Leaderboard
New York
Rank
Level
1
Brian Caulfield
Delmar
98
2
Cody Cao
Huntington Station
93
3
Anthony P
North New Hyde Park
89
4
MathHead
New York
88
5
Spitzer
Delmar
83
6
Mario Cuomo
Delmar
82
7
User
New York
79
8
Henry
New York
79
9
Pataki
Delmar
78
10
User
New York
77
11
User
Batavia
77
12
User
Batavia
77
13
User
Batavia
77
14
User
Batavia
76
15
User
Batavia
76
16
User
Batavia
75
17
User
Batavia
75
18
User
Batavia
75
19
Natalie Schnose
Fairport
75
20
User
Olean
75
21
Kaos
Delmar
75
22
User
Batavia
75
23
User
Batavia
72
24
User
Batavia
72
25
User
Batavia
72
26
User
Batavia
72
27
User
Batavia
71
28
Andrew Ahn
Centereach
71
29
User
New York
70
30
User
Batavia
70
31
Sid J
New York
69
32
User
Batavia
69
33
User
Batavia
69
34
Team America World Police
Delmar
68
35
Yury Namgung
New York
68
36
User
Batavia
68
37
User
Batavia
67
38
User
Batavia
67
39
User
Batavia
66
40
User
Batavia
66
41
Tim walsh
New York
65
42
User
Batavia
65
43
User
Batavia
65
44
Tom Camargo
New York
65
45
User
Batavia
64
46
User
Batavia
64
47
User
Batavia
63
48
Carina
New York
62
49
George Washington
Kerhonkson
62
50
Thomas Tran
New York
62
51
User
Batavia
61
52
User
Batavia
61
53
User
Batavia
61
54
Joe Scarborough
New York
61
55
User
Kings Park
60
56
Need Quant
New York
60
57
User
Kings Park
60
58
Stanislav Perumov
New York
60
59
User
Batavia
60
60
User
Kings Park
59
61
User
Batavia
59
62
User
Batavia
59
63
User
Batavia
59
64
Julia Levcovets
New York
58
65
User
Kings Park
57
66
User
Lindenhurst
57
67
User
New York
57
68
Benjamin Kang
Ithaca
57
69
Benjamin Kang
Ithaca
57
70
User
Batavia
57
71
User
New York
57
72
User
New York
57
73
Epic Gamer
Scarsdale
57
74
User
Lindenhurst
56
75
Christopher D'Onofrio
New York
56
76
User
Batavia
56
77
Pranay Ranjan
New York
56
78
User
New York
56
79
User
Kings Park
56
80
John Appleseed
New York
56
81
User
New York
56
82
User
Batavia
56
83
User
Lindenhurst
56
84
User
Kings Park
55
85
User
New York
55
86
Tyler Stetson
Cheektowaga
55
87
Dan Stewart
New York
55
88
User
Batavia
55
89
nora c
New York
55
90
User
New York
54
91
User
Lindenhurst
54
92
Cuomo
Delmar
54
93
User
New York
54
94
User
New York
53
95
User
Batavia
53
96
User
Kings Park
53
97
User
New York
53
98
Tigerbass
Walton
53
99
User
Batavia
53
100
Shane Lovelady
New York
53
101
User
Kings Park
53
102
User
Batavia
52
103
User
Batavia
52
104
User
Batavia
52
105
Alexander Treibit
New York
52
106
User
New York
52
107
User
Batavia
52
108
User
Kings Park
52
109
User
Country Knolls
51
110
User
Kings Park
51
111
User
New York
51
112
User
Batavia
51
113
User
Batavia
51
114
John Gruska
New York
51
115
User
Batavia
51
116
FARIS
New York
50
117
User
New York
50
118
User
Kings Park
50
119
User
Batavia
50
120
User
New York
50
121
User
New York
50
122
User
Kings Park
50
123
User
Kings Park
50
124
Mihovil Mandic
New York
50
125
User
Batavia
50
126
User
Cheektowaga
50
127
User
New York
49
128
User
New York
49
129
User
New York
49
130
User
Batavia
49
131
User
Batavia
49
132
User
New York
49
133
Derry Chen
New York
49
134
Rangus Dangus
New York
48
135
User
Kings Park
48
136
Anthony Lam
New York
48
137
User
Kings Park
48
138
User
New York
48
139
CC NY
Suffern
48
140
User
Batavia
48
141
Cian C
Smithtown
47
142
User
East Hills
47
143
User
New York
47
144
User
New York
47
145
Alexander
New York
47
146
User
East Hills
47
147
User
New York
47
148
Joseph Cahill
New York
47
149
George Culver
