Leaderboard
California
Rank
Level
1
FelixM
San Diego
93
2
Gordon He
Los Angeles
89
3
user
Oakland
88
4
Andrew Stepp
Los Angeles
87
5
User
Sacramento
86
6
User
Hermosa Beach
85
7
Sneh Patel
Los Angeles
85
8
Timothy So
Westminster
83
9
Иван Мирошниченко
Carmichael
83
10
Sora Ngin
Gilroy
82
11
Vincent Lee
Cerritos
81
12
User
Oakland
79
13
User
Berkeley
79
14
User
Los Angeles
78
15
Shelley Chang
Alameda
77
16
CellPhoneJeff
Seal Beach
76
17
Kevin Nguyen
Santa Clara
73
18
Mason Lien
Cupertino
72
19
Xi Chen
Berkeley
72
20
Matias
Los Angeles
71
21
User
Palo Alto
71
22
User
Irvine
71
23
User
Sacramento
71
24
User
Los Angeles
70
25
User
Santa Clara
70
26
Takahiro Fushimi
Menlo Park
70
27
Sushant Vema
Pleasanton
69
28
User
Westminster
68
29
User
Sonoma
67
30
User
Oakland
67
31
Justin D'Souza
Los Angeles
67
32
User
Cupertino
66
33
User
Los Angeles
65
34
User
Westminster
65
35
User
Westminster
65
36
User
Westminster
65
37
User
Stanford
65
38
Joel Smith
Windsor
64
39
User
Los Angeles
64
40
User
San Jose
63
41
User
San Jose
63
42
User
Palo Alto
62
43
User
Stanford
62
44
User
Santa Ana
62
45
User
Fountain Valley
62
46
User
Cupertino
61
47
User
Los Angeles
61
48
User
Roseville
61
49
User
Concord
61
50
User
Los Angeles
61
51
Katherine Tran
Chula Vista
61
52
pizza
Oakland
60
53
User
Palm Desert
60
54
User
Westminster
59
55
User
Santa Clara
59
56
User
Berkeley
59
57
Amineh Beltran
Oakland
58
58
User
Sacramento
58
59
User
Riverside
58
60
Andrew
Fremont
58
61
Andy Lee
Irvine
58
62
User
Los Angeles
58
63
User
Westminster
57
64
User
Los Angeles
57
65
I got many skills but failing is not one
Mountain House
57
66
User
Westminster
57
67
User
Pleasanton
57
68
User
Rowland Heights
57
69
User
Los Angeles
57
70
Alice
Berkeley
56
71
User
Irvine
56
72
Xander Bystrom
Cypress
56
73
Mark Durbin
Patterson
55
74
User
North Auburn
55
75
User
Santa Ana
55
76
Zhiwei Jing
Berkeley
55
77
Travis Grady
Manhattan Beach
54
78
User
Stanton
54
79
karanbir Singh
Long Beach
54
80
Rohan Jasani
Stanford
54
81
User
Westminster
53
82
User
Westminster
53
83
User
Los Angeles
53
84
Isabelle
Santa Clarita
53
85
Amogh Prakash
Cupertino
52
86
User
Westminster
52
87
Mikko Rouhiainen
Los Angeles
52
88
User
Stockton
52
89
David Salvia
Santa Barbara
52
90
leo naville
Oakland
52
91
Jon Batscha
Simi Valley
52
92
Matt Lunde
Stanford
52
93
User
San Jose
52
94
Joel Kang
Diamond Bar
51
95
User
Los Angeles
51
96
Susie
Santa Ana
51
97
Kaylie
City
51
98
User
Los Angeles
51
99
User
Irvine
51
100
Stephen Kouch
Cherry Valley
51
101
Will Edwards
San Diego
50
102
hey i was the jeff
Seal Beach
50
103
Will Lahera
Beverly Hills
50
104
User
Santa Clara
50
105
Ekta Patel
East Los Angeles
50
106
SPicy
Redwood City
49
107
User
Westminster
49
108
User
Walnut Creek
49
109
User
Cupertino
49
110
User
San Diego
49
111
Emily Trivedi
Los Angeles
49
112
User
San Francisco
48
113
User
Oakland
48
114
User
Riverside
