Leaderboard
California
Rank
Level
1
Timothy So
Westminster
99
2
User
San Francisco
93
3
FelixM
San Diego
93
4
Darren Chen
Walnut
92
5
Gordon He
Los Angeles
89
6
user
Oakland
88
7
Andrew Stepp
Los Angeles
87
8
Oo
Los Angeles
86
9
User
Los Angeles
86
10
User
Sacramento
86
11
User
Manhattan Beach
85
12
Sneh Patel
Los Angeles
85
13
User
Stanford
83
14
Matias
Marina Del Rey
83
15
Иван Мирошниченко
Carmichael
83
16
Nes370
West Sacramento
82
17
Jordan Katz
Oakland
82
18
User
Livermore
81
19
User
Berkeley
81
20
Vincent Lee
Cerritos
81
21
robert hou
Berkeley
79
22
User
Oakland
79
23
User
Berkeley
79
24
User
Los Angeles
78
25
Dean Lee
Santa Clara
77
26
Shelley Chang
Alameda
77
27
User
Sonoma
76
28
Jeff Rangan
Escondido
76
29
CellPhoneJeff
Seal Beach
76
30
ag ar
Santa Clarita
75
31
User
Westminster
75
32
Long John
San Mateo
74
33
User
Westminster
74
34
User
Cupertino
74
35
User
San Diego
73
36
Kevin Nguyen
Santa Clara
73
37
User
San Jose
72
38
User
Westminster
72
39
User
Roseville
72
40
Mason Lien
Cupertino
72
41
Xi Chen
Berkeley
72
42
Cannon Tuttle
San Ramon
71
43
Zavosh
Altadena
71
44
User
San Diego
71
45
User
Palo Alto
71
46
User
Irvine
71
47
User
Sacramento
71
48
User
Los Angeles
70
49
User
Santa Clara
70
50
Takahiro Fushimi
Menlo Park
70
51
User
San Mateo
69
52
User
Stanford
69
53
User
Los Angeles
69
54
Sushant Vema
Pleasanton
69
55
User
Westminster
68
56
User
Oakland
67
57
User
Cupertino
66
58
User
Garden Grove
65
59
User
Foster City
65
60
User
Stanford
65
61
User
San Diego
65
62
I got many skills but failing is not one
Mountain House
65
63
User
Los Angeles
65
64
User
Westminster
65
65
User
Westminster
65
66
User
Westminster
65
67
User
Stanford
65
68
Joel Smith
Windsor
64
69
User
Los Angeles
64
70
Amineh Beltran
Oakland
63
71
User
San Jose
63
72
User
San Jose
63
73
pettni
Pasadena
62
74
User
San Diego
62
75
User
Palo Alto
62
76
User
Stanford
62
77
User
Santa Ana
62
78
User
Fountain Valley
62
79
User
Claremont
61
80
User
San Diego
61
81
User
Claremont
61
82
Hoang
Los Angeles
61
83
User
Los Angeles
61
84
User
Concord
61
85
User
Los Angeles
61
86
Katherine Tran
Chula Vista
61
87
User
Folsom
60
88
pizza
Oakland
60
89
User
Palm Desert
60
90
User
San Diego
59
91
User
Westminster
59
92
User
Santa Clara
59
93
User
Berkeley
59
94
User
Los Angeles
58
95
User
San Francisco
58
96
User
Los Angeles
58
97
User
Sacramento
58
98
User
Riverside
58
99
Andrew
Fremont
58
100
Andy Lee
Irvine
58
101
User
Los Angeles
58
102
Vincent Zhang
San Ramon
57
103
User
Lafayette
57
104
User
Los Angeles
57
105
User
Los Angeles
57
106
User
Westminster
57
107
User
Rowland Heights
57
108
User
Los Angeles
57
109
User
Oakland
56
110
User
San Francisco
56
111
Anthony Lam
Foster City
56
112
User
Culver City
56
113
User
Fresno
56
114
Pandashadow
Cupertino
56
115
User
Los Angeles
56
116
Alice
Berkeley
56
117
User
Irvine
56
118
Xander Bystrom
Cypress
56
119
