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55 | ||
56 | <UL CLASS="ChildLinks"> | |
57 | <LI><A href="node9.html#SECTION000910000000000000000">8.1 Why is Decimal needed?</a> | |
58 | <LI><A href="node9.html#SECTION000920000000000000000">8.2 The <tt class="class">Decimal</tt> type</a> | |
59 | <LI><A href="node9.html#SECTION000930000000000000000">8.3 The <tt class="class">Context</tt> type</a> | |
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64 | ||
65 | <H1><A NAME="SECTION000900000000000000000"> | |
66 | 8 PEP 327: Decimal Data Type</A> | |
67 | </H1> | |
68 | ||
69 | <P> | |
70 | Python has always supported floating-point (FP) numbers, based on the | |
71 | underlying C <tt class="ctype">double</tt> type, as a data type. However, while most | |
72 | programming languages provide a floating-point type, many people (even | |
73 | programmers) are unaware that floating-point numbers don't represent | |
74 | certain decimal fractions accurately. The new <tt class="class">Decimal</tt> type | |
75 | can represent these fractions accurately, up to a user-specified | |
76 | precision limit. | |
77 | ||
78 | <P> | |
79 | ||
80 | <H2><A NAME="SECTION000910000000000000000"> | |
81 | 8.1 Why is Decimal needed?</A> | |
82 | </H2> | |
83 | ||
84 | <P> | |
85 | The limitations arise from the representation used for floating-point numbers. | |
86 | FP numbers are made up of three components: | |
87 | ||
88 | <P> | |
89 | ||
90 | <UL> | |
91 | <LI>The sign, which is positive or negative. | |
92 | </LI> | |
93 | <LI>The mantissa, which is a single-digit binary number | |
94 | followed by a fractional part. For example, <code>1.01</code> in base-2 notation | |
95 | is <code>1 + 0/2 + 1/4</code>, or 1.25 in decimal notation. | |
96 | </LI> | |
97 | <LI>The exponent, which tells where the decimal point is located in the number represented. | |
98 | </LI> | |
99 | </UL> | |
100 | ||
101 | <P> | |
102 | For example, the number 1.25 has positive sign, a mantissa value of | |
103 | 1.01 (in binary), and an exponent of 0 (the decimal point doesn't need | |
104 | to be shifted). The number 5 has the same sign and mantissa, but the | |
105 | exponent is 2 because the mantissa is multiplied by 4 (2 to the power | |
106 | of the exponent 2); 1.25 * 4 equals 5. | |
107 | ||
108 | <P> | |
109 | Modern systems usually provide floating-point support that conforms to | |
110 | a standard called IEEE 754. C's <tt class="ctype">double</tt> type is usually | |
111 | implemented as a 64-bit IEEE 754 number, which uses 52 bits of space | |
112 | for the mantissa. This means that numbers can only be specified to 52 | |
113 | bits of precision. If you're trying to represent numbers whose | |
114 | expansion repeats endlessly, the expansion is cut off after 52 bits. | |
115 | Unfortunately, most software needs to produce output in base 10, and | |
116 | common fractions in base 10 are often repeating decimals in binary. | |
117 | For example, 1.1 decimal is binary <code>1.0001100110011 ...</code>; .1 = | |
118 | 1/16 + 1/32 + 1/256 plus an infinite number of additional terms. IEEE | |
119 | 754 has to chop off that infinitely repeated decimal after 52 digits, | |
120 | so the representation is slightly inaccurate. | |
121 | ||
122 | <P> | |
123 | Sometimes you can see this inaccuracy when the number is printed: | |
124 | <div class="verbatim"><pre> | |
125 | >>> 1.1 | |
126 | 1.1000000000000001 | |
127 | </pre></div> | |
128 | ||
129 | <P> | |
