In most programming languages, container indices start at 0. This is not random or hardware-related; for example Dijkstra's article explains why zero-based indices make sense.
What are the reasons why Mathematica lists start with an index of 1?
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2 Answers
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I think Leonid's answer deserves to be expanded upon. Most other languages are not symbolic, and thus the "variable name" is not something one needs to keep track of --- ultimately the interpreted or compiled code is keeping track of pointers or something. In contrast, in Mathematica the Head of an expression is arbitrary. This is somewhat along the lines of LISP where the first symbol in a list is the procedure which should be applied to the rest of the list. So, in LISP one might write (+ 3 2) which evaluates to 5. Written this way, it's clear that the symbol + occupies the "natural" 0th position, 3 the first, and 2 the second. In Mathematica one would write the equivalent expression as Plus[3,2], so that the 3 is in the first position -- the same position that it would be in in LISP. The fact that some Heads (namely, List) work like vectors for many intents and purposes would break the uniformity of the mapping between a LISP-like language and Mathematica, and worse---break the internal uniformity of Mathematica indexing, if you demand that you should be able to extract the Head of an expression.
This is related to the fact that in some sense, it's the most symmetric thing to do in a symbolic language, if that language is going to support negative indexing and arbitrary Heads. For example if you have
f = F[1,2,3,4,5]
then f[[-1]] evaluates to 5. If you impose "periodic boundary conditions" you might imagine writing the expression f schematically as
F
5 1
4 2
3
so that moving one spot clockwise gives you the first element, one spot counterclockwise gives you the last element, and moving 0 spots gives you the Head.
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8$\begingroup$ Love the hexagonal diagram explaining positive and negative indexing. $\endgroup$– MichaelJun 18, 2015 at 7:03
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7$\begingroup$ This answer has been getting some upvotes lately, and I thought it might be nice to additionally point out that in many shell / scripting languages (like bash), \$0 is a variable which contains the command itself while the arguments start with \$1. $\endgroup$– evanbSep 15, 2015 at 6:03
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$\begingroup$ @evan, that convention carries over to
Slot[];#0refers to the function itself (useful for recursive implementations). $\endgroup$ Dec 22, 2016 at 3:46 -
$\begingroup$ @J.M., true! I think it's true for argc argv, also? I always have to look that up... $\endgroup$– evanbDec 22, 2016 at 7:44
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Some years ago, a friend of mine was in the supermarket with his son, small kid, who asked him to buy some candy. After some resistance, my friend agreed, but told him that should be just one. In the cashier, his son had two candy, and my friend said:
- What is this? haven't we agreed One?
And the answer (very smart) was:
- Yes! Here it is. Zero and One... (His son should be a C programmer)
Well... I believe starting to count lists of discrete items by 1 is much more natural. I don't understand why other languages do it starting by Zero, if it's not a continuum interval.
Mathematica, Matlab and R are the ones I know that follows this convention.
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4$\begingroup$ At least, for coefficients of polynomials and Fourier series, it is certainly sensible to index from zero. But I am definitely part of the group that is fine with Mathematica indexing from 1, leaving 0 for the head and negative indices counting from the end. $\endgroup$ Jun 18, 2015 at 2:44
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8$\begingroup$ His son counted from zero like a C programmer, but he didn't exclude the end number like one. :-) $\endgroup$– MichaelJun 18, 2015 at 4:05
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2$\begingroup$ In lower level languages like C indexing arrays from 0 is more natural because there the indices work as offsets, i.e. if you have a pointer
p, thenp[0]is the same as*(p+0)and the same as*pand the same as0[p]. $\endgroup$– RuslanJun 18, 2015 at 6:31 -
4$\begingroup$ @Ruslan: the rationale for 0-indexing is not limited to lower level pointer arithmetic. The mentioned in the original question Dijkstra's article outlines some of it, namely the convenience of using semi-open intervals [begin, end), implying not including the upper bound in the loop, thus implying starting with 0. Also compare the complexity of arithmetic between base 0 and 1 for using single index for indexing into a 2D or 3D arrays. The case for 0-base is very strong; I am content though with evanb's answer. $\endgroup$– MichaelJun 18, 2015 at 7:01
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2$\begingroup$ To add to Michael's comment, note that our entire decimal number system is based on 0-indexing. When you start counting, every digit except for the very last one is 0. This is especially true in computing, where all digits are "always present" due to fixed storage width. 1-indexing, and its strong manifestation in language ("first", "second", "third", "fourth") is a remnant from the roman times, where there was no such thing as a zero, and consequently mathematics was a whole lot harder. Note how the year 2000 was in the second millenium, but 2001 in the third. $\endgroup$– mic_eJun 18, 2015 at 15:08
lst = {1,2,3}; lst[[0]], which in this case isList. $\endgroup$1<=i<=Nseems to be a quite natural choice for ranges to me... $\endgroup$