Leaderboard
New York
Rank
Level
1
eixsinxicosx
Fairport
94
2
FARIS
New York
86
3
Jonathan G
Katonah
86
4
User
Batavia
85
5
User
New York
85
6
User
Irvington
85
7
User
Batavia
84
8
User
Levittown
83
9
User
Batavia
82
10
User
Batavia
82
11
Astik Gupta
New York
82
12
User
New York
81
13
User
Batavia
81
14
kwikmafs
New York
81
15
User
Batavia
81
16
User
Batavia
81
17
User
Batavia
81
18
User
Batavia
81
19
User
Rochester
81
20
User
New York
81
21
Alexander Treibit
New York
80
22
User
Batavia
80
23
User
Batavia
80
24
User
Batavia
80
25
Le azien
New York
79
26
User
Batavia
79
27
User
Batavia
79
28
User
Batavia
78
29
User
Batavia
78
30
User
Batavia
78
31
User
Batavia
78
32
Jack Wang
Rochester
78
33
User
Batavia
77
34
User
Batavia
77
35
User
Batavia
77
36
User
Batavia
77
37
User
Batavia
77
38
User
Batavia
77
39
User
New York
76
40
User
Batavia
76
41
User
Batavia
76
42
User
Batavia
76
43
User
Batavia
76
44
User
Batavia
75
45
User
Batavia
74
46
User
Batavia
74
47
User
Batavia
73
48
Sejune Kwon
Buffalo
73
49
Jiachen Xu
New York
72
50
User
New York
72
51
User
New York
72
52
User
New York
72
53
User
New York
72
54
User
Batavia
71
55
User
Batavia
71
56
User
Batavia
71
57
User
Batavia
71
58
User
Batavia
70
59
User
Batavia
70
60
User
Batavia
70
61
User
Batavia
69
62
User
Batavia
69
63
kp
New York
69
64
User
New York
69
65
Zhangyi Chen
New York
68
66
Naeem Semerkant
New York
67
67
User
Batavia
67
68
Brian Caulfield
Delmar
65
69
User
New York
65
70
LM
New York
64
71
User
Batavia
63
72
Abe Lusk
New York
63
73
User
Batavia
63
74
Cian C
Smithtown
62
75
User
Batavia
61
76
User
New York
61
77
Yeonwoo Jang
New York
61
78
Raphael Zerah
New York
61
79
Ted Silver
New York
60
80
Teddy
New York
60
81
Neil Chriss
New York
59
82
User
New York
59
83
User
New York
59
84
Amandine Roche
New York
58
85
User
New York
58
86
User
Batavia
58
87
User
New York
58
88
User
New York
58
89
User
Buffalo
58
90
AresTheMars
New York
58
91
User
Kings Park
57
92
Danny Dark
New York
57
93
User
New York
57
94
User
New York
57
95
pigeonrepublic
New York
57
96
User
Batavia
56
97
User
New York
55
98
Karol Rychlik
New York
55
99
Megan Le
Ithaca
55
100
User
New York
54
101
User
Batavia
54
102
Raj Amarnani
New York
54
103
User
New York
54
104
User
New York
54
105
List
New York
53
106
User
Batavia
53
107
User
Batavia
53
108
User
Ithaca
53
109
User
New York
53
110
User
Webster
53
111
Tyler Amend
New York
52
112
User
Batavia
52
113
User
New York
52
114
User
New York
52
115
User
New York
52
116
User
New York
52
117
Ben Sokolowsky
Selden
52
118
User
Ithaca
52
119
User
New York
52
120
User
New York
52
121
Ferdi W
New York
51
122
User
Kings Park
51
123
Jeff Cheng
Croton-On-Hudson
51
124
User
New York
51
125
Sawyer Sacks
Ithaca
51
126
Philip Murphy
New York
51
127
Jozef Putrycz
Candor
51
128
User
Kings Park
50
129
Ray
New York
50
130
Swoon
New York
50
131
User
Batavia
50
132
User
Batavia
50
133
User
Batavia
50
134
User
New York
50
135
Ze Li
New York
50
136
User
Marathon
50
137
User
New York
50
138
Christopher Chi
New York
49
139
User
New York
49
140
User
New York
49
141
User
Stony Brook
48
142
User
Batavia
48
143
User
New York
48
144
User
New York
48
145
User
Batavia
47
146
User
Kings Park
47
147
User
Coram
47
148
User
New York
47
149
User
New York
47
150
User
New York
47
151
Gleb Volkov
New York
47
152
Nick Jiang