48
115
William Neo
Santa Cruz
47
116
User
Riverside
47
117
User
San Diego
47
118
Saransh Gupta
Mountain View
47
119
User
Rancho Cucamonga
47
120
Jihad Dagher
Los Angeles
47
121
User
Carlsbad
47
122
Kellie VanBrunt
Los Angeles
47
123
Jordan Katz
Oakland
46
124
User
Sunnyvale
46
125
Jake Brown
Oakland
46
126
User
Irvine
46
127
Alillia
Stanford
46
128
User
Castro Valley
46
129
User
Los Angeles
46
130
User
Los Angeles
46
131
User
San Francisco
46
132
Diaris Alexander
Los Angeles
45
133
Steezy Wang
Upland
45
134
User
Oakland
45
135
SaumyaS
Palo Alto
45
136
Matias Lopez
Berkeley
45
137
User
Rowland Heights
45
138
trung tran
San Jose
45
139
Dylan
Anaheim
45
140
jade rainbow
Fremont
45
141
User
Walnut Creek
45
142
User
Loomis
45
143
User
Stockton
45
144
User
Los Angeles
45
145
User
Fullerton
44
146
B Thompson
Redwood City
44
147
User
Los Angeles
44
148
User
Beverly Hills
44
149
Derek Wang
Cupertino
44
150
BJJ Boy
Fremont
44
151
User
Stanford
44
152
User
Stanford
44
153
Liam Barnard
Berkeley
44
154
Haruki Kosaki
Chula Vista
44
155
User
Ventura
44
156
Mark Neumann
La Canada Flintridge
44
157
Renee C
Los Angeles
43
158
User
Los Angeles
43
159
User
Berkeley
43
160
User
Berkeley
43
161
User
Huntington Park
43
162
User
Los Angeles
43
163
User
Los Angeles
43
164
Julia Schwieger
Westminster
43
165
User
San Diego
43
166
User
San Carlos
43
167
User
San Jose
43
168
User
Redwood City
43
169
User
La Canada Flintridge
42
170
User
Riverside
42
171
User
San Diego
42
172
Tamur Khan
San Jose
42
173
Caroline T
Montclair
42
174
User
Redwood City
42
175
User
Walnut Creek
42
176
IPlayToWin
Irvine
42
177
User
Los Angeles
42
178
Anaika Bedi
Santa Clara
42
179
Sid R
Dublin
42
180
Daniyar Seisenov
Lake Forest
42
181
User
Los Angeles
41
182
User
Westminster
41
183
User
Los Angeles
41
184
Mohit
San Diego
41
185
User
Livermore
41
186
Chad Hutchinson
Menlo Park
41
187
User
Santa Ana
41
188
User
City
41
189
User
Stanford
41
190
User
Los Angeles
41
191
User
Richmond
41
192
Celina
Irvine
41
193
User
Los Angeles
41
194
William Jones
San Francisco
41
195
User
Los Angeles
40
196
User
Riverside
40
197
User
Irvine
40
198
Garrick Liang
Riverside
40
199
User
Pomona
40
200
User
West Covina
40
201
User
Berkeley
40
202
Rick Nott
Beverly Hills
40
203
User
North Tustin
40
204
Rex Bodoia
Claremont
39
205
K L
San Diego
39
206
User
Riverside
39
207
User
San Luis Obispo
39
208
lirtosiast
San Diego
39
209
User
Long Beach
39
210
User
San Jose
39
211
User
Berkeley
39
212
User
San Francisco
39
213
User
East Los Angeles
39
214
User
Berkeley
39
215
User
Stanford
39
216
Vyassa Baratham
Berkeley
39
217
Kendra
Redwood City
39
218
User
Dublin
39
219
Anthony Lam
Redwood City
38
220
User
Westminster
38
221
User
Elk Grove
38
222
Sol Lee
Los Angeles
38
223
User
Alta Sierra
38
224
User
San Jose
38
225
User
San Jose
38
226
User
Pleasanton
38
227
User
San Francisco
38
228
User
Los Angeles
