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