it is all addition

[miriam_e]

Bear with me on this. At first it all sounds simple and obvious, but it has world-shaking importance that yanked me out of bed and kept me up at 4am this morning.

There are 4 basic mathematical operations, right? Adding, subtracting, multiplying, and division, and you could perhaps say powers (square, cube, etc) and roots are pretty basic too, which makes 6, yeah?

Well, yes, but really, it is all addition.

2 + 3 = 5   that's easy to see. Just get 2 sticks and add another 3 sticks and you're holding 5 sticks.

So, what about multiplication?
2 × 3 = 6   It's still addition. This time you are starting with nothing and adding 2 sticks 3 times, so it is really
2 + 2 + 2 = 6   The multiplication just tells you how many times to do the addition.

How about subtraction?
It is really just addition in reverse. Imagine watching a video of someone adding 3 sticks to 2 sticks and ending up with 5 sticks, but now running the video in reverse. You start with 5 sticks, 3 are taken away leaving 2.
5 - 3 = 2
It is harder to imagine things being taken away, especially if more are being taken away than are there in the first place, resulting in a negative number, so instead of using simple objects, another way of visualising addition is like an ant crawling along a ruler. When she (pretty-much all ants are female) walks forward along the ruler, she's adding a positive number. When she moves backwards, she's adding a negative number... or subtracting.

How about division?
6 ÷ 3 = 2   It asks the question, what number can you add three times to get 6? Imagining the ant walking along the ruler is easier here too, especially if fractional numbers are involved.

Raising to a power or taking a root is similar to multiplication and division -- it is doing multiplication or division a certain number of times. For example, 32 = 9 means adding 3 to itself 3 times (3+3+3), or 3 × 3. Taking it up to third power, 33 = 27 means adding 3 to itself 3 times, then doing that 3 times (3+3+3 + 3+3+3 + 3+3+3), or 3 × 3 × 3.

Using the normal ways of visualising addition (holding sticks or watching an ant on a ruler) don't work very well for taking roots, for example the square root of 9 asks what number can be added to itself its own number of times to make 9. Higher roots sound even weirder. But visualisation can come to the rescue again, as you'll see below.

How you visualise adding numbers can be very powerful.

Multiplication and division can be seen as adding or subtracting using two rulers that are marked with logarithmic scales instead of regular markings. This is how slide rules work. This is more intuitive than you might think. If you walk forward one step, and your friend walks 4 steps, the difference in your position from your starting point seems pretty significant, but if you take 100 steps and your friend takes 104 steps then suddenly the difference in position doesn't seem all that important anymore because we actually think in a kind of logarithmic way.

Slide rules usually have other markings that let you perform squares and cubes and square roots and cube roots too. It is still addition, but how you see it can let you do amazing things.

Why is all this important? Because it seems a young guy named Shinichi Mochizuki in Kyoto has worked out another way of seeing addition which may be very powerful and could let people solve problems that were previously extremely difficult or even impossible. It may also solve the problem of how Fermat's Last Theorem seemed fairly simple to Fermat, yet was only recently solved using advanced maths and with the help of computers in a proof that runs to hundreds of pages long. Fermat may well have seen addition the way Shinichi Mochizuki is able to.

A recent New Scientist news item briefly, and rather tantalisingly, if unsatisfyingly, discussed how exciting this is. The short news item is, surprisingly, not available on their site, so I copied it from the paper version I'd bought and put it here until it is available there. Alternatively you can read about it on phys.org, from where you can also download the pdf files of the proofs, if your mind is way, way more mathematically oriented than mine, or from Shinichi Mochizuki's site. Also a short news item in the New York Times gives an idea of why this may be important yet confounding... even to advanced mathematicians.

But if you think all this is too esoteric for us ordinary folk, don't forget that it was not too many centuries ago that calculus was beyond most of the world's brightest mathematical minds. In my parents' youth it was taught only at university to the mathematically gifted, while today it is routinely taught to very young schoolkids.

So... a new way to see addition that could eventually give us all new insights into our world? Hopefully we will find out soon.

[Darn... no way that I'll get back to sleep now... the sun is up.]