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<html><!-- Mirrored from c-faq.com/misc/btod.html by HTTrack Website Copier/3.x [XR&CO'2008], Sat, 14 Mar 2009 08:02:58 GMT --><head><title>``Converting Binary to Decimal''</title></head><body><p>[This is a message I composed in response to someonewho was asking how to write a function to convert binary numbers to decimal.I have paraphrased the original correspondent's queries.]<p>From: scs@eskimo.com (Steve Summit)<br>Subject: Re: please help<br>Date: Sat, 29 Apr 2000 12:36:07 -0700 (PDT)<br>Message-Id: <200004291936.MAA18560@mail.eskimo.com><p>You wrote:<br><TT>></TT> I am trying to write a function that takes a binary number and returns<br><TT>></TT> its the decimal value. Is there a library function that does this?<p>Sort of; in fact depending on how you think about the question,there are several functions. But those functions don't workquite like you think they do, because you're thinking about thequestion in a different way, one which it turns out doesn't makeas much sense.<p><TT>></TT> The sort of function I'm thinking of is:<br><TT>></TT><pre>> #include <math.h>>> int binary_to_decimal(int number)> {> int decimal = 0;> int i, digit;>> for(i = 0; i < length(number); i++)> {> digit = substring(number, i);> /* where, say, substring(11011, 2) returns 0 */> /* but I'm not sure about this part */>> if(digit == 1)> decimal += pow(2,i);> }>> return decimal;> }</pre><p>You've got the right basic algorithm there, and if your ``number''variable (that is, the input of your function) was a <em>string</em>,you'd have a perfectly useful function. Notice, too, that if youchanged the lines<p><pre> if(digit == 1) decimal += pow(2,i);</pre>to<pre> decimal += digit * pow(2, i);</pre>you could make the program work for any base, just by changingthe value 2 to something else.<p><TT>></TT> but I don't know how to get <TT>substring(number, i)</TT> to work.<p>That's because you're thinking of a ``number'' as a string ofdigits. But integers are <em>not</em> represented internally as stringsof digits.<p>Now, <em>if</em> your function did accept a string of digits (that is,if its input argument were of type string, not int), it would beeasy enough to get your hands on each digit, because of courseit's always easy to get your hand on the individual charactersthat make up a string.<p>(There is no standard ``substring'' function in C, although it'seasy enough to write one, although as its name implies, it'salmost invariably designed to solve the general problem ofextracting substrings from a string, not necessarily singledigits. And, once again, what it would be extracting from isa string variable, not an integer variable.)<p>How were you planning on passing a ``binary'' number to yourfunction? Were you thinking of calling something like this?<p><pre> binary_to_decimal(11011);</pre>In that call, you haven't passed the binary numberone-one-oh-one-one; you've passed the <em>decimal</em> number eleventhousand and eleven. I suppose you could write some code thataccepted decimal numbers with the digits all 0 or 1, andpretended it was a binary number, and ``converted'' it to decimal,but this would be a confusing and not very useful function.(To get at the digits, you'd end up dividing by 10, which wouldbe a strange thing to do to extract bits which you were thinkingof as being in base two.)<p>You were trying to write a function that accepted an int, andthat returned an int. Presumably you were intending one of thoseints to be in ``base 2'' and one of them to be in ``base 10''. Butit really doesn't make sense to talk about what ``base'' an integeris in; an integer is just a <em>number</em>. The ``base'' only matterswhen we write numerals out on paper.<p>Here is an example. How many x's are there on this line?<pre> xxxxxxxxxxxxxxxxx</pre>Is your answer in base two or base ten? The question doesn'tmake much sense. If you said ``seventeen'', your answer is not inbase two or base ten; your answer is in English. Seventeen isthe <em>number</em> of x's on the line.<p>We can represent numbers in different bases. If we representthe number seventeen in base ten, we get the numeral 17. If werepresent it in base two, we get the numeral 10001. But thenumber of x's on the line didn't change when we switched from17 base ten to 10001 base two.<p>I believe that the right way to think of the numbers stored inint variables as just that: numbers. It doesn't really matter,or make sense to ask, what base they're in. (In truth, on amodern computer, integers are virtually always stored in binary,but most of the time, that fact really doesn't concern us.)