New York
46
153
User
Batavia
46
154
User
Batavia
46
155
User
Batavia
46
156
User
Kings Park
46
157
User
New York
46
158
Bani
New York
46
159
User
New York
46
160
VM
New York
46
161
Octavius
Delmar
45
162
User
New York
45
163
Pope Lake
New York
45
164
User
Kings Park
45
165
User
Kings Park
45
166
User
Valhalla
45
167
Aaron Pzena
New York
45
168
User
New York
45
169
User
Saint James
45
170
User
New York
45
171
User
New York
45
172
User
New York
45
173
User
New York
45
174
User
New York
45
175
User
New York
45
176
User
New York
45
177
User
New York
45
178
User
Batavia
44
179
User
Kings Park
44
180
karimfaris
New York
44
181
User
Kings Park
44
182
User
Kings Park
44
183
Brian Gutierrez
New York
44
184
Nicholas Phillips
New York
44
185
User
Batavia
44
186
User
Batavia
44
187
User
Ithaca
44
188
User
New York
44
189
MultiplyByZero
New York
44
190
User
New York
44
191
User
New York
43
192
User
New York
43
193
User
New York
43
194
Ray
New York
43
195
User
Batavia
43
196
User
Valhalla
43
197
User
Kings Point
43
198
User
Elmont
43
199
User
New York
43
200
User
New York
42
201
User
Brighton
42
202
User
Fishers
42
203
Ajay
New York
42
204
User
Levittown
42
205
Anton Njavro
New York
42
206
User
New York
42
207
User
New York
42
208
User
New York
41
209
User
Roslyn Heights
41
210
User
Batavia
41
211
User
Buffalo
41
212
User
New York
41
213
User
Ithaca
41
214
User
New York
41
215
User
New York
41
216
User
New York
41
217
User
Lindenhurst
41
218
User
Lindenhurst
41
219
User
New York
41
220
Charles X
Saratoga Springs
41
221
User
New York
41
222
User
New York
40
223
User
New York
40
224
Anwesha Palit
Buffalo
40
225
User
Getzville
40
226
User
New York
40
227
User
East Ithaca
40
228
User
New York
40
229
Adeel
New York
40
230
User
New York
40
231
User
Rochester
40
232
User
Kings Park
39
233
User
Delmar
39
234
User
Batavia
39
235
User
New York
39
236
User
Woodmere
39
237
User
New York
39
238
User
New York
39
239
User
Annandale-On-Hudson
39
240
User
Ithaca
39
241
User
Ithaca
39
242
User
New York
39
243
User
New York
39
244
Kyle Fram
New York
39
245
K E
New York
38
246
User
Elmont
38
247
User
West Babylon
38
248
User
East Northport
38
249
User
Ithaca
38
250
User
New York
38
251
Rayyan Bhuiyan
New York
38
252
User
Katonah
38
253
User
Batavia
37
254
User
Kings Park
37
255
User
New York
37
256
User
New York
37
257
User
Cortland
37
258
User
Cortland
37
259
User
Pittsford
37
260
User
Ithaca
37
261
User
New York
37
262
User
New York
37
263
User
Batavia
36
264
Matthew Au
New York
36
265
User
Batavia
36
266
User
New York
36
267
User
New York
36
268
User
West Babylon
36
269
User
Buffalo
36
270
User
New York
36
271
User
Riverhead
36
272
User
New York
36
273
User
Rochester
35
274
User
Kings Park
35
275
User
New York
35
276
User
Chatham
35
277
Undergraduate Film Council
Rochester
35
278
User
New York
35
279
Noel Zeyar (Am Nerd)
New York
35
280
User
New York
34
281
User
New York
34
282
User
Kings Park
34
283
User
Kings Park
34
284
User
New York
34
285
User
Batavia
34
286
User
Delmar
34
287
User
New York
34
288
User
Lindenhurst
34
289
User
New York
34
290
User
New York
34
291
Adam Kelly-Penso
New York
34
292
User
New York
34
293
sithole
New York
34
294
Stanley Shao
Ithaca
34
295
User
Ithaca
34
296
User
Kings Park
33
297
User
Kings Park
33
298
User
Kings Park
33
299
User
Centereach
33
300