130 | The inaccuracy isn't always visible when you print the number because | |
131 | the FP-to-decimal-string conversion is provided by the C library, and | |
132 | most C libraries try to produce sensible output. Even if it's not | |
133 | displayed, however, the inaccuracy is still there and subsequent | |
134 | operations can magnify the error. | |
135 | ||
136 | <P> | |
137 | For many applications this doesn't matter. If I'm plotting points and | |
138 | displaying them on my monitor, the difference between 1.1 and | |
139 | 1.1000000000000001 is too small to be visible. Reports often limit | |
140 | output to a certain number of decimal places, and if you round the | |
141 | number to two or three or even eight decimal places, the error is | |
142 | never apparent. However, for applications where it does matter, | |
143 | it's a lot of work to implement your own custom arithmetic routines. | |
144 | ||
145 | <P> | |
146 | Hence, the <tt class="class">Decimal</tt> type was created. | |
147 | ||
148 | <P> | |
149 | ||
150 | <H2><A NAME="SECTION000920000000000000000"> | |
151 | 8.2 The <tt class="class">Decimal</tt> type</A> | |
152 | </H2> | |
153 | ||
154 | <P> | |
155 | A new module, <tt class="module">decimal</tt>, was added to Python's standard | |
156 | library. It contains two classes, <tt class="class">Decimal</tt> and | |
157 | <tt class="class">Context</tt>. <tt class="class">Decimal</tt> instances represent numbers, and | |
158 | <tt class="class">Context</tt> instances are used to wrap up various settings such as | |
159 | the precision and default rounding mode. | |
160 | ||
161 | <P> | |
162 | <tt class="class">Decimal</tt> instances are immutable, like regular Python integers | |
163 | and FP numbers; once it's been created, you can't change the value an | |
164 | instance represents. <tt class="class">Decimal</tt> instances can be created from | |
165 | integers or strings: | |
166 | ||
167 | <P> | |
168 | <div class="verbatim"><pre> | |
169 | >>> import decimal | |
170 | >>> decimal.Decimal(1972) | |
171 | Decimal("1972") | |
172 | >>> decimal.Decimal("1.1") | |
173 | Decimal("1.1") | |
174 | </pre></div> | |
175 | ||
176 | <P> | |
177 | You can also provide tuples containing the sign, the mantissa represented | |
178 | as a tuple of decimal digits, and the exponent: | |
179 | ||
180 | <P> | |
181 | <div class="verbatim"><pre> | |
182 | >>> decimal.Decimal((1, (1, 4, 7, 5), -2)) | |
183 | Decimal("-14.75") | |
184 | </pre></div> | |
185 | ||
186 | <P> | |
187 | Cautionary note: the sign bit is a Boolean value, so 0 is positive and | |
188 | 1 is negative. | |
189 | ||
190 | <P> | |
191 | Converting from floating-point numbers poses a bit of a problem: | |
192 | should the FP number representing 1.1 turn into the decimal number for | |
193 | exactly 1.1, or for 1.1 plus whatever inaccuracies are introduced? | |
194 | The decision was to dodge the issue and leave such a conversion out of | |
195 | the API. Instead, you should convert the floating-point number into a | |
196 | string using the desired precision and pass the string to the | |
197 | <tt class="class">Decimal</tt> constructor: | |
198 | ||
199 | <P> | |
200 | <div class="verbatim"><pre> | |
201 | >>> f = 1.1 | |
202 | >>> decimal.Decimal(str(f)) | |
203 | Decimal("1.1") | |
204 | >>> decimal.Decimal('%.12f' % f) | |
205 | Decimal("1.100000000000") | |
206 | </pre></div> | |
207 | ||
208 | <P> | |
209 | Once you have <tt class="class">Decimal</tt> instances, you can perform the usual | |