User
San Diego
55
120
User
North Auburn
55
121
User
Santa Ana
55
122
Zhiwei Jing
Berkeley
55
123
User
Visalia
54
124
User
West Sacramento
54
125
User
San Jose
54
126
Will Edwards
San Diego
54
127
User
Pleasanton
54
128
User
Stanton
54
129
karanbir Singh
Long Beach
54
130
Rohan Jasani
Stanford
54
131
shane ramsey
Lodi
53
132
SPicy
Redwood City
53
133
User
Westminster
53
134
User
Westminster
53
135
User
Los Angeles
53
136
Isabelle
Santa Clarita
53
137
Jeremy Kintana
Santa Clara
52
138
User
San Diego
52
139
K L
Los Angeles
52
140
User
Los Angeles
52
141
User
Oakland
52
142
Amogh Prakash
Cupertino
52
143
Mikko Rouhiainen
Los Angeles
52
144
User
Stockton
52
145
David Salvia
Santa Barbara
52
146
leo naville
Oakland
52
147
Jon Batscha
Simi Valley
52
148
Matt Lunde
Stanford
52
149
User
San Jose
52
150
BOI
Dublin
51
151
User
Chula Vista
51
152
User
Westminster
51
153
User
San Diego
51
154
User
Santa Monica
51
155
Tom Rafanan
Los Angeles
51
156
Joel Kang
Diamond Bar
51
157
User
Los Angeles
51
158
Susie
Santa Ana
51
159
User
Los Angeles
51
160
User
Irvine
51
161
Stephen Kouch
Cherry Valley
51
162
User
San Jose
50
163
User
San Diego
50
164
User
Pomona
50
165
User
Fairfield
50
166
Renee C
Los Angeles
50
167
hey i was the jeff
Seal Beach
50
168
Will Lahera
Beverly Hills
50
169
User
Santa Clara
50
170
Ekta Patel
East Los Angeles
50
171
User
Chula Vista
49
172
User
Santa Clarita
49
173
User
Westminster
49
174
User
Walnut Creek
49
175
User
Cupertino
49
176
User
San Diego
49
177
Emily Trivedi
Los Angeles
49
178
User
Los Angeles
48
179
User
San Francisco
48
180
User
San Francisco
48
181
User
San Francisco
48
182
User
Oakland
48
183
User
Riverside
48
184
User
Stanford
47
185
User
Stockton
47
186
Ryan Kuroiwa
Los Angeles
47
187
William Neo
Santa Cruz
47
188
User
Riverside
47
189
User
San Diego
47
190
Saransh Gupta
Mountain View
47
191
User
Rancho Cucamonga
47
192
User
Carlsbad
47
193
Kellie VanBrunt
Los Angeles
47
194
Cybil Xu
Santa Clara
46
195
Landon
Indio
46
196
User
Sunnyvale
46
197
Jake Brown
Oakland
46
198
User
Irvine
46
199
Alillia
Stanford
46
200
User
Castro Valley
46
201
User
Los Angeles
46
202
User
Los Angeles
46
203
User
San Francisco
46
204
User
East Los Angeles
45
205
Stephen Atwood
Mountain View
45
206
Dennis
San Francisco
45
207
User
Los Angeles
45
208
User
Oakland
45
209
SaumyaS
Palo Alto
45
210
Matias Lopez
Berkeley
45
211
User
Rowland Heights
45
212
trung tran
San Jose
45
213
Dylan
Anaheim
45
214
jade rainbow
Fremont
45
215
User
Walnut Creek
45
216
User
Loomis
45
217
User
Stockton
45
218
Gavin Liang
San Francisco
44
219
User
San Diego
44
220
User
Fullerton
44
221
B Thompson
Redwood City
44
222
User
Los Angeles
44
223
User
Beverly Hills
44
224
Derek Wang
Cupertino
44
225
BJJ Boy
Fremont
44
226
User
Stanford
44
227
User
Stanford
44
228
Haruki Kosaki
Chula Vista
44
229
User
Ventura
44
230
Mark Neumann
La Canada Flintridge
