Leaderboard
California
Rank
Level
1
K. Gee
Millbrae
98
2
User
Menlo Park
98
3
Jenn Jia
Berkeley
96
4
Viral Jhaveri
Oakland
95
5
SPONDULIX
Berkeley
93
6
Brian Chao
San Diego
90
7
Travis Grady
Manhattan Beach
88
8
Robert H
San Diego
87
9
Sherine Kokado
City
85
10
Andrew W
North Tustin
84
11
User
Mountain View
80
12
User
Glendale
80
13
sirius
San Francisco
79
14
User
Sunnyvale
78
15
User
San Francisco
78
16
User
Berkeley
78
17
Les Miserable
Sacramento
77
18
Ademola Oladosu
Menlo Park
76
19
User
Hermosa Beach
76
20
Gurak
Stanford
75
21
User
Fremont
74
22
User
San Francisco
73
23
Darren Chen
Walnut
72
24
TheProdigyHenry
Garden Grove
72
25
FelixT
San Diego
72
26
User
Marina
70
27
zhiwen xie
Los Angeles
70
28
robert hou
San Jose
69
29
Sam Chinnery
Fresno
69
30
Una Gauper
El Dorado Hills
68
31
William Wang
Berkeley
68
32
Charles Comiter
San Francisco
66
33
Bob Cho
Union City
66
34
Nes370
West Sacramento
66
35
User
Los Angeles
65
36
urjil sanghvi
Berkeley
65
37
User
Visalia
65
38
Oo
Los Angeles
64
39
User
San Diego
64
40
User
San Francisco
63
41
User
Mountain View
63
42
User
Dublin
63
43
User
Stanford
63
44
User
Los Angeles
62
45
Jared Haertel
San Jose
62
46
User
Newport Beach
62
47
User
San Diego
61
48
SPicy
Redwood City
61
49
User
Covina
61
50
Chris Hansen
Redondo Beach
59
51
User
San Francisco
59
52
Dede W
Alhambra
58
53
ag ar
Santa Clarita
58
54
User
Los Angeles
58
55
User
San Francisco
58
56
Tom Solomon
Fontana
58
57
User
San Francisco
57
58
Dean Lee
Santa Clara
57
59
sakshi agte
Fremont
57
60
Melinda Zhang
Foster City
57
61
User
Garden Grove
56
62
noob
Alhambra
56
63
User
Manhattan Beach
56
64
Long John
San Mateo
56
65
User
Manhattan Beach
56
66
Reva Sobti
Fullerton
56
67
FelixM
San Diego
55
68
Cannon Tuttle
San Ramon
55
69
User
Hercules
55
70
User
San Mateo
55
71
User
Livermore
54
72
User
San Francisco
54
73
User
Stanford
54
74
User
Los Angeles
54
75
Zavosh
Altadena
54
76
User
San Francisco
54
77
User
Berkeley
53
78
User
Cupertino
53
79
User
Westminster
53
80
User
San Francisco
53
81
User
Cupertino
52
82
Nelson Li
Oakland
52
83
User
Westminster
52
84
Luke Kim
Palo Alto
52
85
User
Claremont
52
86
pettni
Pasadena
51
87
User
Foster City
51
88
Alexandre Cohen
Berkeley
51
89
User
Stanford
51
90
User
Sonoma
51
91
User
Ontario
51
92
Jeff Rangan
Escondido
51
93
User
Alhambra
51
94
Alexander Welty
Berkeley
51
95
User
Santa Clara
51
96
Jordan Katz
Oakland
50
97
User
Los Angeles
50
98
Devin
San Francisco
50
99
CellPhoneJeff
Seal Beach
50
100
User
San Francisco
50
101
User
Roseville
49
102
User
San Diego
49
103
User
San Jose
49
104
User
Garden Grove
49
105
BOI
Dublin
48
106
S Desai
San Francisco
48
107
Kevin Ramlal
Berkeley
48
108
Mason Lien
Cupertino
48
109
Will Taylor
San Diego
47
110
User
Folsom
47
111
User
San Francisco
47
112
User
Sacramento
47
113
User
Dublin
46
114
User
Los Angeles
46
115
User
San Diego
46
116
Vincent Zhang
