Leaderboard
Worldwide
Rank
Level
1
Debabi
United States
100
2
Algorox
Canada
100
3
SOMASHEKAR A C
India
100
5
Gianni Manginelli
United States
99
8
黃兆宏
Taiwan
99
9
Detective Pikachu
United States
99
10
Spondulix
United States
98
11
fox
China
98
12
Anthony P
United States
98
13
Harryyyy
Taiwan
97
14
Youssef Aly Elsawy
Saudi Arabia
97
15
Christian Balboa
Philippines
97
16
Eugene Park
United States
97
17
User
Ireland
96
18
Robbe
Belgium
96
19
Eddie
Sweden
96
20
WhiteFalcon
United Kingdom
96
21
Ashu
India
96
22
Foster Honeck
United States
95
23
Eskay
India
95
24
Shanon
United States
95
25
Karna Potwar
Germany
95
26
MartimG
Portugal
95
27
Leonid Didyk
Ukraine
95
29
phi
Indonesia
93
30
User
Canada
93
32
User
China
92
34
User
Ireland
91
35
Matthew
United States
91
36
Shields
Hungary
91
37
Calculator
France
91
38
MrRobDoggy
United States
91
39
Tracy
United States
91
40
User
India
91
41
Paulo Cezar
Brazil
91
42
Sandy Chung
China
90
43
imbahero
Germany
90
45
Tyrael
France
90
46
Matthew peters
United Kingdom
90
47
scjennywong
China
90
48
Shivank Khard
India
90
49
RobbieMcDoggie
United States
90
50
Psyl Monst
France
90
51
PapaRobDog
United States
89
52
User
Finland
89
53
Ima Coder
United States
89
54
RobbieDoggie
United States
89
57
Agualva Cacem
Portugal
89
58
Rangga Puraditama
Indonesia
89
59
Alvin
Netherlands
89
60
ade rev
Canada
89
61
CEL-IV
United States
89
62
SirRobDog
United States
88
63
CEL IV
United States
88
64
MrRobDog
United States
88
65
ongzz
Malaysia
88
66
Aurelie Chow
United States
88
67
Hakase
Canada
88
68
User
Netherlands
88
70
RobDog
United States
88
71
User
China
87
72
Sanchi Wankhade
India
87
73
clide
United States
87
75
RobDoggy
United States
87
76
CEL 4
United States
87
77
Florida
United States
87
79
Aekred
Algeria
87
80
batuhan yılmaz
Turkey
87
82
Justin Leemburg
Netherlands
87
83
BerkeyFile444
United Arab Emirates
86
85
Alex Wen
Canada
86
86
RobDogLog
United States
86
87
Lucas Silva
Portugal
86
88
Travis Grady
United States
86
89
ackuhl2@gmail.com
Brazil
86
91
CEL IV
United States
86
94
Gertibird
United States
85
96
Joonas Majuri
Finland
85
97
User
Ukraine
85
98
Samantha Hallgren
United States
85
99
User
France
85
101
User
Netherlands
85
102
Bruno Jahn
Germany
85
103
Vladislav Kubík
Czechia
85
104
User
Belgium
85
106
User
Malaysia
85
107
User
United States
85
108
icecuber
Norway
85
109
User
Peru
85
110
Rich Caputo
United States
84
111
User
India
84
112
User
United States
84
113
Palfi Csaba Sandor
Romania
84
114
User
United States
84
115
hoàn phạm long
Vietnam
84
116
Chan Lok Hin
Malaysia
84
117
User
United States
84
118
Quang Nguyen
Canada
84
120
Ademola Oladosu
United States
84
121
User
Philippines
84
122
Rio SATO
Australia
84
123
User
Colombia
83
124
Joonas Majuri, Jyväskylä
Finland
83
125
User
United States
83
126
User
Netherlands
83
127
Realrova
Mexico
83
129
User
United States
83
130
Cody Goldman
United States
83
131
Nitish Roat
India
83
132
Karen Davtyan
Russia
83
133
JoEp
Czechia
83
134
Callan
Australia
83
135
User
Malaysia
83
136
Daregs
United States
83
137
User
United States
83
138
Ihor Riabchich
Ukraine
83
139
User
Peru
83
140
InSane Samp
Romania
82
141
User
United States
82
142
User
Germany
82
143
User
Singapore
82
144
User
Indonesia
82
145
User
United States
82
146
Terry Zhong
Australia
82
147
Darren Chen
United States
82
148
Alexander Zhang
United Kingdom
82
149
Jonathan
United Kingdom
82
150
Looking for a Junior Trader position: sacha.di-martino18@imperial.ac.uk
United Kingdom
82
151
User
Bangladesh
81
152
User
United States
81
153
Miguel Juncais
Portugal
81
154
Victor
Russia
81
155
Shibasis
India
81
156
User
United States
81
157
Tolga Celebi
Netherlands
81
158
Henry Yapeter
Australia
81
159
John Smith
United States
