Leaderboard
Massachusetts
Massachusetts
United States
Worldwide
Rank
Level
1
User
Boston
99
2
Andrew Ahn
Cambridge
92
3
Rich Caputo
Newton
90
4
Albert Caputo
Hopkinton
86
5
Josh
Boston
85
6
User
Waltham
82
7
User
Boston
79
8
Albert Samaha
Cambridge
79
9
trllisbad
Amherst
78
10
Kevin Li
Cambridge
77
11
User
Boston
75
12
User
Cambridge
72
13
User
Boston
70
14
User
Arlington
69
15
Ima Coder
Wellesley
69
16
Anu
Boston
67
17
User
Cambridge
66
18
Ryan H
Boston
66
19
Anyazelie M. Zéphyaire
Hatfield
64
20
User
Boston
64
21
Stanislav Perumov
Boston
60
22
User
Amherst
58
23
jkuzi
Boston
54
24
User
Cambridge
53
25
User
Canton
52
26
User
Waltham
52
27
Anirudh Jhanji
Needham
51
28
User
Boston
51
29
User
Boston
51
30
User
Boston
51
31
bruh
Quincy
50
32
Isaac Altman Sagan
Waltham
50
33
User
Boston
50
34
User
Boston
50
35
Julia Levcovets
Waltham
49
36
User
Cambridge
49
37
User
Boston
49
38
User
Waltham
48
39
User
Cambridge
48
40
User
Boston
48
41
User
Cambridge
47
42
Edmund Chen
Peabody
47
43
User
Cambridge
47
44
User
Boston
47
45
User
Cambridge
46
46
User
Waltham
46
47
User
Boston
46
48
User
Marlborough
46
49
User
Boston
46
50
User
Newton
46
51
User
Boston
46
52
Christopher Johnson
Grafton
45
53
User
Boston
45
54
User
Cambridge
44
55
Nate Austen
Boston
44
56
User
Boston
43
57
C Smith
Northampton
43
58
User
Boston
43
59
User
Cambridge
42
60
Naveen Arunachalam
Cambridge
42
61
Ilan Barr
Newton
42
62
User
Natick
42
63
User
Worcester
41
64
User
Somerville
41
65
User
Waltham
41
66
User
Wellesley
41
67
User
New Bedford
40
68
samara
Boston
40
69
User
Waltham
40
70
User
Cambridge
40
71
User
Quincy
39
72
User
Arlington
39
73
User
Boston
38
74
Sebastian Rodriguez
Cambridge
38
75
User
Cambridge
38
76
User
Westford
38
77
User
Boston
38
78
User
Boston
38
79
User
Cambridge
37
80
User
Medford
37
81
User
Boston
37
82
User
Boston
37
83
User
Boston
36
84
User
Boston
36
85
User
Wellesley
36
86
User
North Andover
36
87
User
Boston
36
88
Vishal Kumar Sarsani
Amherst
35
89
User
Boston
35
90
Lotus Kemna
Wellesley
35
91
User
Cambridge
35
92
User
Boston
35
93
User
Tewksbury
35
94
User
Longmeadow
34
95
User
Boston
34
96
User
Cambridge
34
97
User
Cambridge
34
98
User
Boston
34
99
User
Cambridge
33
100
User
Cambridge
33
101
Dap
Boston
33
102
User
Cambridge
33
103
User
Boston
33
104
User
Norwood
32
105
arvin hedayaty
Boston
32
106
User
Boston
32
107
User
Cambridge
32
108
User
Brookline
32
109
User
Amherst
32
110
User
Brookline
32
111
User
Cambridge
32
112
User
Boston
32
113
User
Brookline
32
114
User
Boston
32
115
User
Woburn
32
116
User
Newton
31
117
User
Dedham
31
118
User
Cambridge
31
119
User
Arlington
31
120
Prince Towel
Worcester
31
121
KENNY CHEN
Boston
31
122
User
Newton
31
123
User
Boston
31
124
User
Cambridge
31
125
Long Nguyen
Natick
30
126
zorazii
Grafton
30
127
User
Boston
30
128
User
Cambridge
30
129
User
Cambridge
30
130
User
Waltham
30
131
User
Raynham
30
132
User
Boston
30
133
User
Boston
30
134
Kendawg
Boston
29
135
MQYBETH
Cambridge
29
136
User
Milford
29
137
User
Concord
29
138
User
Boston
29
139
User
Quincy
29
140
User
Hopkinton
29
141
User
Worcester
29
142
User
Newton
28
143
User
Boston
28
144
User
Cambridge
28
145
Yaniv Toledano
Cambridge
28
146
Jason Hu Zelin
Deerfield
28
147
User
Cambridge
28
148
User
Boston
28
149
User
Boston
28
150
User
Cambridge
28