User
New York
33
301
User
Delmar
33
302
Jess
Bellmore
33
303
User
New York
33
304
Macklin Fluehr
New York
33
305
James B
New York
33
306
User
Beacon
33
307
dani
New York
33
308
User
Kings Park
32
309
User
New York
32
310
User
Delmar
32
311
User
New York
32
312
User
New York
32
313
User
New York
32
314
User
New York
32
315
User
Buffalo
32
316
User
Getzville
32
317
User
New York
32
318
User
New York
32
319
User
New York
32
320
User
New York
31
321
User
New York
31
322
User
New York
31
323
User
Delmar
31
324
User
New York
31
325
User
New York
31
326
User
Kings Park
31
327
User
Delmar
31
328
User
Binghamton
31
329
User
Delmar
31
330
User
Kings Park
31
331
User
New York
31
332
User
New York
31
333
User
New York
31
334
User
New York
31
335
User
New York
31
336
User
New York
31
337
User
Monsey
31
338
User
Batavia
31
339
User
New York
31
340
Daniel Peters
New York
31
341
User
New York
31
342
User
New York
31
343
User
New York
31
344
User
Getzville
31
345
User
New York
31
346
User
New York
31
347
Tom Camargo
New York
31
348
User
Getzville
31
349
User
New York
31
350
User
New York
31
351
Britton Olle
New York
31
352
User
New York
31
353
User
Johnson City
31
354
Reon Cloete
New York
31
355
User
New York
31
356
User
New York
31
357
User
New York
31
358
User
Great Neck
31
359
User
New York
30
360
Maureen Bies
Levittown
30
361
User
Kings Park
30
362
User
New York
30
363
User
Kings Park
30
364
User
Kings Park
30
365
User
Kings Park
30
366
User
Kings Park
30
367
User
Kings Park
30
368
User
Kings Park
30
369
User
New York
30
370
User
Centereach
30
371
User
New York
30
372
User
New York
30
373
User
New York
30
374
User
New York
30
375
User
West Babylon
30
376
User
New York
30
377
User
New York
30
378
User
New York
30
379
User
New York
30
380
User
New York
30
381
User
New York
30
382
User
New York
30
383
User
New York
30
384
User
Lindenhurst
30
385
User
New York
30
386
User
New York
30
387
User
New York
29
388
User
New York
29
389
User
New York
29
390
User
Kings Park
29
391
User
New York
29
392
User
Kings Park
29
393
User
New York
29
394
User
Ithaca
29
395
User
Delmar
29
396
jim li
New York
29
397
User
New York
29
398
User
New York
29
399
User
New York
29
400
User
New York
29
401
User
Webster
29
402
User
Patchogue
29
403
User
New York
29
404
User
New York
29
405
User
New York
29
406
User
New York
29
407
User
New York
29
408
User
New York
29
409
User
New York
29
410
User
New York
29
411
User
New York
28
412
User
New York
28
413
User
New York
28
414
User
New York
28
415
User
Kings Park
28
416
User
New York
28
417
User
New York
28
418
User
New York
28
419
User
Kings Park
28
420
User
New York
28
421
User
New York
28
422
User
New York
28
423
User
Delmar
28
424
Euphoric Lyra
Latham
28
425
User
Dix Hills
28
426
User
New York
28
427
User
Ithaca
28
428
User
New York
28
429
Yanqi Huang
New York
28
430
User
Syracuse
28
431
User
New York
28
432
User
Port Dickinson
28
433
User
Lindenhurst
28
434
Kirsten Colwell
New York
28
435
Sammy
New York
28
436
User
New Rochelle
28
437
User
Batavia
27
438
User
Batavia
27
439
User
New York
27
440
User
New York
27
441
User
New York
27
442
User
New York
27
443
VALERIYA SHAPOROVA
New York
27
444
dizzydanny
Binghamton
27
445
User
Hauppauge
27
446
User
Port Chester
27
447
User
Kings Park
27
448
User
Roslyn Heights
27
449
User
New York
27
450
Imiz