210 | mathematical operations on them. One limitation: exponentiation | |
211 | requires an integer exponent: | |
212 | ||
213 | <P> | |
214 | <div class="verbatim"><pre> | |
215 | >>> a = decimal.Decimal('35.72') | |
216 | >>> b = decimal.Decimal('1.73') | |
217 | >>> a+b | |
218 | Decimal("37.45") | |
219 | >>> a-b | |
220 | Decimal("33.99") | |
221 | >>> a*b | |
222 | Decimal("61.7956") | |
223 | >>> a/b | |
224 | Decimal("20.64739884393063583815028902") | |
225 | >>> a ** 2 | |
226 | Decimal("1275.9184") | |
227 | >>> a**b | |
228 | Traceback (most recent call last): | |
229 | ... | |
230 | decimal.InvalidOperation: x ** (non-integer) | |
231 | </pre></div> | |
232 | ||
233 | <P> | |
234 | You can combine <tt class="class">Decimal</tt> instances with integers, but not with | |
235 | floating-point numbers: | |
236 | ||
237 | <P> | |
238 | <div class="verbatim"><pre> | |
239 | >>> a + 4 | |
240 | Decimal("39.72") | |
241 | >>> a + 4.5 | |
242 | Traceback (most recent call last): | |
243 | ... | |
244 | TypeError: You can interact Decimal only with int, long or Decimal data types. | |
245 | >>> | |
246 | </pre></div> | |
247 | ||
248 | <P> | |
249 | <tt class="class">Decimal</tt> numbers can be used with the <tt class="module">math</tt> and | |
250 | <tt class="module">cmath</tt> modules, but note that they'll be immediately converted to | |
251 | floating-point numbers before the operation is performed, resulting in | |
252 | a possible loss of precision and accuracy. You'll also get back a | |
253 | regular floating-point number and not a <tt class="class">Decimal</tt>. | |
254 | ||
255 | <P> | |
256 | <div class="verbatim"><pre> | |
257 | >>> import math, cmath | |
258 | >>> d = decimal.Decimal('123456789012.345') | |
259 | >>> math.sqrt(d) | |
260 | 351364.18288201344 | |
261 | >>> cmath.sqrt(-d) | |
262 | 351364.18288201344j | |
263 | </pre></div> | |
264 | ||
265 | <P> | |
266 | <tt class="class">Decimal</tt> instances have a <tt class="method">sqrt()</tt> method that | |
267 | returns a <tt class="class">Decimal</tt>, but if you need other things such as | |
268 | trigonometric functions you'll have to implement them. | |
269 | ||
270 | <P> | |
271 | <div class="verbatim"><pre> | |
272 | >>> d.sqrt() | |
273 | Decimal("351364.1828820134592177245001") | |
274 | </pre></div> | |
275 | ||
276 | <P> | |
277 | ||
278 | <H2><A NAME="SECTION000930000000000000000"> | |
279 | 8.3 The <tt class="class">Context</tt> type</A> | |
280 | </H2> | |
281 | ||
282 | <P> | |
283 | Instances of the <tt class="class">Context</tt> class encapsulate several settings for | |
284 | decimal operations: | |
285 | ||
286 | <P> | |
287 | ||
288 | <UL> | |
289 | <LI><tt class="member">prec</tt> is the precision, the number of decimal places. | |
290 | </LI> | |
291 | <LI><tt class="member">rounding</tt> specifies the rounding mode. The <tt class="module">decimal</tt> | |
292 | module has constants for the various possibilities: | |
293 | <tt class="constant">ROUND_DOWN</tt>, <tt class="constant">ROUND_CEILING</tt>, | |
294 | <tt class="constant">ROUND_HALF_EVEN</tt>, and various others. | |
295 | </LI> | |
296 | <LI><tt class="member">traps</tt> is a dictionary specifying what happens on | |
297 | encountering certain error conditions: either an exception is raised or | |
298 | a value is returned. Some examples of error conditions are | |
299 | division by zero, loss of precision, and overflow. | |
300 | </LI> | |
301 | </UL> | |
