Leaderboard
Worldwide
Rank
Level
1
Detective Pikachu
United States
100
2
Christian Balboa
Philippines
100
3
max level in one hour
United States
100
4
Algorox
Japan
100
6
Miklós Horváth
Switzerland
99
7
Timothy So
United States
99
8
imo miklos
United States
99
9
User
Norway
99
10
Tyler Chiu
Canada
99
11
i got 99 problems but mathematics ain't one
Australia
99
12
Jessey S.
Netherlands
99
13
User
Australia
99
14
QTSon
Vietnam
99
15
Alex Dumitrascu
Romania
99
16
User
India
99
17
Matthew
United States
98
19
Justin Lau
Singapore
98
20
User
Vietnam
98
21
Rajaram
India
98
22
Abhishek Singh
India
98
23
Samuel honestly
South Korea
98
25
Mo
India
97
26
User
France
97
28
Taylor Taylor Yi
New Zealand
97
29
User
Canada
97
30
EpicKira
France
97
31
Dani Díaz
Spain
97
32
User
Vietnam
97
34
User
Israel
97
35
Frank
Greece
97
36
Jenny Wong
China
97
37
fox
China
96
38
alex
Canada
96
39
User
Netherlands
96
40
User
Australia
96
41
CH
Australia
96
42
Shikamaru Nara
Australia
96
43
Milan van de Meeberg
Brazil
96
45
User
Latvia
96
46
AmauryM
France
96
47
User
India
96
48
User
Slovakia
96
49
Zen SH
Japan
96
50
meow meow
South Korea
96
51
Lloyd Plumart
Belgium
95
52
User
India
95
53
OskarTreffner
Estonia
95
54
Sawyer West
United States
95
55
Jan K
Germany
95
56
Axel Brisse
France
95
57
User
Germany
95
58
EASTON SINGER
United States
95
59
Sergio Iommi
Italy
95
60
ackuhl2@gmail.com
Brazil
94
61
The Rarest Frog
United States
94
62
niber
Australia
94
63
timmytanner21
United States
94
64
User
Malaysia
94
66
Jeffrey Huang
United States
94
67
User
Malaysia
94
70
User
Grenada
94
72
Michalis Paizanis
United States
94
73
Dhruv Arora
India
94
74
Milan “youngscience” v.de Meeberg
Brazil
94
75
User
Lithuania
94
76
User
China
94
78
Aden Smith
United States
94
79
User
India
94
80
eixsinxicosx
United States
94
82
phi
Indonesia
93
83
RandomID
France
93
84
Bilal
Canada
93
85
Tyasa
Indonesia
93
88
User
Turkey
93
89
ravenclawstrength 7
United States
93
90
User
Hungary
93
91
FelixM
United States
93
92
Tyson Field
Australia
93
94
User
China
93
97
Queen of England
United Kingdom
93
98
Yung Zawaharz
South Korea
93
99
Jerome Renaux
Belgium
93
100
Gianni Manginelli
United States
92
101
Eddie
Sweden
92
102
Sauske
United States
92
103
User
United States
92
104
Grizzly Bear
United States
92
105
John Ewart
United Kingdom
92
106
Singapore Math Academy
Malaysia
92
107
a b
Finland
92
108
ManInJelly
United States
92
109
User
South Korea
92
111
User
Spain
92
112
User
Saudi Arabia
92
113
WhiteFalcon
United Kingdom
92
114
Aryanto Yoga
Indonesia
92
115
User
India
92
116
User
Vietnam
92
117
User
Vietnam
92
118
User
Netherlands
92
119
Andrew Ahn
United States
92
120
Pserg
Russia
92
122
David Keen
United States
91
123
Rich Caputo
United States
91
124
Olle Rehnquist
Sweden
91
125
User
United States
91
126
User
United States
91
127
User
United States
91
128
Teague Stotlar
United States
91
129
CEL IV
United States
91
130
Andreas Falk
Sweden
91
131
User
India
91
132
Márton
Hungary
91
133
User
Finland
91
134
User
Australia
91
135
User
United States
91
136
Spooner
United States
91
137
MyNameIsMe
India
91
138
Yi Ke Hu
China
91
139
Yellow Ninja
United Kingdom
91
140
Anthony McWilliams
United States
91
141
Andrew Smiciklas
United States
91
142
User
Australia
91
143
Narasimha
India
91
144
Teik seam
Malaysia
91
145
Michele Celentano
Italy
91
146
User
South Korea
91
147
User
Netherlands
91
148
Anand Bahl
United States
91
149
User
Netherlands
91
150
Kate Goloboy
United States
90
151
User
China
90
152
Terry Zhong
Australia
90
153
User
Hungary
90
154
Pochi
Thailand
90
155
Alice
United States
90
156
User
Germany
90
157
cartade
France
90
158
User
France
90
159
Sakboy TB
Netherlands
90
160
Даниил Лазарев
Russia
90
161
Kvothe
United States
90