44
231
NJ
San Diego
43
232
Inc Src
San Francisco
43
233
Michael Luu
Garden Grove
43
234
User
San Diego
43
235
User
Los Angeles
43
236
User
Berkeley
43
237
User
Berkeley
43
238
User
Huntington Park
43
239
User
Los Angeles
43
240
User
Los Angeles
43
241
Julia Schwieger
Westminster
43
242
User
San Diego
43
243
User
San Carlos
43
244
User
San Jose
43
245
User
Redwood City
43
246
Tyler Pak
Fullerton
42
247
User
Chula Vista
42
248
User
Santa Clarita
42
249
Adam Fenyvesi
Saratoga
42
250
User
La Canada Flintridge
42
251
User
Riverside
42
252
User
San Diego
42
253
Tamur Khan
San Jose
42
254
Caroline T
Montclair
42
255
User
Redwood City
42
256
User
Walnut Creek
42
257
IPlayToWin
Irvine
42
258
User
Los Angeles
42
259
Anaika Bedi
Santa Clara
42
260
Sid R
Dublin
42
261
Daniyar Seisenov
Lake Forest
42
262
Steezy Wang
Upland
41
263
Charlie Do
Union City
41
264
User
Pomona
41
265
User
Garden Grove
41
266
User
Los Angeles
41
267
User
Westminster
41
268
User
Los Angeles
41
269
Mohit
San Diego
41
270
User
Livermore
41
271
Chad Hutchinson
Menlo Park
41
272
User
Santa Ana
41
273
User
City
41
274
User
Stanford
41
275
User
Los Angeles
41
276
User
Richmond
41
277
Celina
Irvine
41
278
User
Los Angeles
41
279
User
Berkeley
40
280
Isis Hope McGrew
Anaheim
40
281
User
Los Angeles
40
282
User
Riverside
40
283
User
Irvine
40
284
Garrick Liang
Riverside
40
285
User
Pomona
40
286
User
West Covina
40
287
User
Berkeley
40
288
Rick Nott
Beverly Hills
40
289
User
North Tustin
40
290
User
San Francisco
39
291
User
Santa Monica
39
292
User
San Jose
39
293
User
Rancho Cucamonga
39
294
Indy Behr
Rancho Mirage
39
295
Rex Bodoia
Claremont
39
296
User
Riverside
39
297
User
San Luis Obispo
39
298
lirtosiast
San Diego
39
299
User
Long Beach
39
300
User
San Jose
39
301
User
Berkeley
39
302
User
San Francisco
39
303
User
East Los Angeles
39
304
User
Berkeley
39
305
User
Stanford
39
306
Vyassa Baratham
Berkeley
39
307
Kendra
Redwood City
39
308
User
Dublin
39
309
Reva
Fullerton
38
310
User
Los Angeles
38
311
User
Claremont
38
312
Corruptful Boins
Rocklin
38
313
User
Westminster
38
314
User
Elk Grove
38
315
Sol Lee
Los Angeles
38
316
User
Alta Sierra
38
317
User
San Jose
38
318
User
San Jose
38
319
User
Pleasanton
38
320
User
San Francisco
38
321
User
Los Angeles
38
322
User
South San Francisco
38
323
User
Chula Vista
38
324
User
Santa Ana
38
325
User
Berkeley
38
326
User
Los Angeles
38
327
User
Irvine
38
328
User
Salinas
37
329
Harry Ackley
Menlo Park
37
330
User
Irvine
37
331
Yahaira Aguilar
San Francisco
37
332
User
Los Angeles
37
333
User
San Francisco
37
334
User
Berkeley
37
335
User
Castro Valley
37
336
La Sha
San Francisco
37
337
User
San Ramon
37
338
ThatMathNerd
Nevada City
37
339
User
Sunnyvale
37
340
User
Claremont
37
341
Cameron Lay
La Canada Flintridge
37
342
User
Los Angeles
37
343
User
Menlo Park
37
344
User
Los Angeles
37
345
User
