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