 12,925
 501
People usually say that pictures tell more than 1000 words. Is that still true in mathematics...? I think so. Let me first say what i mean by 'visual proof'. Let's say we have an identity. To prove it's true one may write from a line to more than one page. But what if one was able to write only one formula instead of the whole proof and let the eyes and the mind "figure it out".
Here's what i mean. The sine and cosine addition formulas are a mess to prove in elementary trigonometry. However, i'd say that
[tex] \displaystyle{e^{i\left(x+y\right)} =e^{ix} e^{iy}} [/tex]
is a "visual proof". Basically an agile mind and some healthy eyes would "get it" without feeling the need to grab the pencil & do the calculations involved.
Going further, using the same task (proving the addition formulas for sine & cosine), one could literally come up with a picture, like this one attached below.
So what do you think of my idea? Is it dumb? If not, could you come up with some of your own results...?
Daniel.
Here's what i mean. The sine and cosine addition formulas are a mess to prove in elementary trigonometry. However, i'd say that
[tex] \displaystyle{e^{i\left(x+y\right)} =e^{ix} e^{iy}} [/tex]
is a "visual proof". Basically an agile mind and some healthy eyes would "get it" without feeling the need to grab the pencil & do the calculations involved.
Going further, using the same task (proving the addition formulas for sine & cosine), one could literally come up with a picture, like this one attached below.
So what do you think of my idea? Is it dumb? If not, could you come up with some of your own results...?
Daniel.
Attachments

7.1 KB Views: 290