81
160
User
United States
81
161
Pochi
Indonesia
81
162
User
United States
81
163
Bogdan Boroș
Romania
81
164
Hyein Kim
United Arab Emirates
81
165
Sven Polak
Netherlands
81
166
pupo
Japan
81
167
Justin Lau
Singapore
81
168
Aleksandra Strzelecka
Poland
81
169
User
United States
81
170
User
India
81
171
User
United States
81
172
User
Georgia
81
173
User
India
81
174
User
United States
81
175
Mo
India
81
176
User
China
81
177
User
India
81
178
Kian Tuastomban
Canada
80
179
Mike Orr
United States
80
180
Sanjay Reddy
India
80
181
Lars Holtappels
Netherlands
80
182
User
United States
80
183
ROBIN KUMAR
India
80
184
Tim
Germany
80
185
Michel Zou
France
80
186
ChainLink
United Kingdom
80
187
David Guo
Australia
80
188
User
Mexico
80
189
User
China
80
190
User
United States
80
191
Daniele Transerici
Italy
80
192
Joonas Majuri, Vesanka
Finland
80
193
User
Philippines
80
194
User
Canada
80
195
User
United States
80
196
Emma Carr
United States
80
197
Sachit Mehrotra
India
80
198
User
United States
80
199
User
Netherlands
80
200
Pogo
United States
80
201
The Rarest Frog
United States
79
202
RandomID
France
79
203
User
Malaysia
79
204
Ethan Regal
United States
79
205
User
Malaysia
79
206
User
Australia
79
207
niber
Australia
79
208
Marcos Cabral
Brazil
79
209
timmytanner21
United States
79
210
User
United States
79
211
Ben Brown
United States
79
212
User
United States
79
213
Affan Bhandakya
Canada
79
214
User
Netherlands
79
215
Benjamin Oosterom
Netherlands
79
216
User
India
79
217
User
United States
79
218
Bob Smith
Australia
79
219
User
Peru
79
220
User
United States
79
221
User
United States
79
222
Bibiche
France
79
223
User
India
79
224
Joseph Nguyen
United States
79
225
User
United States
78
226
Joey
United States
78
227
Erik M. L. Johannesen
Denmark
78
228
PinkDraconian
Belgium
78
229
‍강일구[학생](이과대학 지리학과)
South Korea
78
230
Lloyd Plumart
Belgium
78
231
David Keen
United States
78
232
Max Holm
Finland
78
233
Rich Akuna
United States
78
234
Mark Bergman
Australia
78
235
Chengjian Shi
United States
78
236
User
United States
78
237
Dinis Santos
Portugal
78
238
User
India
78
239
User
India
78
240
User
Peru
78
241
User
United States
78
242
Bruno Hatchondo
France
78
243
Sandro
Switzerland
78
244
User
Finland
78
245
Mark L
Estonia
78
246
User
United States
78
247
User
United States
78
248
User
United States
78
249
nurik bissenbay
Kazakhstan
78
250
Carol Liew
Singapore
78
251
User
United States
78
252
User
Italy
77
253
User
Netherlands
77
254
Trym Lyssand
Norway
77
255
Dharmesh Damania
India
77
256
Bilal
Canada
77
257
User
India
77
258
Maciej Jaworski
Poland
77
259
& LEARN PLAY
Vietnam
77
260
Natalie Schnose
United States
77
261
User
India
77
263
Alex Gay
Canada
77
264
OnePiece
Bangladesh
77
265
User
France
77
266
User
Netherlands
77
267
User
United States
77
268
Marcus Hodgart
United Kingdom
77
269
Brian Caulfield
United States
77
270
User
Philippines
77
272
User
United States
77
273
Sachin Dubey
India
77
274
User
United States
77
275
Janek Terlikowski
Poland
77
276
User
Hungary
77
277
User
Malaysia
77
278
RUSHI
India
77
279
Kris Patel
United States
77
280
User
Spain
77
281
Tyasa
Indonesia
77
282
User
Malaysia
77
283
User
India
77
284
FelixT
United States
77
285
User
United States
76
286
Job
Netherlands
76
287
Albert E
Vietnam
76
288
Fr
Kenya
76
289
Olle Rehnquist
Sweden
76
290
User
India
76
291
User
United States
76
292
User
United States
76
293
User
India
76
294
My
Vietnam
76
295
User
United States
76
296
Sauske
United States
76
297
User
United States
76
298
OskarTreffner
Estonia
76
299
User
United States
76
300
User
Singapore
76
301
User
United States
76
302
Michael Sakowski
Norway
76
303
User
Turkey
76
304
User
Malaysia
76
305
Frank Qiang
United States