38
229
User
South San Francisco
38
230
User
Chula Vista
38
231
User
Santa Ana
38
232
User
Berkeley
38
233
User
Los Angeles
38
234
User
Irvine
38
235
Yahaira Aguilar
San Francisco
37
236
User
Los Angeles
37
237
User
San Francisco
37
238
User
Berkeley
37
239
User
Castro Valley
37
240
La Sha
San Francisco
37
241
User
San Ramon
37
242
ThatMathNerd
Nevada City
37
243
User
Sunnyvale
37
244
User
Claremont
37
245
Cameron Lay
La Canada Flintridge
37
246
User
Los Angeles
37
247
User
Menlo Park
37
248
User
Los Angeles
37
249
User
Santa Monica
36
250
User
Riverside
36
251
User
Rocklin
36
252
User
Los Angeles
36
253
User
Los Angeles
36
254
TheCoolTed
Danville
36
255
Justin O'Donnell
San Clemente
36
256
User
Isla Vista
36
257
User
Berkeley
36
258
User
Claremont
36
259
Erika
Palo Alto
36
260
User
Berkeley
36
261
User
Los Angeles
36
262
User
Los Angeles
36
263
User
San Jose
36
264
User
Palo Alto
36
265
User
San Diego
36
266
User
San Diego
36
267
User
San Diego
35
268
James Thiering
Mckinleyville
35
269
User
Riverside
35
270
William Edwards
San Diego
35
271
User
Livermore
35
272
User
Fallbrook
35
273
User
City
35
274
User
Sunnyvale
35
275
Gabriela Smith
Windsor
35
276
User
Los Angeles
35
277
User
Oakland
35
278
User
Eastvale
35
279
User
Los Angeles
35
280
User
North Auburn
35
281
User
Salinas
35
282
User
Cupertino
35
283
User
San Francisco
35
284
User
Alameda
35
285
User
Los Angeles
35
286
Angry DictatorGames
San Jose
34
287
User
Los Angeles
34
288
Jalan Elcano
Modesto
34
289
User
Riverside
34
290
User
Riverside
34
291
User
Alta Sierra
34
292
User
Santa Monica
34
293
User
Irvine
34
294
User
Los Angeles
34
295
User
Pasadena
34
296
User
San Diego
34
297
User
Orange
34
298
yelp
San Francisco
34
299
Sean Leary
Santa Monica
34
300
User
Los Angeles
34
301
Ritvik Banakar
San Jose
34
302
User
Ontario
34
303
User
Pasadena
34
304
User
East Palo Alto
34
305
User
Redlands
34
306
User
Claremont
34
307
User
South San Francisco
34
308
User
Stanford
34
309
User
Baldwin Park
34
310
User
Redwood City
34
311
Steven Hadley
Arcata
34
312
User
Walnut Creek
34
313
User
Albany
34
314
User
Los Angeles
34
315
User
San Diego
34
316
User
San Francisco
34
317
Charlie Do
Union City
33
318
User
Oxnard
33
319
User
San Jose
33
320
Pipi Ding
Cupertino
33
321
Mark Neumann
La Canada Flintridge
33
322
User
Los Angeles
33
323
User
Los Angeles
33
324
User
San Leandro
33
325
User
San Diego
33
326
Harry Ackley
Menlo Park
32
327
User
Fullerton
32
328
J D
San Luis Obispo
32
329
User
Riverside
32
330
User
Riverside
32
331
User
Alta Sierra
32
332
Crystal
Santa Clara
32
333
User
San Francisco
32
334
Caden Pascual
Oakland
32
335
leo naville
Oakland
32
336
User
San Jose
32
337
User
Union City
32
338
User
Los Angeles
32
339
User
Los Angeles
32
340
User
Cerritos
32
341
mcgregian
San Anselmo
32
342
User
Berkeley
32
343
User
San Francisco
32
344
jacob berman
Los Angeles
32
345
User