Culver City
36
346
User
San Diego
36
347
User
Claremont
36
348
User
Hayward
36
349
User
Westminster
36
350
User
Santa Monica
36
351
User
Riverside
36
352
User
Rocklin
36
353
User
Los Angeles
36
354
User
Los Angeles
36
355
TheCoolTed
Danville
36
356
Justin O'Donnell
San Clemente
36
357
User
Isla Vista
36
358
User
Berkeley
36
359
User
Claremont
36
360
Erika
Palo Alto
36
361
User
Berkeley
36
362
User
Los Angeles
36
363
User
Los Angeles
36
364
User
San Jose
36
365
User
Palo Alto
36
366
User
San Diego
36
367
User
San Diego
36
368
User
Diamond Bar
35
369
User
Westminster
35
370
User
Roseville
35
371
User
San Diego
35
372
James Thiering
Mckinleyville
35
373
User
Riverside
35
374
William Edwards
San Diego
35
375
User
Livermore
35
376
User
Fallbrook
35
377
User
City
35
378
User
Sunnyvale
35
379
Gabriela Smith
Windsor
35
380
User
Los Angeles
35
381
User
Oakland
35
382
User
Eastvale
35
383
User
Los Angeles
35
384
User
North Auburn
35
385
User
Salinas
35
386
User
Cupertino
35
387
User
San Francisco
35
388
Joshua Jelley
San Bernardino
34
389
Braedon Hom
San Diego
34
390
Angry DictatorGames
San Jose
34
391
User
Los Angeles
34
392
User
Riverside
34
393
User
Riverside
34
394
User
Alta Sierra
34
395
User
Santa Monica
34
396
User
Irvine
34
397
User
Los Angeles
34
398
User
Pasadena
34
399
User
San Diego
34
400
User
Orange
34
401
yelp
San Francisco
34
402
Sean Leary
Santa Monica
34
403
User
Los Angeles
34
404
Ritvik Banakar
San Jose
34
405
User
Ontario
34
406
User
Pasadena
34
407
User
East Palo Alto
34
408
User
Redlands
34
409
User
Claremont
34
410
User
South San Francisco
34
411
User
Stanford
34
412
User
Baldwin Park
34
413
User
Redwood City
34
414
Steven Hadley
Arcata
34
415
User
Walnut Creek
34
416
User
Albany
34
417
User
Los Angeles
34
418
User
San Diego
34
419
User
Fremont
33
420
User
San Jose
33
421
User
San Jose
33
422
Chez
Fremont
33
423
User
Westminster
33
424
User
Mountain View
33
425
User
San Francisco
33
426
User
Oxnard
33
427
User
San Jose
33
428
Pipi Ding
Cupertino
33
429
Mark Neumann
La Canada Flintridge
33
430
User
Los Angeles
33
431
User
Los Angeles
33
432
User
San Leandro
33
433
Collin Kennedy
Los Angeles
32
434
User
Pleasant Hill
32
435
User
Chula Vista
32
436
User
Westminster
32
437
Marc Mossotti
Los Angeles
32
438
RubberDuck Trinity
Elk Grove
32
439
User
Pasadena
32
440
User
Long Beach
32
441
Steven Goltra
Westminster
32
442
User
Auburn
32
443
Hisham Seddiqee
Concord
32
444
User
Fullerton
32
445
J D
San Luis Obispo
32
446
User
Riverside
32
447
User
Riverside
32
448
User
Alta Sierra
32
449
Crystal
Santa Clara
32
450
User
San Francisco
32
451
Caden Pascual
Oakland
32
452
leo naville
Oakland
32
453
User
San Jose
32
454
User
Union City
32
455
User
Los Angeles
32
456
User
Los Angeles
32
457
User
Cerritos
32
458
mcgregian
San Anselmo
32
459
User
Berkeley
32
460
User
San Francisco
32
461
jacob berman
Los Angeles