76
306
User
India
76
307
Taylor Taylor Yi
New Zealand
76
308
User
United States
76
309
User
United States
76
310
User
Malaysia
76
311
Yury Namgung
United States
76
312
ommim
Ireland
76
313
Simi Chavda
United Kingdom
76
314
User
United States
76
315
William Wang
United States
76
316
User
United States
76
317
User
Peru
76
318
User
United States
76
319
User
Australia
76
320
CH
Australia
76
321
User
United States
76
322
AmauryM
France
76
323
User
Netherlands
76
324
Aviral Somani
United States
76
325
Zohaib
United Kingdom
76
326
User
United States
75
327
CM
China
75
328
User
Canada
75
329
Rahul Jain
United States
75
330
Ondra Daniš
Czechia
75
331
User
Finland
75
332
User
India
75
333
Morgan Bryant
United States
75
334
User
United States
75
335
r4iny_mood
United Arab Emirates
75
336
Elana Greenberg
United States
75
337
Pavalosk
France
75
338
ARKA CHAKRABORTY
India
75
339
Minh Tran Nhat
Vietnam
75
341
GreenSmurf
Germany
75
342
Reggie Bains
United Kingdom
75
343
Moo
Qatar
75
344
User
United States
75
345
User
India
75
346
User
Germany
75
347
Anano Basilaia
Netherlands
75
348
User
Philippines
75
349
Jeffrey Huang
United States
75
350
Rahul Dugar
India
75
351
User
India
75
352
User
India
75
353
User
Romania
75
354
ID ZHAO
Australia
75
355
Karthick Jayaraman
India
75
356
Edisher Kipiani
United Kingdom
75
357
User
United States
74
358
Tanishq Pauskar
United States
74
359
User
Malaysia
74
360
User
Philippines
74
361
User
United States
74
363
James TSANG
China
74
364
Teague Stotlar
United States
74
365
CEL IV
United States
74
366
User
Luxembourg
74
367
Nes370
United States
74
368
User
Lithuania
74
369
User
United States
74
370
User
Finland
74
371
User
Indonesia
74
372
User
Germany
74
373
Jeremy Griffin
United States
74
374
Shrivathsan V
India
74
375
User
United States
74
376
User
Pakistan
74
377
User
Singapore
74
378
marcus dyson
Australia
74
379
Fabrizio Lilliu
Italy
74
380
Giulio
Spain
74
381
User
Bahrain
74
382
Angel Escobar
United States
74
383
Anna
Russia
74
384
Feministen
Norway
74
385
Michalis
United Kingdom
74
386
User
India
74
388
User
United States
74
389
Vinay Indrala
India
74
390
KBTM
India
74
391
Aegwynn
Canada
74
392
Sawyer West
United States
74
393
Robin David
Netherlands
74
395
User
United States
73
396
Trivial
France
73
397
Michael Li
United States
73
398
robert hou
United States
73
399
Grizzly Bear
United States
73
400
Andreas Falk
Sweden
73
401
Gregor Nishino
United States
73
402
Kate Goloboy
United States
73
403
Rupayan Mallick
India
73
404
User
India
73
405
User
United States
73
406
User
United States
73
407
Raiyana Shams
Canada
73
408
Hannah Chu
United States
73
409
User
France
73
410
User
United States
73
411
User
Finland
73
412
User
United States
73
413
Anna Di Lorenzo
Italy
73
414
Pim Kouwenhoven
Netherlands
73
415
Jessica He
United States
73
416
Rarv
Mexico
73
417
User
United States
73
418
Andrew Couser
United States
73
419
User
Australia
73
420
John Ewart
United Kingdom
73
421
ravenclawstrength 7
United States
72
422
el vper
Belgium
72
423
King of the Hill
United States
72
424
Shahid M Salam
Canada
72
425
Myra Lee
United States
72
426
Effaitch
Indonesia
72
427
User
Mexico
72
428
User
China
72
429
User
United States
72
430
User
United States
72
431
bigblackclock
India
72
433
User
Hungary
72
434
Parth Lakhera
India
72
435
Ernst
Netherlands
72
436
User
India
72
437
User
United States
72
438
Koyomin
Canada
72
439
Anya Sanghvi (CK)
India
72
440
CEL-IV
United States
72
441
The Master Procrastinator
United Kingdom
72
442
User
United States
72
444
OWASS
Ghana
72
445
Sai Jahnavi Yeruva
India
72
446
User
Portugal
72
447
EMLJ
Denmark
72
448
User
United States
72
449
User
Italy
72
450
User
Singapore
72
451
Tom Camargo
United States
72