162
matt dowling
United States
90
163
User
United States
90
164
User
Australia
90
165
User
India
90
166
User
India
90
167
Kyle Xiao
United States
90
168
Xeth
Australia
90
169
Morgan Bryant
United States
89
170
r4iny_mood
United Arab Emirates
89
171
Anano Basilaia
Netherlands
89
172
User
United States
89
173
User
China
89
174
Carol Liew
Singapore
89
175
User
India
89
176
Neeraj Thakur
India
89
177
User
Hungary
89
178
Solo
Denmark
89
179
velidi murali krishna
India
89
180
User
United States
89
182
User
Indonesia
89
183
Osten Loo
United States
89
184
Gordon He
United States
89
185
User
Indonesia
88
186
User
France
88
187
Miguel Juncais
Portugal
88
188
User
United States
88
189
CEL-IV
United States
88
190
Bob Smith
Australia
88
191
User
Italy
88
192
CEL IV
United States
88
193
User
Hungary
88
194
Greck
United States
88
195
User
Denmark
88
196
Ryan Zhang
United States
88
197
User
Austria
88
198
ranousse
France
88
199
User
Brazil
88
200
User
Spain
88
201
Adam Chen
United States
88
202
user
United States
88
203
User
Germany
88
204
User
United States
88
205
Eggma Eggma
Canada
88
206
Sachit Mehrotra
India
87
207
Erik M. L. Johannesen
Denmark
87
208
User
United States
87
209
Myra Lee
United States
87
210
King of the Hill
United States
87
211
User
United States
87
212
User
United States
87
213
Ethan Zh
United States
87
214
Vir Brendan Agero
Philippines
87
215
CEL IV
United States
87
216
lionel messi
Vietnam
87
217
User
Singapore
87
218
Elliot Wong
Canada
87
219
劉健聰Kenny
United Kingdom
87
220
Amalgam
Poland
87
221
User
India
87
222
Sergio Tam
Netherlands
87
223
ricardo
Spain
87
224
Peter Andersson
United States
87
225
Tyler Shen
United States
87
226
Andrew Stepp
United States
87
227
Redji_Tissemsilt
Algeria
86
228
Andrew Couser
United States
86
229
Anya Sanghvi (CK)
India
86
230
Ima Coder
United States
86
231
Ernst
Netherlands
86
232
User
United States
86
233
David Guo
Australia
86
234
User
United States
86
235
jolijumper
France
86
236
Michael Lu
United States
86
237
Teow Sin Wei
Malaysia
86
238
Oriane
United States
86
239
Kaden Chen
United States
86
240
User
Portugal
86
241
User
India
86
242
Albert Caputo
United States
86
243
User
Hungary
86
244
User
Philippines
86
245
User
Australia
86
246
User
United States
86
247
Ben Yu
United States
86
248
FARIS
United States
86
249
Maarten Markering
Netherlands
86
250
User
United States
86
251
User
United States
86
252
User
Spain
86
253
Blaziken Boi
United States
86
254
Daniel
Germany
86
255
User
Malaysia
86
256
Empty Suit
Netherlands
86
257
Ben
Australia
86
258
User
United States
86
259
Jonathan G
United States
86
260
User
United States
85
261
User
United States
85
262
Robin David
Netherlands
85
263
User
United States
85
264
Sai Jahnavi Yeruva
India
85
265
EMLJ
Denmark
85
266
Luis Sanchez-Ulloa
United States
85
268
User
Hungary
85
269
User
India
85
270
User
United States
85
271
User
India
85
272
Oleg Aprelikov
United States
85
273
Xingyu Yang
Switzerland
85
274
TJ Cubing
United States
85
275
User
United States
85
276
User
India
85
277
User
France
85
278
User
United States
85
279
User
United States
85
281
Josh
United States
85
282
User
Vietnam
85
283
User
Netherlands
85
284
User
India
85
285
Prof Lishkov
United Kingdom
85
286
ReveHavan
Indonesia
85
287
Nikolas Adiprasetya
Malaysia
85
288
Sneh Patel
United States
85
289
User
United States
85
290
Ellen Khat
United States
85
291
User
Netherlands
84
292
Alberto Piva
Australia
84
293
Bibiche
France
84
294
User
Netherlands
84
295
Fabi
Germany
84
296
User
Malaysia
84
297
KBTM
India
84
298
User
United States
84
299
Benjamin Kang
United States
84
300
Benjamin Kang
United States
84
301
User
India
84
302
Eskay
Germany
84
303
User
Australia
84
304
Joseph Yu
United States
84
305
Dekaz Turuz
