Y'know, multiplication is *hard*

Miscellaneous and Personal Stuff I Must Share
1

:eek:

I never really believed it before, but it is.

I used to think I was pretty good at maths, but now I’m not so sure. I realised I hadn’t done any in a while, so I thought I’d try some out to make sure I hadn’t gotten rusty. I didn’t want to strain myself too much, so I thought I’d go back to basics. Arithmetic.

I started off on addition - I mean, hey. If you can’t do addition you may as well give up maths now, right? It was a little tricky - took my about 15 minutes per problem, but I picked up the pace. By the last problem I was rocketing - it took me barely 5 minutes to solve that one. Go me.

So, having got addition thoroughly under my belt I thought I’d start up on multiplication. Boy was I in for a shock - I took a look at the first problem, and it stopped me cold. Took me almost 20 minutes of staring to figure that one out. In the end it was something pretty stupid - Of course when you multiply one by a number, you get the same number back. Common sense, isn’t it? I did a couple more problems, but they kept just getting harder and harder… Now I’m staring at the last one and I cannot see how to do it for the life of me. :eek: So there you are folks, multiplication is hard.

…

Ok. I think I’ll stop my malicious spread of disinformation. :slight_smile: All of the above is true, but not quite what it seems. I’m a first year maths student, currently in the summer holidays, realised I needed to get to work. So I thought I’d start with brushing up on our last years work. Mainly numbers and sets, because I found that the hardest subject (read: Not totally and mindnumbingly trivial). Realised that the lecturer had left some freaking huge gaps in the subject by saying “You can prove this if you want to. I’m not going to.” so I thought that would be good material to work on.

‘That’ being basic theorems of arithmetic. Simple things like:

a+b=b+a
a+(b+c)=(a+b)+c
a*(b+c)=ab+ac
etc.

And buddha on a pogo stick are some of these hard to prove. Addition was mostly fairly straightforward, as was distributivity of multiplication. The rest are kindof nasty to prove. Right now I’m working on commutativity of multiplication (ab=ba) and am rather stuck.

So, take my word for it, multiplication is hard. (I have yet to prove the maths-is-hard theorem ;)) Don’t doubt it next time some kid tells you that.

2

– Barbie

3

How you go about it depends a lot on what axioms you have to work with and how the operations are defined. But, assuming that you’re starting with the Peano axioms, remember that proof by induction is your friend.

Keep at it. These basic facts can be kind of tedious to prove, but it’s good to know why they’re true. (Particularly when you get to the contexts where they’re not.) And it feels so good when you stop. :wink: I mean, it feels so good when you get it.

–Topologist

4

Yeah, I knew how to do it. It was just figuring out the exact lemmas I needed to prove first, the induction statements I needed to work on, etc. All done in the end 'though. I think I’d have found it much easier had my brain not become rusty. :slight_smile:

Of course now I’m onto the ordering, which isn’t much fun to prove either (not hard, just rather tedious).

5

Horray for friday afternoon calculus.

::implodes::

egg

6

Just yesterday I, for some reason or another, had a bunch of numbers banging about in my noggin trying to get some exercise. I was just pondering multiplications & squares & whatnot, along the lines of 22=4, 33=9, 44=16… and something took me to think about the differences between those squares…
I pondered and pondered over how odd it was that for every square [X], the square of the number one before it [Y] was always Y2 + 1 less. How odd, I thought…

Then, wouldn’t ya know it, some long-forgotten paragraph from some long-forgotten algebra textbook fell from the sky & smashed right into my brain –

(x-1)(x-1) = x^2 - 2X + 1

I felt like calling up Mr. Knudson & saying ‘HEY! I GET IT NOW!’
sigh
I used to think I was good at math.

Okay, you may now return to more interesting mundane & pointless stuff.

7

<looks up at his math… ponders a moment… nods to himself… then looks back again… ponders yet some more… then decides it is far too early on a weekend morning to worry about math.>

8

Darn, Topologist beat me to the hint. Also remember to use the commutativity of addition (ya did prove that already, right? ;)).

9

ultrafilter: Actually I’ve already finished it. I got through it all before topologist posted. :slight_smile: Thanks for the suggestions though, both of you.

And yeah, I did prove all the requisite results about addition before hand, on the grounds that addition is easy. Where I was getting slightly hung up on was proving distributivity - proving it on one side was easy, proving it on the other was a bit harder. Naturally it was the harder one that you needed to prove commutativity.

Bah. I have an excuse. It’s not my strongest area of maths - I do analysis, geometry (the interesting type, not that silly euclidean stuff), calculus, algebra, etc best. Number theory is something of a weak area of mine. I’m trying to rectify this though, as I find logic and set theory to be really immensely neat.

Now if only writing out every little detail in the proofs didn’t make me want to headbutt a brick wall. :slight_smile:

10

Oops. Missed out an important element of that list there: topology. :slight_smile:

(Just for the record, I am most definitely not bad at maths. 5th best mark in a year of about 260 this year. The OP was mostly tongue in cheek).