Origin of mathematical entities

[rhia]

Hi,
There are few mathematical entities which we encounter quite often ..like pi,e,hyperbolic and trigonometrical functions, etc.I understand the significance of pi(=circumference/diameter for any circle).

What about e,hyperbolic functions like cosh,sinh etc?
Then there are prime nos,factorial,logarithm etc ..I wonder why all these were created in the first place.
Any more interesting entities like these?Please tell us their origin and significance.

n00b

 

[JasonRox]

You can learn all of it doing some math or reading books by Isaac Asimov :) .

I don't know very much but e seems to have came about because it is the only number at the base of an exponential function to have a slope of one, when x=0. There's more about it, but this is what I know for now.

Sin and cos are just like pi. They are ratios of angles, or something along those lines. What makes it not so obvious, is that there are so many ratios, where as pi only has one.

Logarithm function is simply the inverse function of a exponential function.

 

[MiGUi]

Actually, the definition of e may be:

$$\int_{0}^{e} \frac{dx}{x} = 1$$

or even this one:

$$e = \lim_{n \rightarrow \infty} \left( 1 + \frac{1}{n} \right)^{n}$$

 

[Gokul43201]

rhia, you talk of two different kinds of entities : (i) numbers like e, pi primes, etc. and (ii) functions like sinh, log, factorial, etc.

You know the significance of pi.

e is the only number (among all possible values of 'a') that satisfies the property that the slope of a^x is equal to its height anywhere (in addition to various other definitions, like those suggested by MiGUi).

Most functions evolved simply as short-hand for operations that were found to be commonly used. Repeated addition was represented as multiplication, to save effort. Likewise, repeated multiplication was represented by an exponent. And just as subtraction and division were made up to represent inverse operations with repect to addition and multiplication, the logarithm performs the inverse task with respect to exponentiation. Trigonometry arose from the properties of right triangles, and complex algebra revealed the simple connection between the trigonometric fonctions and exponential functions. The hyperbolic functions are simply shorthand for another set of exponential functions, only this time, with real exponents, instead of imaginary ones.
The factorial notation arose to make the counting of permutations easier to represent.

 

[robert Ihnot]

I was told when I took junior high printing, that the factorial sign originally looked like an inverted long division sign: $$\underline{\mid 5 }$$ for 5!, and there are still old math books that do it that way. BUT the PRINTERS decided that was too dificult to reproduce, so that they invented another use for the exclamation point.

The same thing may have happened with the superscript sign for squaring:
$$x^2$$, I think. Because, when the computers came out, it was too difficult for most to reproduce, and so contributors seemed to have invented X^2.

 

[Integral]

The same thing may have happened with the superscript sign for squaring:, I think. Because, when the computers came out, it was too difficult for most to reproduce, and so contributors seemed to have invented X^2.

More likely that this stated with higher level programing languages, (Fortran, Basic) where then needed a symbol to represent the operation. The up arrow is the usual symbol for exponential operation. This was in use in the keypunch days before anyone was using a monitor for programming or viewing output. (Keypunch or teletype for input, line printer or teletype for output. The CRT monitor did not see wide use until microcomputers came into existence in the '80s

 

[Janitor]

More likely that this stated with higher level programing languages, (Fortran, Basic) where then needed a symbol to represent the operation...

Ah, the memories! I recall, circa 1980, seeing FORTRAN programs being punched up on Hollerith cards. They warned you to pencil the numbers 1, 2, 3, ... on your stack of cards in case you dropped them and scattered them into jibberish.

 

[robert Ihnot]

More likely that this stated with higher level programing languages, (Fortran, Basic) where then needed a symbol to represent the operation. The up arrow is the usual symbol for exponential operation. This was in use in the keypunch days before anyone was using a monitor for programming or viewing output. (Keypunch or teletype for input, line printer or teletype for output. The CRT monitor did not see wide use until microcomputers came into existence in the '80s

No one had previously ever said anything I remember over the origin of the use of the caret in X^2, which somehow just began showing up on the web until I got used to it. But when I got my computer, it was a Macintosh and it was much superior to the IBM in creating math-type. I don't think at first that IBM could superscript at all, so I guess IBM users were the first to widely use this X^2 originating with programmers.

 

[master_coda]

More likely that this stated with higher level programing languages, (Fortran, Basic) where then needed a symbol to represent the operation. The up arrow is the usual symbol for exponential operation.

The use of '^' for exponents came from Algol-60. The language did use an up-arrow for exponentiation, but the up-arrow in ASCII was later changed to a caret, so it became the exponent symbol instead. This usage was copied by other languages like BASIC and tools like TeX and ended becoming the most common way of representing exponents on a computer.

 

[Gokul43201]

Wasn't Algol an early version of Pascal. Does it still exist ?

 

[master_coda]

Wasn't Algol an early version of Pascal. Does it still exist ?

The ALGOL languages were an early version of Pascal in the the same sense that C++ is an early version of Java. The languages are very, very different, but at the same time it's clear that the later language was created by taking the earlier language and modifying it until the creators had something they liked.

C was also influenced by ALGOL; the basic layout of a C program is similar to the basic layout of an ALGOL-68 program (which is also similar to the layout of a Pascal program). Any language that bases its programs on things like if-blocks, for-blocks, with-blocks, function-blocks, etc. is actually copying a style that came from ALGOL.

Well, you can probably still find a compiler for it somewhere. But ALGOL itself has mostly been abandoned since the 60s. Even then, it was never really used for commercial work, although it was extremely popular for writing pseduocode.

 

[fourier jr]

the graph of the cosh function is the same shape of a hanging chain. one of the bernoullis proved that. it's also the cosine function in hyperbolic geometry, except I think that the $$sin^2 x + cos^2 y = 1$$ identity is instead $$sinh^2 x - cosh^2 y = 1$$, unless I got the sinh & cosh backwards, which is possible.

 

[master_coda]

the graph of the cosh function is the same shape of a hanging chain. one of the bernoullis proved that. it's also the cosine function in hyperbolic geometry, except I think that the $$sin^2 x + cos^2 y = 1$$ identity is instead $$sinh^2 x - cosh^2 y = 1$$, unless I got the sinh & cosh backwards, which is possible.

Your cosh identity is backwards; the identity is $\cosh^2 x - \sinh^2 x=1$.

 

[nolachrymose]

Actually, the "^" symbol has another use - it is the bitwise XOR (exclusive-or) operator in languages such as C, C++, and JavaScript.

 

[master_coda]

Actually, the "^" symbol has another use - it is the bitwise XOR (exclusive-or) operator in languages such as C, C++, and JavaScript.

Of course. '**' is also sometimes uses for exponentiation instead of '^'. You can't really expect the same people who can't decide if move should be spelled 'move', 'mov' or 'mv' to decide on a consistent way of using '^'.

 

[nolachrymose]

Hah, so I remember Assembler! :P