Los Angeles
32
346
User
Sunnyvale
32
347
User
Los Angeles
32
348
User
Bakersfield
32
349
User
Los Angeles
32
350
User
Los Angeles
32
351
User
Concord
31
352
Anterpreet Kaur Josen
San Jose
31
353
Stephen Hansen
Redondo Beach
31
354
User
Redwood City
31
355
User
Carmichael
31
356
User
Berkeley
31
357
User
Lynwood
31
358
User
Lake Elsinore
31
359
User
Orinda
31
360
Eric Perez
Los Angeles
31
361
User
East Palo Alto
31
362
User
Riverside
31
363
K. Rose
Los Angeles
31
364
User
Oakland
31
365
User
San Francisco
31
366
User
Richmond
31
367
User
Mountain View
31
368
User
San Jose
31
369
User
Sacramento
31
370
User
Concord
30
371
User
Berkeley
30
372
User
Long Beach
30
373
N-JOI
San Francisco
30
374
User
Claremont
30
375
User
Los Angeles
30
376
User
San Jose
30
377
User
Santa Monica
30
378
User
Riverside
30
379
User
San Jose
30
380
User
Oakland
30
381
User
Berkeley
30
382
User
Newcastle
30
383
User
East Palo Alto
30
384
User
Los Angeles
30
385
User
San Francisco
30
386
User
Fullerton
30
387
Ory Garmy
Cameron Park
30
388
Henok Tesfamariam
Oakland
30
389
User
Fremont
30
390
User
San Francisco
30
391
User
Los Angeles
30
392
User
Santa Cruz
30
393
Allen Patrick Hosler
Palo Alto
30
394
User
Escondido
30
395
Brian
Fremont
30
396
Isis Hope McGrew
San Diego
29
397
User
Concord
29
398
User
Pomona
29
399
Erique Alcaraz Estrada
Sonoma
29
400
User
La Quinta
29
401
User
Riverside
29
402
Adan Quintana [STUDENT]
Los Angeles
29
403
User
San Luis Obispo
29
404
User
San Diego
29
405
User
San Mateo
29
406
User
Berkeley
29
407
User
Oakland
29
408
Si Jae Teh
Berkeley
29
409
User
Cupertino
29
410
User
San Jose
29
411
User
Mountain View
29
412
User
Cypress
29
413
User
San Jose
29
414
User
City
29
415
User
San Francisco
29
416
User
City
29
417
User
Los Angeles
29
418
User
San Francisco
29
419
User
South San Francisco
29
420
User
Los Angeles
29
421
User
Westminster
29
422
User
Los Angeles
29
423
TINSLEY ZHU
Berkeley
29
424
User
El Dorado Hills
29
425
User
Richmond
29
426
User
Los Angeles
29
427
User
Los Angeles
29
428
CH
Carson
28
429
Hisham Seddiqee
Concord
28
430
Benjamin Webb
Irvine
28
431
User
Rocklin
28
432
User
Tiburon
28
433
User
Loomis
28
434
User
Isla Vista
28
435
User
Saratoga
28
436
User
San Luis Obispo
28
437
User
Stanford
28
438
User
Berkeley
28
439
User
San Jose
28
440
User
Santa Ana
28
441
User
Fremont
28
442
Apurva Shekhar
San Jose
28
443
User
Strawberry
28
444
User
San Carlos
28
445
User
Los Angeles
28
446
User
Los Angeles
28
447
Tammy ⚝
Santa Clara
28
448
User
Cupertino
28
449
User
Thousand Oaks
27
450
User
San Diego
27
451
User
Los Angeles
27
452
User
Los Angeles
27
453
User
Los Angeles
27
454
User
Chula Vista
27
455
User
San Francisco
27
456
User
Mountain View
27
457
User
Palm Springs
27
458
User
Los Angeles
27
459
User
Berkeley
27
460
User
San Ramon
27
461
User
Los Angeles
27
462
User
San Francisco
27
463
User
Woodside
27
464
User