302 | ||
303 | <P> | |
304 | There's a thread-local default context available by calling | |
305 | <tt class="function">getcontext()</tt>; you can change the properties of this context | |
306 | to alter the default precision, rounding, or trap handling. The | |
307 | following example shows the effect of changing the precision of the default | |
308 | context: | |
309 | ||
310 | <P> | |
311 | <div class="verbatim"><pre> | |
312 | >>> decimal.getcontext().prec | |
313 | 28 | |
314 | >>> decimal.Decimal(1) / decimal.Decimal(7) | |
315 | Decimal("0.1428571428571428571428571429") | |
316 | >>> decimal.getcontext().prec = 9 | |
317 | >>> decimal.Decimal(1) / decimal.Decimal(7) | |
318 | Decimal("0.142857143") | |
319 | </pre></div> | |
320 | ||
321 | <P> | |
322 | The default action for error conditions is selectable; the module can | |
323 | either return a special value such as infinity or not-a-number, or | |
324 | exceptions can be raised: | |
325 | ||
326 | <P> | |
327 | <div class="verbatim"><pre> | |
328 | >>> decimal.Decimal(1) / decimal.Decimal(0) | |
329 | Traceback (most recent call last): | |
330 | ... | |
331 | decimal.DivisionByZero: x / 0 | |
332 | >>> decimal.getcontext().traps[decimal.DivisionByZero] = False | |
333 | >>> decimal.Decimal(1) / decimal.Decimal(0) | |
334 | Decimal("Infinity") | |
335 | >>> | |
336 | </pre></div> | |
337 | ||
338 | <P> | |
339 | The <tt class="class">Context</tt> instance also has various methods for formatting | |
340 | numbers such as <tt class="method">to_eng_string()</tt> and <tt class="method">to_sci_string()</tt>. | |
341 | ||
342 | <P> | |
343 | For more information, see the documentation for the <tt class="module">decimal</tt> | |
344 | module, which includes a quick-start tutorial and a reference. | |
345 | ||
346 | <P> | |
347 | <div class="seealso"> | |
348 | <p class="heading">See Also:</p> | |
349 | ||
350 | <dl compact="compact" class="seerfc"> | |
351 | <dt><a href="http://www.python.org/peps/pep-0327.html" | |
352 | title="Decimal Data Type" | |
353 | >PEP 327, <em>Decimal Data Type</em></a> | |
354 | <dd>Written by Facundo Batista and implemented | |
355 | by Facundo Batista, Eric Price, Raymond Hettinger, Aahz, and Tim Peters. | |
356 | </dl> | |
357 | ||
358 | <P> | |
359 | <dl compact="compact" class="seeurl"> | |
360 | <dt><a href="http://research.microsoft.com/~hollasch/cgindex/coding/ieeefloat.html" | |
361 | class="url">http://research.microsoft.com/~hollasch/cgindex/coding/ieeefloat.html</a></dt> | |
362 | <dd>A more detailed overview of the IEEE-754 representation.</dd> | |
363 | </dl> | |
364 | ||
365 | <P> | |
366 | <dl compact="compact" class="seeurl"> | |
367 | <dt><a href="http://www.lahey.com/float.htm" | |
368 | class="url">http://www.lahey.com/float.htm</a></dt> | |
369 | <dd>The article uses Fortran code to illustrate many of the problems | |
370 | that floating-point inaccuracy can cause.</dd> | |
371 | </dl> | |
372 | ||
373 | <P> | |
374 | <dl compact="compact" class="seeurl"> | |
375 | <dt><a href="http://www2.hursley.ibm.com/decimal/" | |
376 | class="url">http://www2.hursley.ibm.com/decimal/</a></dt> | |
377 | <dd>A description of a decimal-based representation. This representation | |
378 | is being proposed as a standard, and underlies the new Python decimal | |
379 | type. Much of this material was written by Mike Cowlishaw, designer of the | |
380 | Rexx language.</dd> | |
381 | </dl> | |
382 | ||
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