25:
319:
480:
with no terms evaluates to 0. Allowing a "sum" with only 1 or 0 terms reduces the number of cases to be considered in many mathematical formulas. Such "sums" are natural starting points in
388:
484:, as well as in algorithms. For these reasons, the "empty sum is zero" extension is standard practice in mathematics and computer programming (assuming the domain has a
424:
478:
451:
223:
196:
169:
231:
35:
93:
65:
600:
72:
655:
633:
50:
137:
where the number of terms is zero. The natural way to extend non-empty sums is to let the empty sum be the
79:
334:
61:
566:
493:
625:
481:
396:
504:
500:
456:
429:
201:
174:
147:
8:
660:
618:
86:
629:
596:
512:
138:
46:
571:
529:
649:
561:
489:
485:
122:
508:
134:
548:. The empty sum convention allows the zero-dimensional vector space
24:
115:
314:{\displaystyle s_{m}=\sum _{i=1}^{m}a_{i}=a_{1}+\cdots +a_{m}}
453:
with only one term evaluates to that one term, while a "sum"
42:
393:
provided that we use the following natural convention:
511:), the value of an empty summation is taken to be its
459:
432:
399:
337:
328:
terms of the sequence. This satisfies the recurrence
234:
204:
177:
150:
617:
472:
445:
418:
382:
313:
217:
190:
163:
647:
593:Practical Foundations for Programming Languages
615:
552:={0} to have a basis, namely the empty set.
523:
51:introducing citations to additional sources
16:Summation where the number of terms is zero
595:. Cambridge University Press. p. 86.
225:, ... be a sequence of numbers, and let
41:Relevant discussion may be found on the
648:
590:
18:
499:For sums of other objects (such as
383:{\displaystyle s_{m}=s_{m-1}+a_{m}}
13:
14:
672:
536:is a linearly independent subset
34:relies largely or entirely on a
23:
609:
584:
1:
577:
532:, a basis of a vector space
488:). For the same reason, the
7:
620:Linear Algebra and Geometry
555:
544:is a linear combination of
540:such that every element of
518:
10:
677:
426:. In other words, a "sum"
113:
567:Iterated binary operation
524:Empty linear combinations
324:be the sum of the first
114:Not to be confused with
616:David M. Bloom (1979).
591:Harper, Robert (2016).
494:multiplicative identity
419:{\displaystyle s_{0}=0}
474:
447:
420:
384:
315:
268:
219:
192:
165:
656:Operations on numbers
475:
473:{\displaystyle s_{0}}
448:
446:{\displaystyle s_{1}}
421:
385:
316:
248:
220:
218:{\displaystyle a_{3}}
193:
191:{\displaystyle a_{2}}
166:
164:{\displaystyle a_{1}}
457:
430:
397:
335:
232:
202:
175:
148:
47:improve this article
492:is taken to be the
470:
443:
416:
380:
311:
215:
188:
161:
513:additive identity
139:additive identity
112:
111:
97:
668:
640:
639:
623:
613:
607:
606:
588:
482:induction proofs
479:
477:
476:
471:
469:
468:
452:
450:
449:
444:
442:
441:
425:
423:
422:
417:
409:
408:
389:
387:
386:
381:
379:
378:
366:
365:
347:
346:
320:
318:
317:
312:
310:
309:
291:
290:
278:
277:
267:
262:
244:
243:
224:
222:
221:
216:
214:
213:
197:
195:
194:
189:
187:
186:
170:
168:
167:
162:
160:
159:
107:
104:
98:
96:
55:
27:
19:
676:
675:
671:
670:
669:
667:
666:
665:
646:
645:
644:
643:
636:
614:
610:
603:
589:
585:
580:
558:
526:
521:
464:
460:
458:
455:
454:
437:
433:
431:
428:
427:
404:
400:
398:
395:
394:
374:
370:
355:
351:
342:
338:
336:
333:
332:
305:
301:
286:
282:
273:
269:
263:
252:
239:
235:
233:
230:
229:
209:
205:
203:
200:
199:
182:
178:
176:
173:
172:
155:
151:
149:
146:
145:
119:
108:
102:
99:
56:
54:
40:
28:
17:
12:
11:
5:
674:
664:
663:
658:
642:
641:
634:
608:
601:
582:
581:
579:
576:
575:
574:
572:Empty function
569:
564:
557:
554:
530:linear algebra
525:
522:
520:
517:
467:
463:
440:
436:
415:
412:
407:
403:
391:
390:
377:
373:
369:
364:
361:
358:
354:
350:
345:
341:
322:
321:
308:
304:
300:
297:
294:
289:
285:
281:
276:
272:
266:
261:
258:
255:
251:
247:
242:
238:
212:
208:
185:
181:
158:
154:
110:
109:
45:. Please help
31:
29:
22:
15:
9:
6:
4:
3:
2:
673:
662:
659:
657:
654:
653:
651:
637:
631:
627:
622:
621:
612:
604:
602:9781107029576
598:
594:
587:
583:
573:
570:
568:
565:
563:
562:Empty product
560:
559:
553:
551:
547:
543:
539:
535:
531:
516:
514:
510:
506:
502:
497:
495:
491:
490:empty product
487:
483:
465:
461:
438:
434:
413:
410:
405:
401:
375:
371:
367:
362:
359:
356:
352:
348:
343:
339:
331:
330:
329:
327:
306:
302:
298:
295:
292:
287:
283:
279:
274:
270:
264:
259:
256:
253:
249:
245:
240:
236:
228:
227:
226:
210:
206:
183:
179:
156:
152:
142:
140:
136:
132:
128:
124:
117:
106:
103:December 2017
95:
92:
88:
85:
81:
78:
74:
71:
67:
64: –
63:
59:
58:Find sources:
52:
48:
44:
38:
37:
36:single source
32:This article
30:
26:
21:
20:
619:
611:
592:
586:
549:
545:
541:
537:
533:
527:
498:
486:zero element
392:
325:
323:
143:
130:
126:
120:
100:
90:
83:
76:
69:
57:
33:
624:. pp.
509:polynomials
131:nullary sum
123:mathematics
62:"Empty sum"
661:0 (number)
650:Categories
635:0521293243
578:References
73:newspapers
360:−
296:⋯
250:∑
135:summation
127:empty sum
43:talk page
556:See also
519:Examples
505:matrices
116:Zero sum
501:vectors
133:, is a
87:scholar
632:
599:
89:
82:
75:
68:
60:
129:, or
125:, an
94:JSTOR
80:books
630:ISBN
597:ISBN
144:Let
66:news
528:In
121:In
49:by
652::
628:.
626:45
515:.
507:,
503:,
496:.
198:,
171:,
141:.
638:.
605:.
550:V
546:B
542:V
538:B
534:V
466:0
462:s
439:1
435:s
414:0
411:=
406:0
402:s
376:m
372:a
368:+
363:1
357:m
353:s
349:=
344:m
340:s
326:m
307:m
303:a
299:+
293:+
288:1
284:a
280:=
275:i
271:a
265:m
260:1
257:=
254:i
246:=
241:m
237:s
211:3
207:a
184:2
180:a
157:1
153:a
118:.
105:)
101:(
91:·
84:·
77:·
70:·
53:.
39:.
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