South Pasadena
27
465
Viridiana Castro
Mariposa
27
466
User
Davis
27
467
User
Berkeley
27
468
Jessica Harrison
Simi Valley
27
469
User
Los Angeles
27
470
User
San Bruno
27
471
User
Los Angeles
27
472
User
Arcadia
27
473
User
Irvine
27
474
User
San Francisco
27
475
User
Carlsbad
26
476
User
Westminster
26
477
Austin Tapola
Thousand Oaks
26
478
User
Santa Monica
26
479
Phoebe
San Diego
26
480
Alexis
Lomita
26
481
Ritch Hansen
Redondo Beach
26
482
User
Burlingame
26
483
User
Lake Forest
26
484
User
Fountain Valley
26
485
User
Santa Clara
26
486
Hunter Jack Maslowsky
Torrance
26
487
User
Clovis
26
488
User
Hacienda Heights
26
489
User
Los Angeles
26
490
NewZD
Oakland
26
491
User
Oakland
26
492
User
San Jose
26
493
User
Walnut Creek
26
494
User
San Francisco
26
495
User
San Diego
26
496
Jessica Wu
San Francisco
26
497
User
Los Angeles
26
498
User
San Francisco
26
499
User
East Los Angeles
26
500
User
San Clemente
26
501
User
North Auburn
26
502
User
Los Angeles
26
503
User
Encinitas
26
504
User
San Francisco
26
505
Sruti Arulmani
Stanford
26
506
User
Stanford
26
507
User
Rancho Cordova
26
508
User
Claremont
26
509
User
Los Angeles
26
510
User
San Jose
26
511
User
Cupertino
26
512
User
Los Angeles
26
513
User
San Francisco
26
514
Nicely Penn
Chula Vista
26
515
User
Placerville
26
516
User
Redwood City
26
517
User
Huntington Beach
26
518
User
San Francisco
26
519
User
Los Angeles
26
520
Pav K
San Bernardino
26
521
User
San Jose
26
522
User
San Diego
25
523
mihir sharma
Eastvale
25
524
Brandon Lim
Los Angeles
25
525
User
Riverside
25
526
User
Los Angeles
25
527
User
San Francisco
25
528
Mischa Fritz
San Francisco
25
529
User
Chino
25
530
User
Los Angeles
25
531
User
Los Angeles
25
532
User
Elk Grove
25
533
User
Fremont
25
534
Hallie Wang (KMS)
Cupertino
25
535
User
Los Angeles
25
536
User
Suisun City
25
537
User
Davis
25
538
User
Stockton
25
539
User
San Francisco
25
540
Lance Battles
Mariposa
25
541
angel mora
Mariposa
25
542
User
Cypress
25
543
User
Pomona
25
544
User
Folsom
25
545
User
Los Angeles
25
546
User
Irvine
25
547
User
Richmond
25
548
User
San Diego
25
549
User
Los Angeles
25
550
User
San Francisco
25
551
User
San Jose
25
552
User
Los Angeles
25
553
User
Hawthorne
25
554
User
San Francisco
25
555
User
Santa Clara
25
556
Andres Alejandro Gonzalez-Sosa
San Diego
24
557
Ryan Kuroiwa
Los Angeles
24
558
Jesmond Phimmachack
San Diego
24
559
Suryansh Gupta
Blackhawk
24
560
User
Irvine
24
561
User
San Diego
24
562
User
San Diego
24
563
User
San Diego
24
564
User
San Francisco
24
565
User
Glendale
24
566
User
Concord
24
567
User
Vineyard
24
568
User
Irvine
24
569
User
Cupertino
24
570
User
Orinda
24
571
User
San Diego
24
572
User
Oakland
24
573
User
San Francisco
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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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