San Ramon
46
117
Amineh Beltran
San Francisco
46
118
User
Oakland
45
119
User
San Diego
45
120
SCart
Fremont
45
121
User
Sacramento
45
122
Tyler Pak
Los Angeles
45
123
Emily Cheng
Carlsbad
44
124
User
Menlo Park
44
125
User
San Francisco
44
126
User
Lafayette
44
127
User
San Jose
44
128
Newton Peng
Garden Grove
44
129
User
Los Angeles
44
130
Timothy So
Westminster
44
131
Rahul Chekuri
Berkeley
44
132
User
Riverside
44
133
User
Los Angeles
44
134
User
San Diego
44
135
Charlie Do
Union City
43
136
User
San Diego
43
137
User
West Sacramento
43
138
Иван Мирошниченко
Sacramento
43
139
User
San Diego
43
140
shane ramsey
Lodi
43
141
User
Westminster
43
142
I got many skills but failing is not one
Mountain House
43
143
User
San Jose
43
144
User
Los Angeles
43
145
User
Claremont
43
146
Landon
Indio
43
147
User
Foster City
42
148
Sushant Vema
Pleasanton
42
149
User
Berkeley
42
150
Shelley Chang
Alameda
42
151
User
East Los Angeles
42
152
user
Oakland
42
153
User
Palo Alto
42
154
Harry Ackley
Palo Alto
42
155
User
Westminster
42
156
Rajath Salegame
La Canada Flintridge
41
157
User
Westminster
41
158
Hoang
Los Angeles
41
159
User
Hacienda Heights
41
160
User
Lake Forest
41
161
Allison Su (LC)
Cupertino
41
162
User
Culver City
41
163
User
Fresno
41
164
Kevin Nguyen
Santa Clara
40
165
User
Los Angeles
40
166
User
Westminster
40
167
User
Sacramento
40
168
User
Hollister
40
169
User
Santa Clara
40
170
User
Sacramento
40
171
J. M.
Oakland
40
172
User
Los Angeles
40
173
User
Riverside
40
174
User
San Francisco
40
175
User
Cupertino
40
176
hi
Berkeley
39
177
User
Mountain View
39
178
User
Cupertino
39
179
User
Chula Vista
39
180
User
Sacramento
39
181
User
Pleasanton
39
182
User
Los Angeles
39
183
User
Mountain View
38
184
User
Westminster
38
185
User
Los Angeles
38
186
Jeremy Kintana
Santa Clara
38
187
User
Artesia
38
188
User
Sacramento
38
189
User
Los Angeles
38
190
User
Mountain View
38
191
Will Edwards
San Diego
38
192
Mariya Goldman
Pasadena
38
193
User
Oakland
38
194
User
San Francisco
37
195
Pandashadow
Cupertino
37
196
User
Mountain View
37
197
User
Westminster
37
198
Gordon He
Los Angeles
37
199
Cybil Xu
Santa Clara
37
200
User
Los Angeles
37
201
User
Palo Alto
37
202
Amy Kermott
Oakland
37
203
User
Mountain View
37
204
User
San Diego
37
205
User
Stanford
37
206
User
Los Angeles
37
207
User
East Los Angeles
37
208
User
Los Angeles
36
209
User
Westminster
36
210
User
Garden Grove
36
211
User
Berkeley
36
212
User
San Jose
36
213
tanya.rizvi@edelweissfin.com
Los Angeles
36
214
Gavin Liang
San Francisco
36
215
User
San Diego
36
216
User
Los Angeles
36
217
User
Los Angeles
35
218
User
Santa Monica
35
219
Emilio Barbosa
San Mateo
35
220
Tom Rafanan
Los Angeles
35
221
User
Los Angeles
35
222
Steven Ircha
San Francisco
35
223
User
Santa Clara
35
224
User
Sacramento
35
225
User
Los Angeles
34
226
User
Oakland
34
227
User
Westminster