32
462
User
Los Angeles
32
463
User
Sunnyvale
32
464
User
Los Angeles
32
465
User
Bakersfield
32
466
User
Los Angeles
32
467
User
Los Angeles
32
468
Chenie
Oakland
31
469
Aaron Chu
San Diego
31
470
User
Carson
31
471
User
Fullerton
31
472
Michael Karr
Stanford
31
473
User
Concord
31
474
Anterpreet Kaur Josen
San Jose
31
475
Stephen Hansen
Redondo Beach
31
476
User
Redwood City
31
477
User
Carmichael
31
478
User
Berkeley
31
479
User
Lynwood
31
480
User
Lake Elsinore
31
481
User
Orinda
31
482
Eric Perez
Los Angeles
31
483
User
East Palo Alto
31
484
User
Riverside
31
485
K. Rose
Los Angeles
31
486
User
Oakland
31
487
User
San Francisco
31
488
User
Richmond
31
489
User
Mountain View
31
490
User
Sacramento
31
491
User
San Diego
30
492
Preethi Merugumala
Fremont
30
493
Arthur T
Oakland
30
494
User
Oakland
30
495
User
Concord
30
496
User
Berkeley
30
497
User
Long Beach
30
498
N-JOI
San Francisco
30
499
User
Claremont
30
500
User
San Jose
30
501
User
Santa Monica
30
502
User
Riverside
30
503
User
San Jose
30
504
User
Oakland
30
505
User
Berkeley
30
506
User
Newcastle
30
507
User
East Palo Alto
30
508
User
Los Angeles
30
509
User
San Francisco
30
510
User
Fullerton
30
511
Ory Garmy
Cameron Park
30
512
Henok Tesfamariam
Oakland
30
513
User
Fremont
30
514
User
San Francisco
30
515
User
Los Angeles
30
516
User
Santa Cruz
30
517
Allen Patrick Hosler
Palo Alto
30
518
User
Escondido
30
519
Brian
Fremont
30
520
User
Irvine
29
521
User
Antioch
29
522
User
San Jose
29
523
User
East Los Angeles
29
524
User
Arcadia
29
525
iris
San Francisco
29
526
Alyssa
San Francisco
29
527
CH
Los Angeles
29
528
User
San Diego
29
529
User
San Jose
29
530
remy leutenegger
Davis
29
531
User
Concord
29
532
User
Pomona
29
533
Erique Alcaraz Estrada
Sonoma
29
534
User
La Quinta
29
535
User
Riverside
29
536
User
San Luis Obispo
29
537
User
San Diego
29
538
User
San Mateo
29
539
User
Berkeley
29
540
User
Oakland
29
541
Si Jae Teh
Berkeley
29
542
User
Cupertino
29
543
User
San Jose
29
544
User
Mountain View
29
545
User
Cypress
29
546
User
San Jose
29
547
User
City
29
548
User
San Francisco
29
549
User
City
29
550
User
Los Angeles
29
551
User
San Francisco
29
552
User
South San Francisco
29
553
User
Los Angeles
29
554
User
Westminster
29
555
User
Los Angeles
29
556
TINSLEY ZHU
Berkeley
29
557
User
El Dorado Hills
29
558
User
Richmond
29
559
User
Los Angeles
29
560
User
Los Angeles
29
561
Counter Kazan
San Jose
28
562
Brandon Winner
Redondo Beach
28
563
User
San Jose
28
564
User
Milpitas
28
565
Georges Ip
Los Angeles
28
566
Colin O'Neill
Pomona
28
567
User
Rancho Santa Margarita
28
568
User
Montclair
28
569
User
Cupertino
28
570
User
Pleasant Hill
28
571
Angelo Gamero Bardinet
Los Angeles
28
572
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 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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