452
User
Canada
72
453
Roshni
India
72
455
Armin Becker
Germany
72
456
Ashish Singh
India
72
457
User
United States
72
458
User
United States
72
459
User
United States
72
460
User
Germany
72
461
Aidan Lytle
United States
72
462
User
Serbia
72
463
User
United States
72
464
Bernd Pfrommer
Germany
72
465
User
Netherlands
72
466
siri
Australia
72
467
User
United States
72
468
User
United States
72
469
urjil sanghvi
United States
72
470
Haining Zhou
United States
72
471
User
Finland
72
473
User
India
72
474
User
Taiwan
72
475
User
United States
72
476
User
Hungary
72
477
Danielle Teter
United States
72
478
User
Malaysia
72
479
User
Finland
72
480
Caleb Strange
United States
72
481
User
United States
72
482
User
Poland
72
483
Kris Patel
United States
72
484
Andrew Rosenthal
United States
72
485
User
United States
72
486
User
United States
72
487
Petr Novák
United Kingdom
72
488
User
India
72
489
Sitaram N
India
72
490
User
United States
72
491
User
United States
72
492
User
United States
72
493
User
United States
72
494
User
India
72
496
User
United States
72
497
User
Colombia
72
498
User
Sweden
72
499
CanterBerryBoi69
United States
72
500
User
Finland
72
501
Leo Bouvier
Cyprus
72
502
Charles Comiter
United States
72
503
User
United States
72
504
User
Romania
72
505
Alice
United States
72
506
jolijumper
France
72
507
User
Netherlands
72
508
User
Spain
72
509
jitender ahuja
India
72
510
User
United States
72
511
Dustin
United States
72
512
Gabe Korer
United States
72
513
User
Peru
71
514
Milan
Brazil
71
515
Haining Zhou
United States
71
516
User
India
71
517
User
United States
71
518
User
United States
71
519
User
India
71
520
Ethan Zh
United States
71
521
User
China
71
522
Eduardo Gallego
Spain
71
523
User
India
71
524
Michael Lu
United States
71
525
User
United States
71
526
User
Indonesia
71
527
User
United States
71
529
Gloudust HD
Spain
71
530
NJOI
Germany
71
531
Brighton Schott
United States
71
532
User
Netherlands
71
533
compte gmail
France
71
534
User
Germany
71
535
Vir Brendan Agero
Philippines
71
536
User
Russia
71
537
User
Vietnam
71
538
CEL IV
United States
71
539
tony huang
United States
71
540
User
Philippines
71
541
Luis Sanchez-Ulloa
United States
71
542
User
United States
71
543
B.Arindam
Canada
70
544
ajinkya awachar
India
70
545
danaeavg@hotmail.com
Mexico
70
546
User
United States
70
547
User
Russia
70
548
Cameron Allen
United States
70
549
User
India
70
550
Teow Sin Wei
Malaysia
70
551
Malik Muhammad Kashif Saeed
Pakistan
70
552
Andrei Puspana
United Kingdom
70
553
User
India
70
554
Márton
Hungary
70
555
User
United States
70
556
User
United States
70
558
User
Russia
70
559
Carina
United States
70
560
User
United States
70
561
Andreas Oikonomou
Spain
70
562
User
Kenya
70
563
Neeraj Thakur
India
70
564
User
United States
70
565
Fridtjov Stokkeland
Norway
70
566
Le Tuan Minh
Vietnam
70
567
Trimitra Mahesa Aditya
Indonesia
69
568
Srikar Satluri
United States
69
569
Alberto Piva
Australia
69
570
Edward Lee
Canada
69
571
bobopop
United States
69
572
listen to JJ Lin
Brunei Darussalam
69
573
Kara Korussi
United States
69
574
User
Argentina
69
575
CEL IV
United States
69
576
User
United States
69
577
User
Australia
69
578
User
United States
69
579
User
India
69
580
User
Finland
69
581
User
Hungary
69
582
Matias
United States
69
583
Filipe de Castro
Brazil
69
584
User
India
69
585
Sachin
Singapore
69
586
rohaN 🐖
United States
69
587
SOPHIE LI
United States
69
588
xXbithcesXx
United States
69
589
MODOZ
United States
69
590
User
Netherlands
69
591
User
India
69
592
reloan
China
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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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