Wikipedia:Reference desk/Archives/Mathematics/2008 June 1

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June 1

what's the diffrent with prism and Pyramid

--24.78.51.208 (talk) 00:43, 1 June 2008 (UTC)

Try looking at our prism (geometry) and pyramid (geometry) articles. 70.121.187.109 (talk) 01:30, 1 June 2008 (UTC)

Division in matrices

could you please explain why division in matrices is not defined Kasiraoj (talk) 07:14, 1 June 2008 (UTC)

(Fixed leading space causing prefomatted text section.) Well, if we tried to define matrix division as multiplication by an inverse, we'd run into a couple of problems. First, many matrices don't have inverses. Second, matrix multiplication does not commute, so it would be important to distinguish whether you did it as left multiplication or right multiplication by the inverse. I don't see any particular benefit to defining such an operation, but go for it if you want; you'll just have to explain it where you use it since it isn't a commonly used notation. --Prestidigitator (talk) 07:59, 1 June 2008 (UTC)

As long as you stick to square matrices, I don't think the lack of commutativity is a problem. A non-singular square matrix has a unique inverse - it's both a left-inverse and the right-inverse. If you have non-square matrices, it's obviously a problem - the left and right inverses have to be different sizes, so they can't be the same. The main problem is that not all matrices have inverses (that is, ones with determinant zero). If you restrict yourself to non-singular, square matrices, then division is perfectly well defined - that's why $GL_{n}(\mathbb {R} )$ is a group (the general linear group). --Tango (talk) 11:55, 1 June 2008 (UTC)

A group doesn't have "division" - it has multiplication and inverses, which invertible matrices surely do. But what would "A divided by B" be - the matrix X such that BX = A or that XB = A? Put differently, will it be $AB^{-1}$ or $B^{-1}A$? -- Meni Rosenfeld (talk) 11:58, 1 June 2008 (UTC)

Division is just the name we give to multiplication by an inverse. In a non-commutative situation, you have to specify "on the left" or "on the right", just as you do with multiplication. Using your logic, matrix multiplication isn't defined either, since "the product of A and B" could be either AB or BA - we don't consider that a problem, we just need to be more precise in our notation and terminology. There are notations that work for dividing on different sides, for example you could use / and \, so that A divided by B on the left would be B\A, A divided by B on the right would be A/B. It's generally easier to write B^-1A and AB^-1, though. --Tango (talk) 12:41, 1 June 2008 (UTC)

The status of division is different from that of multiplication. Given two matrices A and B, there are just two ways you can multiply them - AB and BA, but four ways to divide them - $AB^{-1},\ B^{-1}A,\ A^{-1}B,\ BA^{-1}$. Put differently, multiplication is a binary operation on $GL_{n}(\mathbb {R} )$ (that is, a function from $GL_{n}(\mathbb {R} )\times GL_{n}(\mathbb {R} )$ to $GL_{n}(\mathbb {R} )$). You need to specify the first operand and the second operand, and that's it. Of course the outcome can change if you switch the operands. "Division" is actually two distinct binary operations - left division and right division. For each of them you need, again, to specify the two operands. In short, there is an operation called "matrix multiplication", but no operation called "matrix division". -- Meni Rosenfeld (talk) 13:07, 1 June 2008 (UTC)

Ok, so there is no single operation called "matrix division", there is however such a thing is dividing matrices, it just requires a little more information than for multiplying them. --Tango (talk) 14:20, 1 June 2008 (UTC)

It could be noted that the two division operations for matrices are only formally different in that they provide an isomorphic structure. I mean, every result for left division has a symmetrical result for right division. In more general settings left and right division might act completely differently. GromXXVII (talk) 23:16, 1 June 2008 (UTC)

Sine Operations

I was wondering: if i was lost in the dessert without a calculator, and my life depended on finding the side opposite to angle b, using the trigonometric functions of sine cosine and tangent, how would i do that? Or, how do you do sine without a calculator? --Xtothe3rd (talk) 17:24, 1 June 2008 (UTC)

Try Trigonometric function#Computation. --Prestidigitator (talk) 18:00, 1 June 2008 (UTC)

Factorial

How do you differentiate the factorial function? Also how do you integrate it? I suspect that I am too much of an amateur in maths to understand the second answer because integrals.com couldn't do it but I'll ask anyway. Thanks 92.4.5.56 (talk) 18:29, 1 June 2008 (UTC)

The factorial function, strictly speaking, is defined only on the natural numbers, so you can't differentiate it. But you can differentiate the gamma function -- start there. --Trovatore (talk) 18:33, 1 June 2008 (UTC)