<p>Here's one way to convince yourself that ints in C are justnumbers. Suppose you have an int variable<pre> int i;</pre>and you print it out using <TT>printf</TT>:<p><pre> printf("%d\n", i);</pre><p>Now, as you probably know, <TT>printf</TT> can also print things out inbase 8 and base 16. So we could also write<p><pre> printf("%o\n", i);</pre>or<pre> printf("%x\n", i);</pre><p>Do we have to match the <TT>printf</TT> format we use (<TT>%d</TT>, <TT>%o</TT>, or <TT>%x</TT>) tothe base of the number we're printing? No, because as I wassaying, the int variable <TT>i</TT> is just a number; it doesn't have anyinherent base. (Or, if it does have an inherent base, it's basetwo, meaning that all three <TT>printf</TT> conversions involve a conversionfrom base 2 to some other base.)<p>In fact, one answer to the question ``how do I convert a binarynumber to decimal?'' in C is, ``print it out using <TT>printf %d</TT>''.The answer only works if you think of ints as being base twointernally, and it probably isn't the answer you expected if youwere trying to convert binary to decimal explicitly, but it is avery real answer. (If the binary number you had was a string of0's and 1's, then what you really wanted to do was just toconvert that string to an int, perhaps so that you could printit back out using %d.)<p>If we believe (as I do) that int variables are ``just numbers'',then the only times it makes sense to talk about what basethey're in are when we print them out for people in the realworld to look at on paper or on the screen, or when we read themin from people in the real world who are poking at keyboardscontaining keys with the digits 0 through 9 on them. In thosecases, just about all of the conversions are taken care for usautomatically, without our hardly being aware of it:<p><OL><li>When we write a C program containing a line like<pre> int i = 1234;</pre>the C compiler automatically converts those decimaldigits <TT>1 2 3 4</TT> to an integer. If we want to, we canenter integer constants in base 8 or base 16:<pre> int j = 0123; int k = 0x123;</pre>and the compiler does octal or hexadecimal conversion,as appropriate.<p><li>When we print an integer, using <TT>printf</TT> or the like, it'sprinted in base 10 if we use <TT>%d</TT>, or in base 8 or 16 ifwe use <TT>%o</TT> or <TT>%x</TT>.<p><li>When we request numeric input from the user, and wechoose to use <TT>scanf</TT> or the like, we again select the baseusing the format character, <TT>%d</TT>, <TT>%o</TT>, or <TT>%x</TT>.<p><li>When we have a string that we want to convert to aninteger, we can call the <TT>atoi</TT> function:<pre> char *str = "123"; int i = atoi(str);</pre><TT>atoi</TT> always does decimal conversion.<p><li>Finally, if we have a string in an arbitrary base, wecan convert it to an integer using the <TT>strtol</TT> function:<pre> char *str2 = "1234"; i = strtol(str2, NULL, 7);</pre>This converts the string <TT>"1234"</TT>, interpreted as a numeralin base 7, to an integer, and stores it in <TT>i</TT>. (If wethen printed <TT>i</TT> back out using <TT>%d</TT>, we'd see the decimalnumeral 466, which is 1234 base 7.)</OL><p>So <TT>strtol</TT> is the other standard C library function that's oftenuseful in realistic base conversion problems, and it's one of theonly standard C library functions that lets you specify anarbitrary base. (You can also specify the base as 0, in whichcase <TT>strtol</TT> will use the same rules the C compiler uses, namelythat a leading 0 indicates base 8 and a leading <TT>0x</TT> indicates base16. It turns out that <TT>scanf</TT> can do this, too, using the <TT>%i</TT>format.)<p>You'll notice that every standard function (or other languagefeature) that seems to have anything to do with base conversionis either converting from an integer to a string or printedrepresentation, or is converting from a string or printedrepresentation to an integer. You won't find a ``base conversion''function that ``converts'' from integer to integer, because as I'veexplained, that ``conversion'' doesn't make sense. Integers arenot stored internally as strings of digits. If, however, youwant to think about the ``base'' of a numeral, that's a conceptwhich only makes sense (in fact, which is defined) in terms ofstrings of digits.<p>If you'd like to truly understand base conversion, in thecontexts where it really comes up when writing modern computerprograms, here are a few functions for you to write.<p><OL><li>Write a function which accepts a string of digits andconverts it to an integer, interpreting the string ofdigits as a numeral in base 10. This function willactually be a lot like the ``conversion'' function you weretrying to write, except that (a) it accepts a string, notan int; (b) it will be easy to access the digits (thatis, to perform the ``substring'' operation), because astring in C is just an array of characters, and arraysubscripting is easy; and (c) you'll be converting frombase 10, not base 2. (Also, you can get away withoutcalling <TT>pow()</TT> at all, if you arrange things right. Ifyou convert the digits in left-to-right order, all youhave to do is keep a running sum, and at each digit,multiply the previous sum by 10 before adding in the newdigit.)