Leaderboard
Worldwide
Rank
Level
1
Detective Pikachu
United States
100
2
MartimG
Portugal
100
4
Debabi
United States
100
5
Spondulix
United States
100
6
Christian Balboa
Philippines
100
7
Callan
Australia
100
9
Przegrywacz
Poland
99
11
Miklós Horváth
Switzerland
99
12
Timothy So
United States
99
13
imo miklos
United States
99
14
User
Norway
99
15
Tyler Chiu
Canada
99
16
i got 99 problems but mathematics ain't one
Australia
99
17
Jessey S.
Netherlands
99
18
User
Australia
99
19
QTSon
Vietnam
99
20
Alex Dumitrascu
Romania
99
21
User
India
99
22
Rangga Puraditama
Indonesia
98
23
ongzz
Malaysia
98
24
Matthew
United States
98
27
Justin Lau
Singapore
98
28
User
Vietnam
98
29
Rajaram
India
98
30
Abhishek Singh
India
98
31
Samuel honestly
South Korea
98
32
User
France
97
33
Bruno Jahn
Germany
97
34
User
Malaysia
97
35
Mo
India
97
37
Taylor Taylor Yi
New Zealand
97
38
User
Canada
97
39
EpicKira
France
97
40
Dani Díaz
Spain
97
41
User
Vietnam
97
43
User
Israel
97
44
Frank
Greece
97
45
Jenny Wong
China
97
46
Ashu
India
96
47
M Yu
Netherlands
96
48
fox
China
96
49
Shanon
United States
96
50
Alex Wen
Canada
96
51
ackuhl2@gmail.com
Brazil
96
52
WhiteFalcon
United Kingdom
96
53
User
Netherlands
96
54
User
Australia
96
55
CH
Australia
96
56
Shikamaru Nara
Australia
96
58
User
Latvia
96
59
User
India
96
60
User
Slovakia
96
61
Zen SH
Japan
96
62
meow meow
South Korea
96
63
Lloyd Plumart
Belgium
95
64
User
India
95
65
OskarTreffner
Estonia
95
66
Sawyer West
United States
95
67
Jan K
Germany
95
68
Axel Brisse
France
95
69
User
Germany
95
70
EASTON SINGER
United States
95
71
Sergio Iommi
Italy
95
72
Eddie
Sweden
94
73
The Rarest Frog
United States
94
74
niber
Australia
94
75
timmytanner21
United States
94
76
User
Malaysia
94
78
Jeffrey Huang
United States
94
79
User
Malaysia
94
82
User
Grenada
94
84
Michalis Paizanis
United States
94
85
Dhruv Arora
India
94
86
Milan “youngscience” v.de Meeberg
Brazil
94
87
User
Lithuania
94
88
User
China
94
90
Aden Smith
United States
94
91
User
India
94
92
eixsinxicosx
United States
94
93
phi
Indonesia
93
95
Shields
Hungary
93
96
User
China
93
98
Henry Yapeter
Australia
93
99
RandomID
France
93
100
Joey
United States
93
101
Bilal
Canada
93
102
Tyasa
Indonesia
93
103
User
United States
93
104
User
United States
93
105
User
Turkey
93
106
ravenclawstrength 7
United States
93
107
User
Hungary
93
108
FelixM
United States
93
109
Tyson Field
Australia
93
110
DoMo
United States
93
111
User
China
93
112
User
United States
93
113
User
United States
93
114
Queen of England
United Kingdom
93
115
Yung Zawaharz
South Korea
93
116
Jerome Renaux
Belgium
93
117
Sanchi Wankhade
India
92
118
Quang Nguyen
Canada
92
119
User
Malaysia
92
120
Darren Chen
United States
92
121
User
Finland
92
122
Bob Smith
Australia
92
123
Sauske
United States
92
124
User
United States
92
125
Grizzly Bear
United States
92
126
John Ewart
United Kingdom
92
127
Singapore Math Academy
Malaysia
92
128
a b
Finland
92
129
ManInJelly
United States
92
130
User
South Korea
92
132
User
Spain
92
133
User
Saudi Arabia
92
134
Aryanto Yoga
Indonesia
92
135
User
India
92
136
User
Vietnam
92
137
User
Vietnam
92
138
User
Netherlands
92
139
Andrew Ahn
United States
92
140
Pserg
Russia
92
141
Ima Coder
United States
91
142
Rich Caputo
United States
91
143
User
United States
91
144
Miguel Juncais
Portugal
91
145
User
United States
91
146
User
India
91
147
Ben Brown
United States
91
148
User
Finland
91
149
David Keen
United States
91
150
User
India
91
151
Olle Rehnquist
Sweden
91
152
User
United States
91
153
User
United States
91
154
User
United States
91
155
Teague Stotlar
United States
91
156
CEL IV
United States
91
157
Andreas Falk
Sweden
91
158
Márton
Hungary
91
159
User
Finland
91
160