East Greenbush
47
150
User
Batavia
47
151
User
Lindenhurst
47
152
User
Batavia
46
153
User
Kings Park
46
154
Neil Chriss
New York
46
155
User
Batavia
46
156
User
Kings Park
46
157
User
New York
45
158
User
New York
45
159
User
Brighton
45
160
User
New York
45
161
User
East Hills
45
162
User
Batavia
45
163
User
Lindenhurst
44
164
User
Batavia
44
165
User
New York
44
166
User
Batavia
44
167
User
Batavia
44
168
Zhangyi Chen
New York
44
169
R H
New York
44
170
User
Lindenhurst
44
171
User
New York
43
172
User
New York
43
173
User
Batavia
43
174
Shantanu Laghate
New York
43
175
User
New York
43
176
User
New York
43
177
User
Batavia
43
178
User
Batavia
43
179
User
New York
42
180
User
New York
42
181
User
Lindenhurst
42
182
User
New York
42
183
User
Batavia
42
184
User
New York
42
185
User
New York
42
186
User
Batavia
42
187
User
New York
42
188
User
Ithaca
41
189
User
New York
41
190
Gage Czito
New York
41
191
User
Kings Park
41
192
User
Getzville
41
193
User
New York
41
194
Sameer Lal
Ithaca
41
195
User
New York
41
196
kwikmafs
New York
41
197
User
Batavia
41
198
Nicholas Phillips
New York
40
199
User
Batavia
40
200
User
Batavia
40
201
User
Kings Park
40
202
User
Batavia
40
203
User
Batavia
40
204
User
New York
40
205
User
New York
40
206
User
Batavia
40
207
User
Batavia
40
208
User
Batavia
40
209
User
Batavia
40
210
Nick Jiang
New York
39
211
User
Batavia
39
212
Tyler Amend
New York
39
213
User
Batavia
39
214
User
New York
39
215
User
New York
39
216
Melissa Dunn
New York
39
217
User
Batavia
39
218
User
Batavia
39
219
User
Batavia
39
220
User
Batavia
39
221
User
New York
39
222
User
New York
39
223
User
Batavia
39
224
User
Batavia
39
225
User
Lindenhurst
38
226
User
Batavia
38
227
User
Batavia
38
228
User
New York
38
229
User
Batavia
38
230
User
Batavia
38
231
Ranti Odujinrin
New York
38
232
DareDeviL
New York
38
233
Lisa Fijal
Country Knolls
38
234
User
New York
38
235
User
Batavia
38
236
User
Batavia
38
237
User
New York
38
238
User
Batavia
38
239
User
Batavia
38
240
User
Batavia
38
241
User
New York
37
242
User
Batavia
37
243
User
Batavia
37
244
User
Rochester
37
245
User
Batavia
37
246
User
New York
37
247
User
Batavia
37
248
User
New York
37
249
User
Batavia
37
250
Azuma Isao
New York
37
251
User
Washingtonville
37
252
User
Kings Park
37
253
User
Lindenhurst
36
254
User
New York
36
255
User
Batavia
36
256
Karol Rychlik
New York
36
257
User
New York
36
258
User
Batavia
36
259
User
New York
36
260
Jack Wang
Rochester
36
261
Danny Dark
New York
36
262
User
Batavia
36
263
User
Batavia
35
264
User
Batavia
35
265
Jiachen Xu
New York
35
266
User
Batavia
35
267
Kel Bat
New York
35
268
User
Albany
35
269
User
New York
35
270
Hara gallar
New York
35
271
User
New York
35
272
Robert Brosnan
New York
35
273
User
Roslyn Heights
35
274
User
Hauppauge
34
275
User
New York
34
276
User
New York
34
277
User
Batavia
34
278
User
Batavia
34
279
User
Batavia
34
280
User
Batavia
34
281
User
Batavia
34
282
User
New York
34
283
User
New York
33
284
User
Batavia
33
285
User
New York
33
286
User
Batavia
33
287
User
New York
33
288
User
New York
33
289
User
Kings Park
33
290
User
New York
33
291
User
New York
33
292
Jiles Alexander
New York
33
293
User
Delmar
33
294
Vibhav R Joopelli
New York
33
295
Curious Case
New York
33
296
User
Batavia
32
297
User
Batavia
32
298
AJANTA BARI
New York
32
299
User
New York
32
300
User
New York
32
301