151
User
Boston
28
152
User
Boston
28
153
Katharine Bruce
Cambridge
28
154
Thorven
Boston
27
155
User
Boston
27
156
User
Boston
27
157
User
Cambridge
27
158
User
Amherst
27
159
User
Boston
27
160
User
Cambridge
27
161
User
Cambridge
27
162
User
Cambridge
27
163
User
Waltham
27
164
User
Waltham
27
165
User
Newton
27
166
User
Waltham
26
167
User
Stoneham
26
168
John Appleseed
Winchester
26
169
User
Cambridge
26
170
User
Brookline
26
171
User
Boston
25
172
User
Wellesley
25
173
User
Boston
25
174
User
Boston
25
175
User
Brookline
25
176
User
Boston
25
177
User
Cambridge
25
178
User
Boston
25
179
User
Cambridge
25
180
User
Wayland
25
181
User
Boston
25
182
User
Boston
24
183
User
Cambridge
24
184
User
Cambridge
24
185
User
Arlington
24
186
Christopher Painter
Boston
24
187
User
Billerica
24
188
User
Waltham
24
189
User
Newton
24
190
User
Boston
24
191
User
Boston
24
192
User
Boston
23
193
User
Fitchburg
23
194
User
Boston
23
195
User
Boston
23
196
User
Boston
23
197
User
Boston
23
198
User
Boston
23
199
User
Cambridge
23
200
User
Natick
23
201
User
Cambridge
23
202
R Arora
Maynard
23
203
Jan Smith
East Bridgewater
23
204
User
Boston
23
205
User
Andover
22
206
User
Marblehead
22
207
User
Boston
22
208
User
North Reading
22
209
User
Gardner
22
210
User
Worcester
22
211
S M
Northampton
22
212
User
Arlington
22
213
Geoffrey the Sicko Mode
Boston
22
214
User
Boston
22
215
Anvi Khatri
Sudbury
22
216
User
Swansea
22
217
User
Natick
22
218
User
Natick
22
219
User
Waltham
22
220
User
Freetown
22
221
User
Boston
22
222
User
Framingham
22
223
User
Cambridge
22
224
User
Boston
22
225
User
Boston
21
226
User
Boston
21
227
User
Fitchburg
21
228
User
Lexington
21
229
User
Boston
21
230
User
Natick
21
231
User
Boston
21
232
User
Milton
21
233
User
Cambridge
21
234
User
Chelsea
21
235
User
Tewksbury
21
236
User
Cambridge
21
237
User
East Bridgewater
21
238
User
Boston
21
239
User
Holliston
21
240
User
Holyoke
21
241
User
Boston
21
242
User
Needham
21
243
User
Chelsea
20
244
User
Brookline
20
245
User
Marblehead
20
246
Samhitha Satish kumar
Westford
20
247
User
Boston
20
248
User
Westfield
20
249
User
Fitchburg
20
250
User
Fitchburg
20
251
User
Boston
20
252
User
Boston
20
253
User
Cambridge
20
254
User
Boston
20
255
User
Winchester
20
256
User
Winchester
20
257
User
Randolph
20
258
User
West Boylston
20
259
User
Cambridge
20
260
User
Concord
20
261
User
Fall River
20
262
User
Fall River
20
263
User
Newton
20
264
User
Boston
20
265
Harsha Paladugu
Arlington
20
266
Ruth
Holyoke
20
267
User
Boston
20
268
worth smith
Boston
19
269
User
Boston
19
270
User
Fitchburg
19
271
User
Southbridge
19
272
User
Cambridge
19
273
User
Westford
19
274
User
Pittsfield
19
275
User
Amherst
19
276
User
Millis
19
277
User
Natick
19
278
User
Boston
19
279
User
Acton
19
280
User
Boston
19
281
User
Boston
19
282
Jennifer Sawyer
Framingham
19
283
User
Harvard
19
284
User
Boston
19
285
User
Milton
19
286
Yahir
Boston
18
287
User
Amherst
18
288
User
Cambridge
18
289
User
Beverly
18
290
MAX ALLARD
Franklin
18
291
User
Tyngsborough
18
292
User
Hadley
18
293
User
Andover
18
294
User
Boston
18
295
User
Boston
18
296
User
Holyoke
18
297
User
North Attleborough
17
298
User
Cambridge
17
299
User
Hingham
17
300
User
Fitchburg
17
301
User
Boston
17
302
User
Boston
17
303
User
Boston
17
304
User