34
228
User
Stockton
34
229
User
Stanton
34
230
User
Sacramento
34
231
User
Riverside
34
232
User
Menlo Park
34
233
User
Westminster
34
234
User
Burlingame
34
235
User
Sacramento
34
236
Pyarimunshi Munshi
San Diego
34
237
Cynthia E
Santa Clara
34
238
User
San Francisco
34
239
Saumya Singhal
Palo Alto
34
240
User
San Francisco
34
241
Amogh Prakash
Cupertino
34
242
Xi Chen
Berkeley
33
243
Blasted Xcel
San Jose
33
244
User
San Francisco
33
245
Sereena
Los Angeles
33
246
Trevor Myers
Tustin
33
247
Stephen Atwood
Mountain View
33
248
User
San Francisco
33
249
Dennis
San Francisco
33
250
User
Costa Mesa
32
251
User
Tracy
32
252
NJ
San Diego
32
253
Mikko Rouhiainen
Los Angeles
32
254
User
San Jose
32
255
User
Oakland
32
256
User
Rocklin
32
257
User
Westminster
31
258
Ryan Kuroiwa
Los Angeles
31
259
User
Hacienda Heights
31
260
Steezy Wang
Upland
31
261
Joel Kang
Diamond Bar
31
262
User
Stockton
31
263
Inc Src
San Francisco
31
264
Vincent Lee
Cerritos
31
265
User
San Diego
31
266
User
Stanford
31
267
Urvi Sharma
Newark
31
268
Sean Leary
Santa Monica
31
269
User
San Jose
31
270
User
Folsom
30
271
User
Hacienda Heights
30
272
User
San Jose
30
273
Michael Luu
Garden Grove
30
274
User
Berkeley
30
275
User
Palo Alto
30
276
David Salvia
Santa Barbara
30
277
User
San Jose
30
278
User
Chula Vista
30
279
SP
Los Angeles
30
280
User
Sunnyvale
30
281
User
Chino
30
282
User
San Diego
30
283
User
Los Angeles
30
284
Joel Smith
Windsor
30
285
User
Stanford
30
286
User
Santa Rosa
30
287
User
San Jose
29
288
User
Salinas
29
289
User
San Francisco
29
290
User
Los Angeles
29
291
User
Culver City
29
292
User
North Auburn
29
293
Susie
Santa Ana
29
294
User
San Francisco
29
295
User
Santee
29
296
User
Santa Clarita
29
297
User
San Francisco
29
298
User
Los Angeles
29
299
Aaron Gonzales
Costa Mesa
29
300
User
San Diego
29
301
User
San Jose
29
302
User
Sacramento
29
303
User
Berkeley
29
304
User
Pomona
29
305
User
Richmond
29
306
User
Santa Clara
28
307
pizza
Oakland
28
308
Brandon Winner
Redondo Beach
28
309
User
Irvine
28
310
User
Pomona
28
311
User
San Jose
28
312
User
Garden Grove
28
313
User
Los Angeles
28
314
User
Sacramento
28
315
Adam Fenyvesi
Saratoga
28
316
Treeskrub
Garden Grove
28
317
User
Rancho Cucamonga
28
318
Grace
San Jose
28
319
Indy Behr
Rancho Mirage
28
320
User
Daly City
28
321
User
San Diego
28
322
User
Berkeley
28
323
User
San Francisco
28
324
User
San Ramon
28
325
User
Claremont
28
326
Albert Wu
San Diego
28
327
User
Hercules
28
328
User
North Auburn
28
329
User
Foster City
27
330
sc
Los Angeles
27
331
User
Santa Ana
27
332
User
San Jose
27
333
User
Santa Rosa
27
334
User
Berkeley
27
335
User
Norwalk
27
336
User
Berkeley
27
337
William Neo
Santa Cruz
27
338
User
Riverside
27
339
User
Claremont
27
340
User
Fairfield
27
341
User
Los Angeles
27
342
User
Hayward
27
343
Alice
Berkeley
27
344
User
Irvine
27
345
User
Santa Ana
27