New York
27
451
User
New York
27
452
User
New York
27
453
User
New York
27
454
User
New York
27
455
User
New York
27
456
User
New York
27
457
User
New York
27
458
User
Ithaca
27
459
User
New York
27
460
User
New York
27
461
User
New York
27
462
User
New York
27
463
User
New York
27
464
sdar
Ithaca
26
465
User
New York
26
466
User
New York
26
467
User
New York
26
468
User
Kings Park
26
469
User
New York
26
470
User
Kings Park
26
471
User
Kings Park
26
472
User
Kings Park
26
473
User
Kings Park
26
474
User
Kings Park
26
475
User
Kings Park
26
476
Michelle Howe
New York
26
477
User
New York
26
478
User
Delmar
26
479
User
New York
26
480
User
Delmar
26
481
User
Delmar
26
482
User
Delmar
26
483
User
Delmar
26
484
User
Delmar
26
485
User
Delmar
26
486
User
New York
26
487
User
New York
26
488
User
New York
26
489
User
New York
26
490
Jacob Spivack
Johnson City
26
491
User
New York
26
492
User
New York
26
493
User
Jericho
26
494
User
New York
26
495
User
Port Chester
26
496
User
New York
26
497
User
West Babylon
26
498
User
New York
26
499
User
Fairmount
26
500
User
Webster
26
501
User
New York
26
502
User
New York
26
503
User
New York
26
504
User
New York
26
505
User
Ithaca
26
506
User
New York
26
507
User
New York
26
508
User
Lindenhurst
26
509
User
New York
26
510
User
New York
26
511
User
New York
26
512
User
New York
26
513
User
New York
26
514
User
New York
26
515
User
Lindenhurst
26
516
User
New York
26
517
George Koulouris
New York
25
518
xiabing feng
New York
25
519
User
Ithaca
25
520
User
New York
25
521
User
New York
25
522
User
Kings Park
25
523
User
Kings Park
25
524
User
Kings Park
25
525
User
New York
25
526
User
New York
25
527
User
Huntington Station
25
528
User
New York
25
529
Connie Lu
New York
25
530
User
New York
25
531
User
New York
25
532
User
Batavia
25
533
User
New York
25
534
User
New York
25
535
User
Marathon
25
536
User
Cohoes
25
537
User
Lindenhurst
25
538
User
New York
25
539
User
New York
25
540
User
New York
25
541
User
New York
25
542
User
New York
25
543
User
Ithaca
25
544
jason
New York
25
545
User
Stony Point
25
546
User
New York
25
547
User
New York
25
548
User
Lindenhurst
25
549
User
Spring Valley
25
550
User
New York
25
551
User
Brinckerhoff
24
552
Irina Zheng
New York
24
553
User
New York
24
554
User
Kings Park
24
555
User
Kings Park
24
556
User
Delmar
24
557
User
Batavia
24
558
User
Kings Park
24
559
User
Delmar
24
560
User
Galeville
24
561
Andrew
New York
24
562
User
Getzville
24
563
User
Ithaca
24
564
User
New York
24
565
User
Ithaca
24
566
User
New York
24
567
meng cai
New York
24
568
User
New York
24
569
User
Suffolk County
24
570
User
Batavia
23
571
User
New York
23
572
User
New York
23
573
Shahriar Chowdhury
New York
23
574
User
New York
23
575
User
Kings Park
23
576
User
Oceanside
23
577
User
New York
23
578
User
Marathon
23
579
User
Buffalo
23
580
User
Delmar
23
581
User
Rochester
23
582
User
New York
23
583
User
New York
23
584
User
New York
23
585
User
New York
23
586
User
New York
23
587
User
New York
23
588
User
Rochester
23
589
User
New York
23
590
User
New York
23
591
User
New York
23
592
User
New York
23
593
User
New York
23
594
User
New York
23
595
User
Ithaca
23
596
User
New York
23
597
User
White Plains
22
598
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A Web App for Mathematics Training