Germany
84
306
Alexandre Leonetti
United States
84
307
User
United States
84
308
Andreas Forsgren
Norway
84
309
User
Turkey
84
310
Cory Huelskamp
United States
84
311
User
India
84
312
User
Spain
84
313
User
United States
84
314
UserChitti
United States
84
315
User
Turkey
84
316
User
Australia
84
317
User
Germany
84
318
Phoomraphee Luenam
Switzerland
84
319
User
Canada
84
320
Hui Cai
United States
84
321
Abhilash IITP
India
84
322
User
United States
84
323
Максим Федосеев
Russia
84
324
Thomas Joyce
Ireland
84
325
User
United States
83
326
Matias
United States
83
327
SunnyB
Belgium
83
328
siri
Australia
83
329
User
Spain
83
330
defform
Ukraine
83
331
Angel Escobar
United States
83
332
User
United States
83
333
User
Netherlands
83
334
User
United States
83
335
Jack Treanor
United Kingdom
83
336
User
India
83
337
Crispin Zou
Australia
83
338
User
Malaysia
83
339
User
Australia
83
340
User
China
83
341
Mihaly Gajdos
Hungary
83
342
User
Italy
83
344
Иван Мирошниченко
United States
83
345
User
Italy
83
346
chemspirit_live
Russia
83
347
Kevin Copeland
United States
83
348
Stephen Mellor
United Kingdom
83
349
Vivek Nautiyal
India
83
350
balsu
United States
83
351
moormaster
Germany
83
352
User
Spain
83
353
User
United States
83
354
User
United States
83
355
User
United States
83
356
Bipin Bihari che15(IIT bhu)
India
83
357
User
United States
83
358
User
Germany
83
359
User
Canada
83
360
Travis Grady
United States
82
361
User
Germany
82
362
User
Spain
82
363
User
United States
82
364
Edward Lee
Canada
82
365
ommim
Ireland
82
366
Chengjian Shi
United States
82
367
Daregs
United States
82
368
User
Netherlands
82
369
User
India
82
370
User
United States
82
371
Jordan Katz
United States
82
372
User
Japan
82
373
User
India
82
374
User
Finland
82
375
Seth Robles
Germany
82
376
James TSANG
United Kingdom
82
377
Tyler Chiu
Canada
82
378
camden
United States
82
379
User
United States
82
380
coy95 coy95
France
82
381
User
Canada
82
382
User
Netherlands
82
383
Achintya Singh
United Kingdom
82
385
Mark Wo
United States
82
386
User
United States
82
387
User
United States
82
388
User
Netherlands
82
389
Mohamed Lashuel
Saudi Arabia
82
390
Freddy Fan
United Kingdom
82
391
Daneo
United States
82
392
Davide Accordini
Italy
82
393
User
Germany
82
394
User
Vietnam
82
395
Angel Cheikh
Lebanon
82
396
shubham bansal
India
82
397
Astik Gupta
United States
82
398
ajinkya awachar
India
82
399
Hana
Philippines
82
400
Yi Shi
Canada
82
401
ooer
Germany
82
402
User
United States
82
403
Filip Vesovic
Greece
82
404
User
United States
82
405
User
New Zealand
82
406
Elle C
Australia
82
407
User
United States
82
408
John Laboe
United States
82
409
User
Netherlands
81
410
MODOZ
United States
81
411
Mike Orr
United States
81
412
Victor
Russia
81
413
User
Switzerland
81
414
User
United States
81
415
Malik Muhammad Kashif Saeed
Pakistan
81
416
Francois
Australia
81
417
User
Japan
81
419
User
United States
81
420
User
Singapore
81
421
User
India
81
422
User
United States
81
423
User
United States
81
424
User
United States
81
425
User
Japan
81
426
Palfi Csaba Sandor
Romania
81
427
DKNY
United States
81
428
Timothy Gates
United States
81
429
Ruben Bieze
Netherlands
81
430
User
Malaysia
81
431
George K
United Kingdom
81
432
Александр Мелентьев
Russia
81
433
User
United States
81
434
kwikmafs
United States
81
435
User
Germany
81
436
User
United States
81
437
User
United States
81
438
User
Ukraine
81
439
User
United States
81
440
User
United States
81
441
User
United States
81
442
Marcell Szilvási
Hungary
81
443
User
United States
81
444
User
United States
81
445
User
United States
81
446
Shin
United States
81
447
Chuusei
Brazil
81
448
User
Canada
81
449
P McC
United Kingdom
81
450
Jakmac
Australia
81
451
User