User
Australia
91
161
User
United States
91
162
Spooner
United States
91
163
MyNameIsMe
India
91
164
Yi Ke Hu
China
91
165
Yellow Ninja
United Kingdom
91
166
Anthony McWilliams
United States
91
167
Andrew Smiciklas
United States
91
168
User
Australia
91
169
Narasimha
India
91
170
Teik seam
Malaysia
91
171
Michele Celentano
Italy
91
172
User
South Korea
91
173
User
Netherlands
91
174
Anand Bahl
United States
91
175
User
Netherlands
91
176
Terry Zhong
Australia
90
177
User
United States
90
178
Pochi
Indonesia
90
179
User
United States
90
180
User
United States
90
181
User
India
90
182
User
United States
90
183
Dinis Santos
Portugal
90
184
User
India
90
185
Kate Goloboy
United States
90
186
User
China
90
187
User
Hungary
90
188
Alice
United States
90
189
User
Germany
90
190
cartade
France
90
191
User
France
90
192
Sakboy TB
Netherlands
90
193
Даниил Лазарев
Russia
90
194
Kvothe
United States
90
195
matt dowling
United States
90
196
User
United States
90
197
User
Australia
90
198
User
India
90
199
User
India
90
200
Kyle Xiao
United States
90
201
Xeth
Australia
90
202
User
China
89
203
ROBIN KUMAR
India
89
204
User
United States
89
205
Mark Bergman
Australia
89
206
Carol Liew
Singapore
89
207
User
United States
89
208
User
India
89
209
Morgan Bryant
United States
89
210
r4iny_mood
United Arab Emirates
89
211
Anano Basilaia
Netherlands
89
212
User
United States
89
213
User
China
89
214
User
India
89
215
Neeraj Thakur
India
89
216
User
Hungary
89
217
Solo
Denmark
89
218
User
United States
89
220
User
Indonesia
89
221
Gordon He
United States
89
222
User
Malaysia
88
223
User
Malaysia
88
224
Erik M. L. Johannesen
Denmark
88
225
User
France
88
226
RUSHI
India
88
227
User
Indonesia
88
228
User
United States
88
229
CEL-IV
United States
88
230
User
Italy
88
231
CEL IV
United States
88
232
User
Hungary
88
233
Greck
United States
88
234
User
Denmark
88
235
Ryan Zhang
United States
88
236
User
Austria
88
237
ranousse
France
88
238
User
Brazil
88
239
User
Spain
88
240
Adam Chen
United States
88
241
user
United States
88
242
User
Germany
88
243
User
United States
88
244
Eggma Eggma
Canada
88
245
Zohaib
United Kingdom
87
246
User
India
87
247
James TSANG
China
87
248
Sachit Mehrotra
India
87
249
User
United States
87
250
Myra Lee
United States
87
251
King of the Hill
United States
87
252
User
United States
87
253
User
United States
87
254
Ethan Zh
United States
87
255
Vir Brendan Agero
Philippines
87
256
CEL IV
United States
87
257
lionel messi
Vietnam
87
258
User
Singapore
87
259
劉健聰Kenny
United Kingdom
87
260
Amalgam
Poland
87
261
User
India
87
262
Sergio Tam
Netherlands
87
263
ricardo
Spain
87
264
Peter Andersson
United States
87
265
Tyler Shen
United States
87
266
Andrew Stepp
United States
87
267
User
United States
86
268
sven_polak
Netherlands
86
269
Job
Netherlands
86
270
User
Turkey
86
271
User
United States
86
272
Tanishq Pauskar
United States
86
273
marcus dyson
Australia
86
275
Michael Li
United States
86
276
User
United States
86
277
Redji_Tissemsilt
Algeria
86
278
Andrew Couser
United States
86
279
Anya Sanghvi (CK)
India
86
280
Ernst
Netherlands
86
281
User
United States
86
282
David Guo
Australia
86
283
User
United States
86
284
jolijumper
France
86
285
Michael Lu
United States
86
286
Teow Sin Wei
Malaysia
86
287
Kaden Chen
United States
86
288
User
Portugal
86
289
User
India
86
290
Albert Caputo
United States
86
291
User
Hungary
86
292
User
Philippines
86
293
User
Australia
86
294
User
United States
86
295
Ben Yu
United States
86
296
FARIS
United States
86
297
Maarten Markering
Netherlands
86
298
User
United States
86
299
User
United States
86
300
User
Spain
86
301
Blaziken Boi
United States
86
302
Daniel
Germany
86
303
User
Malaysia
86
304
