Suppose we have two rationals, $\frac{a}b$ and $\frac{c}d$. I daresay anyone with some form of mathematical education would disagree that our result would be $\frac{a}b\times \frac{c}d=\frac{ac}{bd}$. That is, we multiply our numerators to get the result numerator and we multiply the denominator to get the result denominator. However, mathematically speaking, why do we do this?  If we consider the canonical definition of multiplication (i.e. repeated addition) in the case where we have two integers, say $3$ and $4$, we would come up with $(3\times 4) = (4 + 4 + 4)$. If this is so (and hopefully it is) what does it mean mathematically to add $\frac{c}d$ to itself $\frac{a}b$ times?  After doing a bit of research, all I've managed to come up with is this image.   Unfortunately, I don't see how this has anything to do with multiplication and seems to me like little more than a tool for teaching grade schoolers.  This also led me to think about how we multiply decimal numbers. For example, $(0.2) \times (0.4) = 0.08$ (i.e. $2\times 4$ with the decimal moved over a number of places equal to the number significant digits past the decimal).  My intuition tells me that the answer to one of these will provide the answer to the other.  Lastly, excuse me if this question seems silly, but I've pondered it quite a bit and can't come up with anything mathematically rigerous.    EDIT: I changed all literal values to be arbitrary values (as they should have been originally).  EDIT 2: Please note that I am PERFECTLY CAPABLE of multiplying fractions together in any way shape or form. This is NOT a post about how to multiply fractions. This post is about fundamentally understanding what it means to multiply a pair of fractions together.
Why multiplying fractions is equal to multiply the tops, multiply the bottoms? $$\frac{a}{b}\times \frac{c}{d}=\frac{a\times c}{b \times d},$$ And why $$\frac{a}{b}\times \frac{c}{c}=\frac{a}{b},$$ Also why $$\frac{a}{b}+\frac{c}{b}=\frac{a+c}{b}.$$ I understand it, but I want a mathematical approach as a math student proves it. Also I want to know the mathematics topic of this question (number theory, logic, etc). A full answer is not necessary. Just a reference.
Why $\sqrt{-1 \cdot {-1}} \neq \sqrt{-1}^2$? How do I check which shell I am using? I read that terminal is nothing but shell, and Unix provides different flavors of shells:   Bourne shell (sh) C shell (csh) TC shell (tcsh) Korn shell (ksh) Bourne Again shell (bash)   Questions:   When I open a terminal window, which shell is opened by default? How do I check how many shells are installed? How do I change the shell used from my account? Sheafs and closed immersion In my object record type i have created formula field. Now i want my formula field to work as below.  If Basement__c = NA and Property__r.Basement__c = NA OR  Basement__c = Crawl and Property__r.Basement__c = Crawl OR  Basement__c = Slab and Property__r.Basement__c = Slab Then it will return Type__c(formula field) = "MAtch"  And (Another rule) Basement__c DOES NOT contain any of the following "N/A,Crawl,Slab,None and Property__r.Basement__c Does not contain any of this then formula field will return "HAs basement"  How can i achieve this? Please help. Mac Mini (Mid-2011), 16GB SSD, 16GB RAM, El Capitan (v10.11.6)  The Mac ran fine until I upgraded to Sierra. The latest Mac OS conflicted with most of the specialized apps I use every day, So I immediately returned to El Capitan.  Activity monitor shows that two "Installer/Installer.app" items are always running and are they major users of all of my available memory. Consequently, all my others apps become unresponsive.  I run Disk Utility, and the Installer.app show up there as .dmg files; but I cannot erase, delete, or unmount them. 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Sturm-Liouville Questions In thinking about Sturm-Liouville theory a bit I see I have no actual idea what is going on.  The first issue I have is that my book began with the statement that given  $$L[y] = a(x)y'' + b(x)y' + c(x)y = f(x)$$  the problem $L[y] \ = \ f$ can be re-cast in the form $L[y] \ = \ \lambda y$.  Now it could be a typo on their part but I see no justification for the way you can just do that!  More importantly though is the motivation for Sturm-Liouville theory in the first place. The story as I know it is as follows:  Given a linear second order ode   $$F(x,y,y',y'') = L[y] = a(x)y'' + b(x)y' + c(x)y = f(x)$$  it is an exact equation if it is derivable from a differential equation of one order lower, i.e.  $$F(x,y,y',y'') = \frac{d}{dx}g(x,y,y').$$  The equation is exact iff  $$a''(x) - b'(x) + c(x) = 0. $$  If $F$ is not exact it can be made exact on multiplication by a suitable integrating factor $\alpha(x)$.  This equation is exact iff  $$(\alpha(x)a(x))'' - (\alpha(x)b(x))' + \alpha(x)c(x) = 0 $$  If you expand this out you get the Adjoint operator  $$L^*[\alpha(x)] \ = \ (\alpha(x)a(x))'' \ - \ (\alpha(x)b(x))' \ + \ \alpha(x)c(x) \ = 0 $$  If you expand $L^*$ you see that we can satisfy $L \ = \ L^*$ if $a'(x) \ = \ b(x)$ &amp; $a''(x) \ = \ b'(x)$ which then turns $L[y]$ into something of the form  $$L[y] \ = \ \frac{d}{dx}[a(x)y'] \ + \ c(x)y \ = \ f(x).$$  Thus we seek an integratiing factor $\alpha(x)$ so that we can satisfy this &amp; the condition this will hold is that $\alpha(x) \ = \ \frac{1}{a(x)}e^{\int\frac{b(x)}{a(x)}dx}$  Then we're dealing with:  $$\frac{d}{dx}[\alpha(x)a(x)y'] \ + \ \alpha(x)c(x)y \ = \ \alpha(x)f(x)$$  But again, by what my book said they magically re-cast this problem as  $$\frac{d}{dx}[\alpha(x)a(x)y'] \ + \ \alpha(x)c(x)y \ = \ \lambda \alpha(x) y(x)$$  Then calling  $$\frac{d}{dx}[\alpha(x)a(x)y'] \ + \ ( \alpha(x)c(x)y \ - \ \lambda \alpha(x) )y(x) \ = \ 0$$  a Sturm-Liouville problem.  My question is, how can I make sense of everything I wrote above? How can I clean it up &amp; interpret it, like at one stage I thought we were turning our 2nd order ode into something so that it reduces to the derivative of a first order ode so we can easily find first integrals then the next moment we're pulling out eigenvalues &amp; finding full solutions - what's going on? I want to be able to look at $a(x)y'' \ + \ b(x)y' \ + \ c(x)y \ = \ f(x)$ &amp; know how &amp; why we're turning this into a Sturm-Liouville problem in a way that makes sense of exactness &amp; integrating factors, thanks for reading! How to get div height to auto-adjust to background size? How do I get a div to automatically adjust to the size of the background I set for it without setting a specific height (or min-height) for it? 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