Mathematics can ask you to remember things that have no obvious connection to common sense. Either because it’s arbitrary (the name of a function in respect to what it computes) or because you haven’t quite figured all the details out. In either cases, a few mnemonics are always useful!
The first series of mnemonics you’ll learn is to remember constants. Constants like and have special importance in modern mathematics—beyond plain creepy piphilia.
The notation for is due to William Jones who first used it in print in 1707. He chose probably because it had some mnemonic connection to perimeter, which make sense given the context. The notation was adopted and popularized by Euler, which in turn introduced the notation for the base of the natural logarithm. It appeared in print in 1736.
The discipline of remembering digits of either—and of in particular—has risen to Olympic levels. For every day use knowing the values to 6 or 16 digits is probably quite sufficient1. To remember to 20 digits, the following English poem does the trick:
141592 I wish I could determine Pi
65358 Eureka, cried the great inventor
9793 Christmas pudding, Christmas pie
23846 Is the problem’s very center
And, indeed, . For , we have the same kind of poem:
2718281 We require a mnemonic to remember
Trigonometric functions such as sine, cosine, and tangent are somewhat named arbitrarily. Maybe not tangent, because, more than the two others, it suggest something of a straight line, a slope. Fortunately, there’s a rather simple—if Japanese-sounding—mnemonic for those. Take a deep breath and say it out loud: SOHCAHTOA!
Indeed, sine is opposite over hypotenuse, cosine is adjacent over hypotenuse, and tangent is opposite over adjacent:
Let us look at a mitsudomoe crest:
You can imagine it spinning clockwise:
If you read the products clockwise, the answer is the next vector clockwise. Indeed, .
The same works if you use it counter-clockwise: the answer is the next vector in counter-clockwise order: .
You can recompute the identities using the definition of cross-product based on the matrix determinant:
But that’s a lot of work.
Mnemonics are always useful, if you manage to remember them. I mean it seriously. I think it’s easier to remember to 6 digits than to remember a 20-word poem. The clockwise/counter-clockwise analogy is a lot easier to remember if you’re a visual like I am.
The Hidari Mitsudomoe images are based on a work released under the creative commons. The derived works complies with the license. The hiragana ‘sohcahtoa’ calligraphy is mine, I retain full rights on this image.
1 6–7 decimal digits corresponds to the safe precision of IEEE 754 float, 16 corresponds to double.