United States
81
452
User
India
81
453
User
Vietnam
81
454
User
Italy
81
455
User
United States
81
456
User
Canada
81
457
David
Netherlands
81
458
Vincent Lee
United States
81
459
User
United States
81
461
User
Vietnam
81
463
User
Armenia
81
464
User
India
81
465
Kelly Chin KW
Malaysia
81
466
Carlos Alberto Neto
Netherlands
81
467
User
Thailand
81
468
User
Denmark
81
469
User
Canada
81
470
Davis Zhang
United States
81
471
Kevin Trang
Australia
81
472
Mahya
Australia
81
473
Kent A
United States
81
474
User
United States
81
475
User
United States
81
476
James Hu
United States
81
477
Cristian Rios
Argentina
81
478
Wang Ziyuan
Australia
81
479
User
United States
81
480
Longinus torwaldzki
Poland
81
481
Anna
Russia
80
482
Crack Eggs
Australia
80
483
User
Australia
80
484
User
Taiwan
80
485
User
New Zealand
80
486
Filipe de Castro
Brazil
80
487
Satya swarup Jena
India
80
488
Raiyana Shams
Canada
80
489
Eeshan Zele
India
80
490
User
Canada
80
491
User
United States
80
492
Alexander Treibit
United States
80
493
User
Hungary
80
494
Eyedropping
Russia
80
495
Iwona P
Poland
80
496
User
United States
80
497
Tayip
United Kingdom
80
498
User
South Korea
80
499
Uroš Jarc
Slovenia
80
500
User
India
80
501
Kavishaur Dwarika
South Africa
80
502
User
United States
80
503
Jeong Augustine
Canada
80
504
justin silver
United States
80
505
User
United States
80
506
JC
Canada
80
507
User
United States
80
508
Victor Harmsen
Netherlands
80
509
User
India
80
510
Timmy
Germany
80
511
User
Malaysia
80
512
Pavels Stecenko
Netherlands
80
514
User
Turkey
80
515
User
Germany
80
516
rdu
Canada
80
517
User
Portugal
80
518
User
United States
80
519
Jordan Wells
United States
80
520
User
Italy
80
521
Shae Wang
United States
80
522
User
India
80
523
compte gmail
France
79
524
User
Netherlands
79
525
User
United States
79
526
TimC
Ireland
79
527
bigblackclock
India
79
528
nacho pacheco
Argentina
79
529
kutas
Turkey
79
530
Kaka
Australia
79
531
User
United States
79
533
Heath Vincent
United States
79
534
robert hou
United States
79
535
Nikhar Patel
United States
79
536
Stanislav Perumov
United States
79
537
User
India
79
538
User
Ireland
79
539
User
Slovenia
79
540
User
Switzerland
79
541
Le azien
United States
79
542
Jiawei Huang
United States
79
543
Fritsie boiker
Netherlands
79
544
User
United States
79
545
Monica
India
79
546
User
United States
79
547
User
United States
79
548
Fer G
France
79
550
User
United States
79
551
User
United States
79
552
User
United States
79
553
Morgan
United Kingdom
79
554
User
United States
79
555
User
United States
79
556
User
Netherlands
79
557
User
United States
79
558
Albert Samaha
United States
79
559
Lukas Baker
United States
78
560
Sachin
Singapore
78
561
Bogdan Boroș
Romania
78
562
S H
Germany
78
563
Michel Zou
France
78
564
User
Germany
78
565
Ken Fang
Australia
78
566
Alexander Pham
Australia
78
567
User
United States
78
568
ARKA CHAKRABORTY
India
78
569
Tobias Nack
Germany
78
570
User
United States
78
571
User
United States
78
572
Zhonyas
Philippines
78
574
Ibrahim Sulaiman
United Kingdom
78
575
Eduardo Gallego
Spain
78
576
Filip Konarik
Czechia
78
577
User
Canada
78
578
Jagat
India
78
579
User
Australia
78
580
Pi Monster
United States
78
581
User
India
78
582
George
United Kingdom
78
583
User
India
78
584
Léo Bouvier
France
78
585
User
Netherlands
78
586
User
United States
78
587
User
China
78
588
trllisbad
United States
78
589
User
Netherlands
78
590
Chris Rodriguez
United States
78
591
Frankburton
United Kingdom
78
592
User
India
78
593
User
United States
78
594
User
Brazil
78
595
User
Mexico
78
596
User
United States
78
597
User
United States
78
598
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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