Do you want to be fast at mental math? Many people do, but the options for doing the necessary exercises are simply too cumbersome for all but the most dedicated of trainees. In physical fitness, many people are interested in training their bodies but allocating the time, energy, and money for it is a significant obstacle. Likewise, lugging around books and whatnot for math practice is a threshold that just doesn't meet the standards of modern life.

Training yourself to be skilled at mental math needs to be quick and convenient. mathtrainer.org is a web app that works in your browser rather than a program you have to download and install on your computer or phone. This allows users to try and use the app without having to install new software. As a web app, updates are also much simpler. There is no need to download endless updates—the website will always be the most current version.

You can access a web app from any device connected to the internet and equipped with a web browser, including smartphones, tablets, and desktop computers. Moreover, you are free to use whichever browser you prefer, including Google Chrome, Safari, Firefox, and others. Google Chrome is the recommended browser for the best maths training since it tends to lead the pack in supporting the latest web technologies.

Math Trainer is designed to offer a similar experience regardless of what you’re using to access it, whether it be Android, iOS, Windows, or another operating system. Though an on-screen touch keyboard will appear on mobile devices, you may prefer to use the app on a desktop with a keyboard. Hopefully the advantages of a web app for convenient mathematics training are apparent.

Another part of making the app easy to use is eliminating the need for signing up and logging in. Users can get started with their math training as soon as they click the start button on this page. After progressing to higher levels in the app, your progress is automatically saved so long as your return to the site through the same browser.


What's a Mental Math Tip?

A mental math tip is a sequence of steps that can be taken to solve a math problem in your head. Click the arrow below to see an example for the following problem:

÷
984
3
328

A tip like this one is available for every problem in Math Trainer, so there's always help if you get stuck. With enough practice, you'll be able to predict what the tips will say—you'll have learned mental math!


Get Better at Mental Math

The ability to quickly perform mental calculations offers advantages in certain circumstances. But even without applications, getting better at mental math is a great way to stimulate one’s mind. It develops better number sense and intuition for quantifying the world around us. Practicing mental calculation will strengthen your foundation for learning more advanced maths topics.

Nonetheless, the tangible benefits of improving at mental math are many. It is certainly expected that educated people are able to do simple arithmetic without having to pull out a calculator. An inability to do so may reflect poorly on you, while being well-practiced in mental calculation will leave your contemporaries impressed. In many scientific and technical circles, mental math ability is even more highly regarded.

For students, mental calculation speed will often have a direct impact on math and science test scores. At all grade levels, it is not sufficient to know how to solve math problems when tests have a time limit on them. The highest-scoring test takers are able to answer questions both correctly and efficiently. Improving mental math skills will only benefit a student’s academic career.

Calculating the solution to an arithmetic problem in your head is often faster than pulling out a device to tell you the answer. For example, figuring out how much to tip a server at a restaurant is a straightforward arithmetic problem that many people are unable to perform without a calculator. By training your brain to solve basic math problems, you can save time in situations like these.

Mental math can also be relied upon when calculation devices are not available. Even with the conveniences of modern life, we occasionally find ourselves without access to our cell phones or other capable devices. A mind skilled in mental math is always available to you.

Finally, getting better at mental math enables a quick estimate and sanity check on results obtained from calculators. While computers are extremely reliable at solving math problems, there is always the risk of incorrectly inputting the problem to the computer. By getting better at mental mathematics, you will develop an intuition for whether the results of calculators make sense.

In fact, the ability to estimate is often sufficient to avoid using calculators altogether. While the use of computers is widespread, estimation is an increasingly valued skill in many industries. There are many situations where complex math will eventually be required, but a preliminary estimate is needed quickly. A major boost to productivity!


Use a Math Trainer

Mental math ability is a lot like physical fitness training. You may be out of shape in the beginning, but with diligent training you can and will improve. Initially you might not enjoy the exercise, but you will reap significant rewards for your effort. As you become more fit, you’ll begin to enjoy the activity much more. If you are serious about it, your mental calculation fitness could become a source of energy, galvanizing you to face the challenges of life with enthusiasm.

In physical training, you break down the fibers in your muscles during a workout session. Your muscles actually sustain tiny tears during resistance training exercises. While you rest afterwards, your body repairs the damage, rebuilding the fibers thicker and stronger.

A similar process is believed to occur for cognitive tasks. A 2016 study found "extensive evidence that brain-training interventions improve performance on the trained tasks".1 Therefore you can expect training your brain to answer mental math questions will lead to improved performance over time.

In the context of physical fitness, a "trainer" often refers to a trained professional who guides the workout and recovery process. Personal trainers are tasked with assessing a trainee's level of ability, prescribing an exercise regimen, and offering feedback as the training goes along. The word "trainer" could also refer to a system that automates the role of a personal trainer. Many aerobic exercise machines today offer interactive training programs with feedback and analysis of a user's performance.

A math trainer is needed for optimal math fitness. Like in physical fitness, the trainer should be compatible with users at a variety of skill levels and should guide them to the next level. It should give an accurate assessment of a user's strengths and weakness, as well as offer helpful feedback on where to focus one's efforts. Learning the ropes of mental maths with a math trainer should be a seamless, rewarding journey to ever-greater abilities.


Mental Calculation

Mental calculation, or mental math, is performing arithmetical calculations without the aid of tools or supplies. As opposed to using a calculator or pencil and paper, mental math is performed entirely in one’s head.

People use mental calculation when computation aides are not available, when it is faster to do so, or when they wish to practice, show off, or participate in mental math competitions. Most people perform basic mental calculation using elementary arithmetic on a daily basis. An inability to calculate mentally is a serious obstacle to many common tasks.

