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