doing proofs. In particular, we focus on how to start proofs, and how and when to use definitions and known results. With practice, students should become fluent in these routine aspects of writing proofs, and this will allow them to focus instead on the more creative and interesting aspects of constructing proofs.

Part I is suitable for anyone with a knowledge of elementary algebra (including odd numbers, multiples of eight and the binomial theorem for expanding powers of (a+b)), and functions from
the set of real numbers to itself (odd functions, even functions, multiplication and composition of functions). This recording is the tenth class in Dr Joel Feinstein's G12MAN Mathematical Analysis module. Further module materials are available for download from The University of Nottingham open courseware site: http://unow.nottingham.ac.uk/resources/resource.aspx?hid=c6c045f6-286d-6b9f-b... and on iTunes U: http://itunesu.nottingham.ac.uk/albums/71.rss

Category:

Education

Tags:

• Mathematics
• Proofs
• Direct_Proofs
• Joel_Feinstein
• Definitions
• Odd_Numbers
• Multiples_Eight
• Odd_Functions
• Even_Functions
• Pure_Maths
• Pure Mathematics
• mathematical analysis
• real numbers
• calculus
• mathematics
• sequences
• limits
• functions
• Euclidian spaces
• differentiation
• integration

[Bl4nkB0x]

@JoelFeinstein That's brilliant! I tried something similar: trying to show that for all integers m, 8(2m^4+4m^3+3m^2+m) is even. I tried to show separately for when m is odd and even, instead of separating the terms and seeing it's quite obvious to prove.

Thanks for your response.

[JoelFeinstein]

@Bl4nkB0x The proof given shows that when n=2m+1, then n^4-1= 8(2m^4+4m^3+3m^2+m). To prove divisibility by 16, you need to show that 2m^4+4m^3+3m^2+m is even. This is obvious for the first two terms (2m^4+4m^3), so you are left with the problem of proving that, for all integers m, 3m^2+m is even. Again this is obvious if m is even. But if m is odd, then so is 3m^2, and so 3m^2+m is a sum of two odd integers, hence is even.

[Bl4nkB0x]

How would you prove for divisible by 16?

[JoelFeinstein]

@elninovaquero This is the definition of "odd integer" we are working with. The even integers are integers of the form 2m for some other integer m, while the odd integers are integers of the form 2m+1 instead. (Equivalently, you can ask for remainder 1 when you divide by 2, but that is the same thing.)

[elninovaquero]

How did you get the 2m+1??