Newton
17
305
User
Boston
17
306
User
Amherst
17
307
User
Boston
17
308
User
Boston
17
309
User
Boston
16
310
User
Quincy
16
311
User
Boston
16
312
User
Beverly
16
313
User
Cambridge
16
314
User
Marblehead
16
315
User
Boston
16
316
User
Natick
16
317
User
Waltham
16
318
User
Waltham
16
319
User
Quincy
16
320
User
Cambridge
15
321
User
Chelmsford
15
322
User
Amherst
15
323
User
Melrose
15
324
User
Cambridge
15
325
User
Cambridge
15
326
User
Natick
15
327
User
Natick
15
328
User
Cambridge
15
329
User
Fall River
15
330
User
Sherborn
15
331
User
Holyoke
15
332
User
Revere
15
333
User
Waltham
15
334
User
Lexington
14
335
User
Cambridge
14
336
User
Cambridge
14
337
User
Cambridge
14
338
User
Boston
14
339
User
Worcester
14
340
User
Sudbury
14
341
User
Natick
14
342
User
Newton
14
343
I Z
Cambridge
14
344
User
Dracut
14
345
User
Hingham
13
346
User
Boston
13
347
User
Boston
13
348
User
Winchester
13
349
User
Boston
13
350
User
North Attleborough
13
351
User
Winchester
13
352
User
Boston
13
353
User
Cambridge
13
354
User
Springfield
13
355
User
Pepperell
13
356
User
Peabody
13
357
User
Boston
13
358
User
Boston
13
359
User
Barnstable
12
360
User
Cambridge
12
361
Peter Sweeney
Boston
12
362
User
Manchester-By-The-Sea
12
363
User
Somerville
12
364
User
Cambridge
12
365
User
Lexington
12
366
User
Cambridge
12
367
Providinq
Hull
12
368
User
Boston
12
369
User
Boston
12
370
User
Cambridge
11
371
User
Natick
11
372
User
Boston
11
373
User
Framingham
11
374
User
Quincy
11
375
Sarthak Tripathi
Wilmington
11
376
User
Chicopee
11
377
User
Wellesley
11
378
User
Boston
11
379
User
Cambridge
11
380
User
Woburn
11
381
User
Natick
11
382
User
Leominster
11
383
User
Cambridge
11
384
User
Boston
11
385
User
Cambridge
11
386
User
Franklin
11
387
User
Boston
11
388
User
Harwich
11
389
User
Boston
11
390
User
Boston
11
391
User
Boston
11
392
User
Wellesley
10
393
User
Springfield
10
394
User
Cambridge
10
395
User
Boston
10
396
User
Winchester
10
397
User
Weymouth
10
398
User
Natick
10
399
User
Boston
10
400
User
Somerville
10
401
Michael Pasqua
Boston
10
402
User
Boston
10
403
User
Chelsea
10
404
User
Cambridge
10
405
User
Boston
10
406
User
Lowell
9
407
User
Boston
9
408
AM
Framingham
9
409
User
Boston
9
410
User
Natick
9
411
User
Natick
9
412
User
Cambridge
9
413
User
Boston
9
414
User
Westford
9
415
User
Cambridge
9
416
User
Cambridge
9
417
User
Hardwick
9
418
User
Boston
9
419
User
Cambridge
9
420
User
Waltham
9
421
User
Holyoke
9
422
User
Marlborough
9
423
User
Natick
8
424
Brianna DiFronzo
Chicopee
8
425
User
Boston
8
426
User
Boston
8
427
User
Quincy
8
428
User
Cambridge
8
429
User
Burlington
8
430
User
Cambridge
8
431
User
Winchester
8
432
User
Newton
8
433
User
Natick
8
434
User
Randolph
8
435
User
Brookline
8
436
User
Boston
8
437
User
Cambridge
8
438
User
Cambridge
8
439
User
Sudbury
8
440
User
Boston
8
441
User
Boston
7
442
User
Boston
7
443
User
Medford
7
444
User
Fitchburg
7
445
User
Fitchburg
7
446
User
Fitchburg
7
447
Padre
Boston
7
448
User
Quincy
7
449
User
Boston
7
450
User
Westford
7
451
User
Boston
7
452
User
Boston
7
453
User
Boston
7
454
User
Winchester
7
455
User
Boston
7
456
User
Winchester
7
457
User
Winchester
7
458
User
Cambridge
7
459
User
Cambridge
7
460
Michael Pasqua
Boston
7
461
User
Cambridge
7
462
User
Hamilton
7
463
User