346
User
Fremont
27
347
Joshua Jelley
San Bernardino
27
348
Renee C
Los Angeles
27
349
User
San Francisco
27
350
User
Concord
27
351
Andrew
Fremont
26
352
User
Los Angeles
26
353
User
Fremont
26
354
Isis Hope McGrew
Anaheim
26
355
User
Menlo Park
26
356
User
San Francisco
26
357
User
Fullerton
26
358
User
Sacramento
26
359
User
San Jose
26
360
Corruptful Boins
Rocklin
26
361
User
San Jose
26
362
User
San Diego
26
363
User
Westminster
26
364
User
Foster City
26
365
Test Demo
Ontario
26
366
Zhiwei Jing
Berkeley
26
367
Monica LW
Rocklin
26
368
Collin Kennedy
Los Angeles
26
369
Braedon Hom
San Diego
26
370
Chez
Fremont
26
371
User
Sacramento
26
372
User
Santa Clara
25
373
User
Los Angeles
25
374
User
Pleasant Hill
25
375
User
South Pasadena
25
376
B Thompson
Redwood City
25
377
User
Diamond Bar
25
378
User
Los Angeles
25
379
User
Oakland
25
380
User
Cupertino
25
381
User
Sunnyvale
25
382
Chenie
Oakland
25
383
User
Westminster
25
384
User
Sacramento
25
385
User
Manhattan Beach
25
386
User
Los Angeles
25
387
User
Los Angeles
25
388
Sneh Patel
Los Angeles
25
389
GoMath
San Francisco
25
390
User
Los Angeles
25
391
leo naville
Oakland
25
392
User
Westminster
25
393
User
Irvine
24
394
User
La Canada Flintridge
24
395
User
Long Beach
24
396
hey i was the jeff
Seal Beach
24
397
User
Corona
24
398
User
Pittsburg
24
399
User
Chula Vista
24
400
amner
Isla Vista
24
401
User
Mountain View
24
402
Mohit
San Diego
24
403
User
San Jose
24
404
User
Placerville
24
405
User
Fountain Valley
24
406
User
Foster City
24
407
User
San Francisco
24
408
User
Westminster
24
409
User
Riverside
24
410
User
Westminster
24
411
User
Bakersfield
24
412
Justin Amakor
Lynwood
24
413
User
Westminster
24
414
User
Roseville
24
415
User
Rocklin
24
416
User
Concord
24
417
Andy Lee
Irvine
24
418
Rick Ashley
Rancho Cucamonga
24
419
Marc Mossotti
Los Angeles
24
420
User
Westminster
24
421
RubberDuck Trinity
Elk Grove
24
422
Yahaira Aguilar
San Francisco
23
423
User
Mountain View
23
424
User
Irvine
23
425
User
Davis
23
426
Dylan Smallcomb
San Francisco
23
427
Rit Vik
Milpitas
23
428
User
Manhattan Beach
23
429
User
Fremont
23
430
User
Antioch
23
431
User
Elk Grove
23
432
User
Riverside
23
433
Sagayamary Sagayaradj
Davis
23
434
Hayden Jacobs
Los Angeles
23
435
User
Stanford
23
436
User
Pasadena
23
437
User
Los Angeles
23
438
User
San Diego
23
439
User
San Jose
23
440
Floxer Green
Alameda
23
441
User
Irvine
23
442
Ima Unicorn 🦄
Fullerton
23
443
User
Milpitas
23
444
User
Riverside
23
445
Garrick Liang
Riverside
23
446
User
San Diego
22
447
User
Irvine
22
448
User
San Francisco
22
449
User
Livermore
22
450
User
San Francisco
22
451
Aaron Chu
San Diego
22
452
User
East Los Angeles
22
453
Preethi Merugumala
Fremont
22
454
User
Arcadia
22
455
User
San Francisco
22
456
User
Los Angeles
22
457
Counter Kazan
San Jose
22
458
User
Oakland
22
459
User
Los Angeles
22
460
User
