What's the most used symbol for "defined to be equal to", at least in your experience (and I'm sure there are a lot of experienced people here)? Also, which one do you think is the "the most right" of them, in the sense of making the most amount of sense (no pun intended)? The ones I frequently see in literature, papers and articles on the Web are '$\equiv$', '$:=$' and '$=_{def}$'. The first one I come across a lot, though for me it's still "reserved" for modular arithmetic. The second one seems like it's come straight out of some programming language and the last one I frequently see in philosophy papers on logic and the like. What are your thoughts on this?
[Math] What’s “the most right” symbol to use for “defined to be equal to”
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Related Solutions
There are many formulas in which an operation is repeated over and over. Addition and multiplication are so common that we have corresponding summation and product notations. These are convenient ways to give compact expressions for various formulae.
For example: Instead of writing $1^2+2^2+3^2+\cdots+n^2$ we can write $\sum\limits_{i=1}^n i^2$
The first place most students see summation notation used in any serious manner is in calculus when Riemann sums are defined. A Riemann sum approximates areas under curves using rectangles.
For example: If we want to approximate the area under $y=x^2$ where $0 \leq x \leq 3$ using 6 rectangles. Then we break the interval $[0,3]$ into 6 pieces: $[0,1/2]$, $[1/2,1]$, $\dots$, $[5/2,3]$. If we use $y=0^2$ as the height of the first rectangle, $y=(1/2)^2$ as the height of the second, and so forth, then noting that the width of each of these rectangles is $1/2$ we have that the area under the parabola is approximately $\frac{1}{2}(0)^2+\frac{1}{2}(1/2)^2+\frac{1}{2}(1)^2+\frac{1}{2}(3/2)^2+\frac{1}{2}(2)^2+\frac{1}{2}(5/2)^2$. Summation notation lets us write this much more compactly as $\sum\limits_{i=0}^5 \frac{1}{2}\cdot \left(\frac{i}{2}\right)^2$. Now what if we wished to approximate with $n$ rectangles? We have $\sum\limits_{i=0}^{n-1} \frac{3}{n}\cdot \left(\frac{3i}{n}\right)^2$ (nice and compact).
Similarly mathematicians use product notation to express repeated multiplications.
For example: $\prod\limits_{i=1}^4 i^2=1^2\cdot2^2\cdot3^2\cdot4^2$.
Or you may be familiar with the factorial function: $n! = \prod\limits_{i=1}^n i = 1\cdot 2\cdot 3 \cdots n$
By the way, the analogy with for loops is a good one. However, for loops allow you to repeat a plethora of operations. Summation and product notations are far more specialized (summations and products can both be computed using for loops). Also, not all for loops increment the index by 1 each time. Most programming languages allow for more complicated increments. In the same way summations and products are sometimes done over sets (other then ${1,2,\dots,n}$).
For example: If $S = \{1,2,4,8\}$, then $\sum\limits_{x \in S} \frac{1}{x} = \frac{1}{1} + \frac{1}{2} + \frac{1}{4} + \frac{1}{8}$.
In calculus (often second semester calculus) one learns about series where you are essentially summing over infinite sets [summing over infinite sets gets a little tricky -- there are issues of convergence and problems when shuffling the order of the summation around].
Best Answer
$$:=$$ is the commonest symbol to denote "is equal by definition."
Note that $$\equiv$$ is used to denote an algebraic identity: this means that the equation is true for all permitted values of its variables. Rarely, however, it may denote a definition, so it's best to use this symbol only for congruences or identities.
In short: $$:=$$ is the most widespread (presumably as it's the easiest to typeset) "by definition" symbol .
Other symbols used to denote a definition include $$\stackrel{\triangle}= \quad , \stackrel{\text{def}}= \quad, \stackrel{\cdot}= \quad .$$
Whilst there's no amibguity in the latter three symbols, you try typing
\stackrel{\triangle}=every single time you make a definition, as opposed to the much-shorter:=. You'll then see why the latter of these two is most widespread in this context.