In U.S. schools, mental calculation is taught only for the most elementary arithmetic, such as single-digit addition and multiplication of two numbers between 0 and 12. To solve addition problems involving multiple digits, you are taught to add columns of digits from right to left, carrying the tens digit if the column sum exceeds 9. For example, how would you approach this addition problem?

Example of two-digit
addition problem

If you were trained like many of us were, you’d add the right column to obtain 12. Since that’s two digits, you’d write the 2 under the right column and carry the 10 by writing a 1 above the left column. Finally, you’d add the two tens digits and the carried 1 to obtain the answer, 52.

To solve an addition problem mentally, it’s best to add the columns from left to right. In our example, you could add the tens digit of the second number, 30, to the first number, 14, to obtain 44. This is easier than the full problem because you’re just doing one mental calculation and tacking on the 4 from the 14 as the singles digit. Then you’d add the remaining ones digit of the second number, 8, to 44 to arrive at the answer, 52.

Which approach seems simpler to you? Can you do the first approach without pulling out a pencil and paper? It turns out the same advantages of left-to-right addition apply to much larger numbers as well. It’s unlikely that difficult addition problems can be solved right to left without needing to write it all out, which of course is more time consuming.

Mental math should be distinguished from the memorization of math facts such as multiplication tables. A foundation of memorized answers to simple math problems will make mental math easier, but performing maths in your head requires both memorized facts and the manipulation of numbers and operations to solve problems. This combination of skill and memory allows us to solve far more complex math questions than can be answered with readily-memorized math facts.

Many mental math tricks are specific to particular numbers or types of problems, usually dependent on the base of the number system used. In the decimal numeral system, for example, it is trivially easy to multiply by 10—just add a 0 to the end of the number. This mental math trick wouldn’t work in the hexadecimal numeral system, though, because the base is 16 instead of 10.

Therefore mental calculation is the ability to manipulate complex arithmetic problems in such a way that they can be resolved using simple memorized math facts.


Arithmetic

Arithmetic is the branch of mathematics concerning basic number operations: addition, subtraction, multiplication, and division. As kids, we are taught to do arithmetic because real-world math problems depend on a mastery of elementary arithmetic. Higher-level study of arithmetic and the integers, or whole numbers, is known as number theory.

Though the math kids initially study is arithmetic, the word is rarely used in this context anymore. Originally it comes from the Greek arithmos, meaning “number”. It has however been included in the “three Rs” of elementary Western education: reading, writing, and arithmetic.

There is evidence prehistoric humans were using arithmetic as hunter-gatherers. Archaeologists have uncovered a tally stick, believed to be over 20,000 years old, which may exhibit the earliest known sequences of prime numbers. An understanding of prime numbers, which are only divisible by themselves and the number 1, requires knowledge of the operation in arithmetic known as division.

From tally marks came base-10 numerals such as those used in Egypt over 5,000 years ago. Number systems based on 10 probably arose because humans have ten “digits” as fingers on their hands (or toes on their feet). A later advance in arithmetic was positional notation, which allowed the same symbols to represent different magnitudes depending on their position in the written number. These numeral systems allowed complex arithmetic to be communicated, recorded, and applied to the challenges faced by our ancestors.

The basic operation of arithmetic is addition. It combines two or more numbers into one, the sum of the terms. The terms can be added in any order, which is known as the commutative property of arithmetic. On a number line, the sum of two numbers is the total distance from zero covered by both numbers.

The inverse arithmetical operation of addition is subtraction. It finds the difference between two numbers. Subtraction is not commutative because the order of the numbers determines whether the answer is positive or negative. On a number line, the difference between two numbers is the distance between their positions.

A second basic operation of arithmetic is multiplication, which scales a number by another number. This second number is called a factor. Like addition, multiplication is commutative—you can change the order of the factors and still get the same answer. Multiplication on a number line can be viewed as adding the first number a total number of times equal to the second factor.

Finally, division is an arithmetical operation that is essentially the inverse of multiplication. Rather than scaling a number, it is divided into a number of pieces equal to the second number. Dividing by the number 0 is not defined in arithmetic because dividing something into zero pieces is impossible.

Basic arithmetic allows us to evaluate the answers to an unlimited number of mathematical expressions. Arithmetical expressions can be purely mathematical, as in 2 + 2, or they can represent quantities in the physical world, such as two items plus two more. Understanding the laws of arithmetic is tremendously powerful.


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