Los Angeles
22
461
Stephen Kouch
Cherry Valley
22
462
User
San Francisco
22
463
User
Pomona
22
464
User
Long Beach
22
465
User
Rowland Heights
22
466
User
San Jose
22
467
User
Los Angeles
22
468
User
San Luis Obispo
22
469
User
Seaside
22
470
User
San Diego
22
471
Jake Brown
Oakland
22
472
iris
San Francisco
22
473
User
Camp Pendleton North
22
474
Tamur Khan
San Jose
22
475
Cherry H
Poway
22
476
Steven Goltra
Westminster
22
477
User
Chula Vista
22
478
Caroline T
Montclair
22
479
User
Santa Monica
22
480
SaumyaS
Palo Alto
22
481
User
San Jose
22
482
lirtosiast
San Diego
22
483
Saransh Gupta
Mountain View
22
484
Angry DictatorGames
San Jose
22
485
User
Carson
21
486
User
Berkeley
21
487
User
Berkeley
21
488
User
Los Angeles
21
489
User
Auburn
21
490
Matias Lopez
Berkeley
21
491
Arthur T
Oakland
21
492
User
Irvine
21
493
User
Fullerton
21
494
User
Oxnard
21
495
User
Los Angeles
21
496
User
Milpitas
21
497
User
Los Angeles
21
498
User
Los Angeles
21
499
Hisham Seddiqee
Concord
21
500
User
Mountain View
21
501
User
Oakland
21
502
User
Santa Ana
21
503
User
Los Angeles
21
504
Michael Karr
Stanford
21
505
User
Oakland
21
506
User
Rowland Heights
21
507
User
Pomona
21
508
User
Elk Grove
21
509
User
Los Angeles
21
510
James Thiering
Mckinleyville
21
511
trung tran
San Jose
21
512
User
Santa Cruz
21
513
User
Los Angeles
21
514
User
Stanford
21
515
User
Rancho Cucamonga
21
516
User
Los Angeles
21
517
User
Los Angeles
21
518
User
Yucca Valley
21
519
User
Huntington Beach
21
520
User
Los Angeles
21
521
Alyssa
San Francisco
21
522
User
Riverside
21
523
User
San Francisco
21
524
User
Rocklin
20
525
User
San Jose
20
526
User
Riverside
20
527
William Edwards
San Diego
20
528
Sol Lee
Los Angeles
20
529
User
Concord
20
530
Chad Hutchinson
Menlo Park
20
531
martina cerina
San Francisco
20
532
User
West Covina
20
533
Brandan Tran
San Diego
20
534
User
Los Angeles
20
535
CH
Los Angeles
20
536
User
Berkeley
20
537
Arthur
Brea
20
538
User
San Diego
20
539
User
Irvine
20
540
User
San Diego
20
541
Yunfan Yang
Alhambra
20
542
User
San Clemente
20
543
User
San Francisco
20
544
User
San Jose
20
545
User
San Jose
20
546
John Smith
Compton
20
547
User
San Jose
20
548
User
Los Angeles
20
549
User
San Jose
20
550
User
San Francisco
20
551
Takahiro Fushimi
Menlo Park
20
552
Georges Ip
Los Angeles
20
553
User
Santa Clara
20
554
Rianna Chu
Monrovia
20
555
Anterpreet Kaur Josen
San Jose
20
556
User
Livermore
20
557
User
San Diego
20
558
User
Los Angeles
20
559
User
Concord
20
560
User
Long Beach
20
561
User
Los Angeles
20
562
User
Sunnyvale
20
563
User
Sacramento
20
564
User
San Jose
20
565
User
Hawthorne
19
566
User
Fresno
19
567
Richella
Chula Vista
19
568
Katherine Tran
Chula Vista
19
569
User
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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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