<p>If you write this function, you'll essentially havereimplemented the standard library <TT>atoi()</TT> function.<p>One other problem you may encounter is dealing withdigits as characters. Your input is a string like<TT>"1234"</TT>, so the digits are characters, like <TT>'1'</TT>.But the value of the character <TT>'1'</TT> in ASCII is <em>not</em> 1.But it turns out that converting digit characters totheir appropriate digit values is easy: just subtract thecharacter set value of the character <TT>'0'</TT>, whatever that is.<TT>'0' - '0'</TT> is obviously 0, no matter what value <TT>'0'</TT> has.Similarly, <TT>'1' - '0'</TT> is 1. So if you have a charactervariable <TT>c</TT>, containing one of the digit characters <TT>'0'</TT>through <TT>'9'</TT>, then <TT>c - '0'</TT> is the decimal value of thatdigit.[The point here is that you do <em>not</em> need to know whatthe character set value of the character <TT>'0'</TT> is;all you need to do is subtract the character constant <TT>'0'</TT>.]<p><li>Write a function which accepts a string of digits and aninteger value <TT>b</TT>, and converts the string to an integer,interpreting the string of digits as a numeral in base <TT>b</TT>.Basically, all you'll have to do is take your functionfrom #1 and change ``10'' to ``<TT>b</TT>''. When you're done, you'llhave reimplemented part of the standard library <TT>strtol()</TT>function.[And you can use it, if you like, to perform ``binary to decimal conversion'',by calling it with a base of 2.As we've discussed, this is actually ``binary to number conversion'',not ``binary to decimal''.]<p><li>Write a function which accepts an integer and creates astring of digits forming the base 10 representation ofthe integer. (Your compiler may already have such afunction, perhaps called ``<TT>itoa</TT>'', although it's not aStandard function.) As you write this function, you'llessentially be reimplementing the task which <TT>printf</TT>performs when it prints something using <TT>%d</TT>.<p>It's actually easy enough to compute the digits of anumber's base-10 representation: you can take <TT>i % 10</TT> topick off the rightmost digit, and then say <TT>i = i / 10</TT> tothrow away the rightmost digit (since you already pickedit off), and then say <TT>i % 10</TT> again to pick off what hadbeen the next-to-rightmost digit, etc.<p>One tricky part of writing this function is that it givesyou the digits the wrong way around: it picks them off inright-to-left order, while you probably wanted to printthem out, or store them in string, in left-to-rightorder. There are various ways around this problem.(Another tricky part is converting digit values tocharacters, but here all you have to do is use theinverse of the trick mentioned under #1 above: if <TT>d</TT>is a digit value you've computed, then <TT>d + '0'</TT> is thecharacter value you want.)<p><li>Write a function which accepts an integer and creates astring of digits forming the base <TT>b</TT> representation of theinteger, where <TT>b</TT> is another int argument to the function.(Some compilers come with an <TT>itoa</TT> function which alsoaccepts a base.) Basically, all you'll have to do istake your function from #3 and change ``10'' to ``<TT>b</TT>''.</ol><p>Finally, if you do write these functions, and get them to work,take a look: your functions in #1 and #2 convert <em>from</em> base 10or base <TT>b</TT>, and your functions in #3 and #4 convert <em>to</em> base 10or base <TT>b</TT>. But what base do your functions in #1 and #2 convert<em>to</em>? What base do your functions in #3 and #4 convert <em>from</em>?You can't tell; all they really convert to and from is ``ints''.The base (if it's even meaningful to ask what it is) is implicit;the details of the internal representation of integers are alltaken care of for you by your C compiler, the machine code itgenerates, and by your machine's CPU. You did a certain amountof multiplying or dividing by 10 or <TT>b</TT>, but the ``other'' base wasessentially built in to the multiplication and divisionoperators. If you had to ``port'' the code you wrote from amachine that used binary integers internally to a hypotheticalone that used decimal integers internally, you wouldn't have tochange a thing. All of these facts underlie my claim that itdoesn't make much sense to ask what base these ints are in;they're just numbers.<p>See also the comp.lang.c FAQ list, question 20.10 (and 13.1,while you're at it).<p><address><a href="http://www.eskimo.com/~scs/">Steve Summit</a><br><a href="mailto:scs@eskimo.com">scs@eskimo.com</a></address></body><!-- Mirrored from c-faq.com/misc/btod.html by HTTrack Website Copier/3.x [XR&CO'2008], Sat, 14 Mar 2009 08:02:58 GMT --></html>⌨️ 快捷键说明
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