Empty Suit
Netherlands
86
305
Ben
Australia
86
306
User
United States
86
307
Jonathan G
United States
86
308
User
Hungary
85
309
User
Spain
85
310
Travis Grady
United States
85
311
Frank Qiang
United States
85
312
Bibiche
France
85
314
User
Germany
85
315
Robin David
Netherlands
85
316
User
India
85
317
User
United States
85
318
User
United States
85
319
Sai Jahnavi Yeruva
India
85
320
EMLJ
Denmark
85
321
Luis Sanchez-Ulloa
United States
85
322
User
Hungary
85
323
User
India
85
324
User
United States
85
325
User
India
85
326
Oleg Aprelikov
United States
85
327
Xingyu Yang
Switzerland
85
328
TJ Cubing
United States
85
329
User
United States
85
330
User
France
85
331
User
United States
85
332
User
United States
85
334
Josh
United States
85
335
User
Vietnam
85
336
User
Netherlands
85
337
User
India
85
338
Prof Lishkov
United Kingdom
85
339
ReveHavan
Indonesia
85
340
Nikolas Adiprasetya
Malaysia
85
341
Sneh Patel
United States
85
342
User
United States
85
343
Ellen Khat
United States
85
344
Alvin
Netherlands
84
345
InSane Samp
Romania
84
346
User
Netherlands
84
347
Victor
Russia
84
348
Psyl Monst
France
84
349
Marcos Cabral
Brazil
84
350
User
Australia
84
351
ommim
Ireland
84
352
Angel Escobar
United States
84
353
Le azien
United States
84
354
KBTM
India
84
355
Jessica He
United States
84
356
User
Mexico
84
357
User
Netherlands
84
358
Alberto Piva
Australia
84
359
User
Netherlands
84
360
Fabi
Germany
84
361
User
Malaysia
84
362
User
United States
84
363
Benjamin Kang
United States
84
364
Benjamin Kang
United States
84
365
User
India
84
366
Eskay
Germany
84
367
User
Australia
84
368
Joseph Yu
United States
84
369
Dekaz Turuz
Germany
84
370
Alexandre Leonetti
United States
84
371
User
United States
84
372
Andreas Forsgren
Norway
84
373
User
Turkey
84
374
Cory Huelskamp
United States
84
375
User
India
84
376
User
Spain
84
377
User
United States
84
378
UserChitti
United States
84
379
User
Turkey
84
380
User
Australia
84
381
User
Germany
84
382
Phoomraphee Luenam
Switzerland
84
383
User
Canada
84
384
Hui Cai
United States
84
385
Abhilash IITP
India
84
386
User
United States
84
387
Максим Федосеев
Russia
84
388
Thomas Joyce
Ireland
84
389
Tracy Lenahan
United States
83
390
User
Georgia
83
391
JoEp
Czechia
83
392
User
United States
83
393
Mike Orr
United States
83
394
User
Belgium
83
395
Raiyana Shams
Canada
83
396
Brian Caulfield
United States
83
397
siri
Australia
83
398
Chengjian Shi
United States
83
399
User
United States
83
400
User
Poland
83
401
User
United States
83
402
Matias
United States
83
403
SunnyB
Belgium
83
404
User
Spain
83
405
defform
Ukraine
83
406
User
United States
83
407
User
Netherlands
83
408
User
United States
83
409
Jack Treanor
United Kingdom
83
410
User
India
83
411
Crispin Zou
Australia
83
412
User
Australia
83
413
User
China
83
414
Mihaly Gajdos
Hungary
83
415
User
Italy
83
417
Иван Мирошниченко
United States
83
418
User
Italy
83
419
chemspirit_live
Russia
83
420
Kevin Copeland
United States
83
421
Stephen Mellor
United Kingdom
83
422
Vivek Nautiyal
India
83
423
balsu
United States
83
424
moormaster
Germany
83
425
User
Spain
83
426
User
United States
83
427
User
United States
83
428
User
United States
83
429
Bipin Bihari che15(IIT bhu)
India
83
430
User
United States
83
431
User
Germany
83
432
User
Canada
83
433
Nes370
United States
82
434
User
United States
82
435
User
Pakistan
82
436
Michalis
United Kingdom
82
437
Daregs
United States
82
438
Rarv
Mexico
82
439
User
United States
82
440
User
Germany
82
441
Milan
Brazil
82
442
User
United States
82
443
User
United States
82
444
Edward Lee
Canada
82
445
User
Netherlands
82
446
User
India
82
447
User
United States
82
448
Jordan Katz
United States
82
449
User
Japan
82
450
User