Burlington
7
464
User
Danvers
7
465
User
Boston
7
466
User
Waltham
7
467
User
Boston
7
468
User
North Andover
7
469
User
Boston
7
470
User
Holyoke
7
471
User
Taunton
7
472
Abijit Aji
Boston
7
473
User
Boston
7
474
User
Attleboro
7
475
User
Cambridge
7
476
User
Holyoke
7
477
User
Amherst
6
478
User
Boston
6
479
User
Boston
6
480
Eamon
Lee
6
481
User
Concord
6
482
User
Quincy
6
483
User
Boston
6
484
User
Boston
6
485
User
Boston
6
486
User
Framingham
6
487
Maksuda Lillah
Cambridge
6
488
User
Boston
6
489
User
Wakefield
6
490
User
West Springfield
6
491
User
Boston
6
492
User
Boston
6
493
User
Boston
6
494
User
Boston
6
495
User
Walpole
6
496
User
Brookline
6
497
User
Randolph
6
498
User
Brookline
6
499
User
Boston
6
500
User
Malden
6
501
User
Leominster
6
502
User
Wakefield
6
503
User
Holyoke
6
504
User
Malden
6
505
Mark Tran
Boston
5
506
Eric Kruglak
Waltham
5
507
User
Longmeadow
5
508
User
Tyngsborough
5
509
User
Westfield
5
510
User
Woburn
5
511
User
Boston
5
512
User
Cambridge
5
513
User
Boston
5
514
User
Milford
5
515
User
Springfield
5
516
User
Boston
5
517
User
Lowell
5
518
User
Fitchburg
5
519
User
Fitchburg
5
520
User
Sandwich
5
521
User
Fitchburg
5
522
User
Bedford
5
523
User
Somerville
5
524
User
Boston
5
525
User
Brookline
5
526
User
Cambridge
5
527
User
Cambridge
5
528
User
Boston
5
529
User
Boston
5
530
User
Clinton
5
531
User
Boston
5
532
User
Wellesley
5
533
User
Wilmington
5
534
User
Boston
5
535
User
Winchester
5
536
User
Framingham
5
537
User
Boston
5
538
User
Boston
5
539
User
Cambridge
5
540
User
Hopkinton
5
541
User
Holyoke
5
542
User
Amherst
5
543
User
Cambridge
5
544
User
Cambridge
5
545
User
Needham
5
546
User
Waltham
5
547
Li
Boston
5
548
User
Cambridge
5
549
User
Marshfield
5
550
User
Williamstown
5
551
User
Milton
5
552
User
Holyoke
5
553
User
Boston
5
554
User
Holyoke
5
555
User
Holyoke
5
556
User
Holyoke
5
557
User
Holyoke
5
558
User
Newton
4
559
User
Waltham
4
560
User
Boston
4
561
User
Boston
4
562
User
Boston
4
563
User
Newton
4
564
User
Amherst
4
565
User
Waltham
4
566
User
Cambridge
4
567
User
Gardner
4
568
User
Gardner
4
569
User
Fitchburg
4
570
User
Fitchburg
4
571
Isabelle Servant
Fall River
4
572
User
Boston
4
573
User
Deerfield
4
574
User
Boston
4
575
User
Franklin
4
576
Michael Chu
Boston
4
577
User
Templeton
4
578
User
Northborough
4
579
User
Newton
4
580
User
Tewksbury
4
581
User
Boston
4
582
User
Winchester
4
583
User
Boston
4
584
User
Boston
4
585
User
Randolph
4
586
User
Boston
4
587
User
Lowell
4
588
User
Medford
4
589
User
Cambridge
4
590
User
Worcester
4
591
User
Cambridge
4
592
User
Cambridge
4
593
User
Boston
4
594
User
Marshfield
4
595
User
Newton
4
596
User
Holyoke
4
597
User
Holyoke
4
598
User
Newton
4
599
User
Holyoke
4
600
User
Acton
4
601
User
Marlborough
3
602
User
Arlington
3
603
User
Boston
3
604
User
Boston
3
605
User
Boston
3
606
User
Boston
3
607
A A
Framingham
3
608
User
Gardner
3
609
User
Fitchburg
3
610
User
Fitchburg
3
611
User
Fitchburg
3
612
User
Fitchburg
3
613
User
Fitchburg
3
614
User
Fitchburg
3
615
User
Fitchburg
3
616
User
Fitchburg
3
617
User
Brookline
3
618
User
Waltham
3
619
User
Brookline
3
620
User
Medford
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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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