User
Batavia
32
302
User
Batavia
32
303
List
New York
32
304
User
Batavia
32
305
User
Batavia
32
306
User
New York
32
307
User
Cheektowaga
32
308
LM
New York
32
309
thmurphy1993@gmail.com
New York
32
310
User
Batavia
31
311
User
New York
31
312
User
New York
31
313
Naeem Semerkant
New York
31
314
User
New York
31
315
User
New York
31
316
User
New York
31
317
User
Batavia
31
318
User
New York
31
319
User
New York
31
320
User
New York
31
321
Paterson
Delmar
31
322
User
Batavia
30
323
User
Batavia
30
324
User
Batavia
30
325
User
New York
30
326
User
Kings Park
30
327
User
New York
30
328
User
New York
30
329
User
East Islip
30
330
tumi
New York
29
331
User
New York
29
332
Michael C Wang
New York
29
333
User
Lagrangeville
29
334
User
Kings Park
29
335
User
Kings Park
29
336
User
Batavia
29
337
Fred Dintenfass
New York
29
338
User
New York
29
339
User
New York
29
340
User
New York
29
341
William Aramony
Delmar
29
342
User
New York
29
343
User
New York
29
344
User
New York
29
345
User
New York
29
346
User
Levittown
29
347
User
Batavia
29
348
User
Batavia
29
349
User
Batavia
29
350
User
New York
29
351
User
New York
28
352
User
Kings Park
28
353
User
Batavia
28
354
User
New York
28
355
User
Batavia
28
356
User
New York
28
357
User
New York
28
358
Eric Gavrilov
New York
28
359
User
New York
28
360
User
New York
28
361
User
Batavia
28
362
User
New York
28
363
User
New York
28
364
User
New York
28
365
User
Country Knolls
28
366
Abe Lusk
New York
28
367
User
Deer Park
28
368
Pope Lake
New York
28
369
User
Great Neck
28
370
User
Fairmount
28
371
User
Wellsville
28
372
User
New York
28
373
User
New York
27
374
User
New York
27
375
User
New York
27
376
User
New York
27
377
User
Brighton
27
378
User
New York
27
379
User
Kings Park
27
380
practice
New York
27
381
Ted Silver
New York
27
382
User
New York
27
383
Sejune Kwon
Buffalo
27
384
User
New York
27
385
User
New York
27
386
User
New York
27
387
User
Batavia
26
388
User
Batavia
26
389
User
Ballston
26
390
User
New York
26
391
Astik Gupta
New York
26
392
Ajit
New York
26
393
CC
New York
26
394
User
New York
26
395
User
New York
26
396
User
Batavia
26
397
User
Binghamton
26
398
User
New York
26
399
User
New York
26
400
User
Batavia
26
401
User
New York
25
402
User
Kings Park
25
403
Yeonwoo Jang
New York
25
404
Raphael Zerah
New York
25
405
Andy
New York
25
406
Riley Griffin
New York
25
407
User
New York
25
408
User
New York
25
409
User
White Plains
25
410
Louis Abrassard
New York
25
411
User
New York
25
412
Z Set
New York
25
413
User
Coram
25
414
User
New York
25
415
User
Rochester
25
416
Liam Springer
New York
25
417
User
New York
25
418
User
New York
25
419
User
Rochester
25
420
Ray
New York
25
421
User
New York
25
422
User
Kings Park
25
423
User
Batavia
25
424
User
Kings Park
25
425
User
Batavia
25
426
Xonana Scrubb
New York
25
427
User
New York
24
428
User
Irvington
24
429
karimfaris
New York
24
430
Ray
New York
24
431
Swoon
New York
24
432
User
New York
24
433
User
New York
24
434
Jeff Cheng
Croton-On-Hudson
24
435
User
Kings Park
24
436
User
New York
24
437
User
Batavia
24
438
User
New York
24
439
User
Mount Kisco
24
440
User
New York
24
441
User
New York
24
442
User
Brighton
23
443
User
New York
23
444
Oswin G
Getzville
23
445
User
Kings Park
23
446
User
New York
23
447
User
New York
23
448
User
New York
23
449
mehak malik
New York
23
450
User
New York
23
451
User
Batavia