Finland
82
451
Seth Robles
Germany
82
452
Tyler Chiu
Canada
82
453
User
United States
82
454
coy95 coy95
France
82
455
User
Canada
82
456
User
Netherlands
82
457
Achintya Singh
United Kingdom
82
459
User
United States
82
460
User
United States
82
461
User
Netherlands
82
462
Mohamed Lashuel
Saudi Arabia
82
463
Freddy Fan
United Kingdom
82
464
Daneo
United States
82
465
Davide Accordini
Italy
82
466
User
Germany
82
467
User
Vietnam
82
468
Angel Cheikh
Lebanon
82
469
shubham bansal
India
82
470
Astik Gupta
United States
82
471
Hana
Philippines
82
472
Yi Shi
Canada
82
473
ooer
Germany
82
474
User
United States
82
475
Filip Vesovic
Greece
82
476
User
United States
82
477
User
New Zealand
82
478
Elle C
Australia
82
479
User
United States
82
480
John Laboe
United States
82
481
User
Netherlands
81
482
Anna
Russia
81
483
Bogdan Boroș
Romania
81
484
Sachin Dubey
India
81
485
Alex Gay
Canada
81
486
User
United States
81
488
Armin Becker
Germany
81
490
User
United States
81
491
User
United States
81
492
Haining Zhou
United States
81
493
User
Sweden
81
494
User
Russia
81
495
User
United States
81
496
Malik Muhammad Kashif Saeed
Pakistan
81
497
MODOZ
United States
81
498
User
Switzerland
81
499
User
United States
81
500
Francois
Australia
81
501
User
Japan
81
503
User
United States
81
504
User
Singapore
81
505
User
India
81
506
User
United States
81
507
User
United States
81
508
User
United States
81
509
User
Japan
81
510
Palfi Csaba Sandor
Romania
81
511
DKNY
United States
81
512
Timothy Gates
United States
81
513
Ruben Bieze
Netherlands
81
514
User
Malaysia
81
515
Александр Мелентьев
Russia
81
516
User
United States
81
517
kwikmafs
United States
81
518
User
Germany
81
519
User
United States
81
520
User
United States
81
521
User
Ukraine
81
522
User
United States
81
523
User
United States
81
524
User
United States
81
525
Marcell Szilvási
Hungary
81
526
User
United States
81
527
User
United States
81
528
User
United States
81
529
Shin
United States
81
530
Chuusei
Brazil
81
531
User
Canada
81
532
P McC
United Kingdom
81
533
Jakmac
Australia
81
534
User
United States
81
535
User
India
81
536
User
Vietnam
81
537
User
Italy
81
538
User
United States
81
539
User
Canada
81
540
David
Netherlands
81
541
Vincent Lee
United States
81
542
User
United States
81
544
User
Vietnam
81
546
User
Armenia
81
547
User
India
81
548
Kelly Chin KW
Malaysia
81
549
Carlos Alberto Neto
Netherlands
81
550
User
Thailand
81
551
User
Denmark
81
552
User
Canada
81
553
Davis Zhang
United States
81
554
Kevin Trang
Australia
81
555
Mahya
Australia
81
556
Kent A
United States
81
557
User
United States
81
558
User
United States
81
559
James Hu
United States
81
560
Cristian Rios
Argentina
81
561
Wang Ziyuan
Australia
81
562
User
United States
81
563
Longinus torwaldzki
Poland
81
564
User
Luxembourg
80
565
pupo
Japan
80
566
Michel Zou
France
80
567
User
United States
80
568
Shrivathsan V
India
80
569
User
Vietnam
80
570
Haining Zhou
United States
80
571
User
Russia
80
572
Andreas Oikonomou
Spain
80
573
Tim-p
Germany
80
574
Sachin
Singapore
80
575
Filipe de Castro
Brazil
80
576
Crack Eggs
Australia
80
577
TimC
Ireland
80
578
ARKA CHAKRABORTY
India
80
579
User
Australia
80
580
User
Taiwan
80
581
User
New Zealand
80
582
Satya swarup Jena
India
80
583
Eeshan Zele
India
80
584
User
Canada
80
585
User
United States
80
586
Alexander Treibit
United States
80
587
User
Hungary
80
588
Eyedropping
Russia
80
589
Iwona P
Poland
80
590
User
United States
80
591
Tayip
United Kingdom
80
592
User
South Korea
80
593
Uroš Jarc
Slovenia
80
594
User
India
80
595
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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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