23
452
User
Batavia
23
453
User
New York
23
454
User
Stony Brook
23
455
User
Carmel Hamlet
23
456
User
New York
23
457
User
New York
23
458
User
New York
23
459
User
New York
23
460
User
Batavia
23
461
Brian Gutierrez
New York
22
462
User
New York
22
463
Penelope Chan
North New Hyde Park
22
464
User
New York
22
465
User
Batavia
22
466
User
Deer Park
22
467
User
Kings Park
22
468
User
Valhalla
22
469
User
New York
22
470
User
Kings Park
22
471
Aaron Pzena
New York
22
472
User
New York
22
473
Napoleon
New York
22
474
User
New York
22
475
User
New York
22
476
kp
New York
22
477
User
New York
22
478
User
New York
22
479
Isaac Freedman
New York
22
480
User
New York
22
481
User
Buffalo
22
482
User
Batavia
21
483
User
Lindenhurst
21
484
User
Utica
21
485
Reedfreida 12
Webster
21
486
User
New York
21
487
User
New York
21
488
User
New York
21
489
User
Buffalo
21
490
User
New York
21
491
User
Valhalla
21
492
User
New York
21
493
Bani
New York
21
494
User
New York
21
495
User
New York
21
496
User
Batavia
21
497
User
Rouses Point
21
498
User
New York
21
499
User
Syracuse
21
500
User
New York
21
501
User
New York
21
502
Rachel Barry
Mamaroneck
21
503
User
New York
21
504
User
New York
21
505
User
New York
20
506
User
New York
20
507
User
New York
20
508
User
Batavia
20
509
User
New York
20
510
User
Batavia
20
511
User
New York
20
512
User
Batavia
20
513
User
Roslyn Heights
20
514
User
New York
20
515
Matthew Au
New York
20
516
User
Myers Corner
20
517
User
New York
20
518
User
New York
20
519
User
New York
20
520
User
New York
20
521
User
Kings Point
20
522
User
New York
20
523
User
New York
20
524
User
Kings Park
20
525
Anwesha Palit
Buffalo
20
526
User
Batavia
20
527
User
New York
20
528
User
Kings Park
20
529
User
Batavia
20
530
User
Delmar
19
531
User
New York
19
532
User
New York
19
533
Aman M.
Smithtown
19
534
User
New York
19
535
User
Brighton
19
536
User
New York
19
537
User
New York
19
538
User
Fishers
19
539
User
New York
19
540
User
Rye
19
541
ProW YT
Walton Park
19
542
User
New York
19
543
User
Rochester
19
544
User
New York
19
545
User
Batavia
19
546
POOOO
Delmar
19
547
User
Kings Park
19
548
User
Batavia
19
549
Ajay
New York
19
550
User
Batavia
19
551
User
Cheektowaga
19
552
User
New York
19
553
User
Syosset
19
554
User
Levittown
19
555
User
New York
19
556
User
New York
19
557
User
New York
19
558
User
Akron
19
559
User
New York
19
560
User
Kings Park
19
561
User
Kings Park
19
562
Sawyer Sacks
Ithaca
19
563
Ben Sokolowsky
Selden
19
564
Fx
New York
19
565
User
New York
18
566
User
Batavia
18
567
User
New York
18
568
Meundies
New York
18
569
User
Batavia
18
570
User
New York
18
571
User
Batavia
18
572
User
New York
18
573
User
Woodmere
18
574
User
New York
18
575
User
Batavia
18
576
User
Delmar
18
577
User
Shelter Island
18
578
User
New York
18
579
User
New York
18
580
User
Batavia
18
581
User
Batavia
18
582
K E
New York
18
583
Raj Amarnani
New York
18
584
User
New York
18
585
User
Kings Park
18
586
User
Batavia
18
587
User
New York
18
588
User
Great Neck
18
589
User
New York
18
590
User
Ithaca
18
591
Abby
Cicero
18
592
User
New York
18
593
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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 math 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 math, 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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