Leaderboard
Worldwide
Rank
Level
1
Callan
Australia
100
2
Matthew peters
United Kingdom
100
3
Cyril Guzeev
Russia
100
4
Eddie
Sweden
100
5
Yu Kai Tan
Singapore
100
6
Algorox
Canada
100
7
Leonid Didyk
Ukraine
100
8
Logi SMITH
Australia
100
9
Aryaseto
Indonesia
100
10
Migs de las Llagas
Philippines
100
11
Owen Summers
United States
100
12
Trym Lyssand
Norway
99
13
M Yu
Netherlands
99
14
Tom Scotney
United Kingdom
99
15
Milan van de Meeberg
Brazil
99
16
ade rev
Canada
99
17
dotop
Peru
99
18
Joonas Majuri, Jyväskylä
Finland
99
19
Kevin Wang
United States
99
20
Zizz
Hungary
99
21
Aarush Sahoo
China
98
22
tony
Colombia
98
23
Gerd Wichers
Netherlands
98
24
K. Gee
United States
98
25
Debabi
United States
98
26
Detective Pikachu
United States
98
27
Brian Caulfield
United States
98
29
Miklos
Austria
98
30
Sachin Dubey
India
98
31
Giulio
Italy
98
32
Maciej Jaworski
Poland
98
34
Balazs
Netherlands
98
37
Tracy
United States
97
38
Miguel Juncais
Portugal
97
39
Pavalosk
France
97
40
Diego Caligiore
Italy
97
42
Rio SATO
Australia
97
44
Youssef Aly Elsawy
Saudi Arabia
96
45
François Ardèche
France
96
46
Shampoo
Norway
96
47
Gianni Manginelli
United States
96
48
User
Netherlands
96
49
Maxence Tricheur
France
96
50
User
Netherlands
96
51
ricky
Netherlands
96
52
batuhan yılmaz
Turkey
96
53
Eugene Park
United States
96
54
SOMASHEKAR A C
India
96
55
Anaik Gill
United Arab Emirates
96
58
Jenn Jia
United States
96
59
Aarush
United Kingdom
96
60
User
Lebanon
96
61
John Smith
Canada
95
62
Hakase
United States
95
64
imbahero NO one
Germany
95
65
Marcus Hodgart
United Kingdom
95
66
Woodrow Wilson
Germany
95
67
Santa Claus
United Kingdom
95
68
Leonid Didyk
Ukraine
95
69
🐧🐧🐧🐧🐧🐧🐧🐧🐧🐧🐧🐧🐧🐧🐧🐧🐧🐧🐧🐧🐧🐧🐧🐧🐧🐧🐧🐧🐧🐧🐧🐧🐧🐧🐧🐧🐧🐧🐧🐧🐧🐧🐧🐧🐧🐧🐧
Australia
95
70
Nicolas
Australia
95
71
James TSANG
China
95
72
Viral Jhaveri
United States
95
73
User
Netherlands
95
74
Philip Speegle
United States
95
75
User
India
95
76
Karen Davtyan
Russia
95
77
Tejas ___
India
94
80
Robert Hansen
United States
94
82
scjennywong
China
94
83
Wouter Huisman
Netherlands
94
84
Ashu
India
94
85
Alan Shenkerman
United States
93
86
Hakase
Canada
93
88
User
Netherlands
93
90
User
Germany
93
91
Maayan Shav Artza
Israel
93
92
Cucumba
Estonia
93
93
Cody Cao
United States
93
94
User
Lebanon
93
95
Dekaz Turuz
Germany
93
98
Chris Weston
Australia
93
99
Michael Bosworth
United States
93
100
SPONDULIX
United States
93
101
User
United States
93
102
Sean McCarthy
United Kingdom
93
103
Liam godly
Australia
93
104
ryan n
Australia
93
105
User
Canada
93
106
CYPHER =3
United Arab Emirates
92
107
Dāvis Šterns
Latvia
92
109
RobDogLovesYou
United States
92
110
SuperRobDog
United States
92
111
Mário Gomes
Portugal
92
112
User
Canada
92
113
SeeRobDogRun
United States
92
114
Shanon
China
92
115
GoRobDogGo
United States
92
116
Steven Goldberg
Australia
92
117
Wei Xi
Australia
92
118
Rajrishi Bhaisare
India
92
119
RobDogRocks
United States
92
120
RobDogSharkDoDo
United States
92
121
Maciej Kopanski
Poland
91
122
Calculator
France
91
123
User
United States
91
124
Shivank Khard
India
91
125
Kyle
Netherlands
91
126
User
India
91
127
User
India
91
128
Hello There
United States
91
129
User
United States
91
130
RobCoughDog
United States
91
131
Cr7pro
Singapore
91
132
ishaan grewal
United States
91
133
Joonas Majuri
Finland
91
134
Karthick Jayaraman
India
91
135
rishabh gupta
India
90
136
João Gomes
Portugal
90
137
User
United States
90
138
User
United States
90
139
Brian Chao
United States
90
140
User
United States
90
141
Eduardo Escamilla
Mexico
90
142
User
United States
90
143
Dikshant Chawla
India
90
144
Richard Ramirez
United States
90
145
User
Mexico
90
146
Sabrina
Canada
89
147
Sharon
United States
89
148
User
United States
89
149
Jeff
Netherlands
89
150
User
Ireland
89
151
User
United States
89
152
黃兆宏
Taiwan
89
153
Anthony P
United States
89
154
HiepQH
Vietnam
89
155
Teow Sin Wei
Malaysia
89
156
Giovanni Casalegno
Italy
89
157
Phung Dang Khoi
Vietnam
89
158
Shields
Hungary
89
159
Smith
Germany
89
160
door glass
United States
88
161
User
China
88
162
User
India
88
163
fox
China
88
164
ReynaertPJ
Netherlands
88
165
User
Greece
88
166
HoanNguyen
Vietnam
88
167
Janez Novak
Slovenia
88
168
MathHead
United States
88
169
Etash Guha
United States
88
170
User
India
88
171
Kaushik Samantaray
India
88
172
Busy Bee
Australia
88
173
User
India
88
174
Ihor Riabchich
Ukraine
88
175
Hannah Chu
United States
88
176
Travis Grady
United States
88
177
Shrootz
United Kingdom
88
178
Yannick Vels
Netherlands
87
179
Jiunhao Lau
Singapore
87
180
Samuel Ogunsanya
United States
87
181
Marco Julian Hummel
Germany
87
182
Harryyyy
Taiwan
87
184
User
Netherlands
87
185
Nitin Shukla
India
87
186
Julian Vidal Polania
Russia
87
187
User
Russia
87
188
J.D. Slattery
United States
87
189
Robert H
United States
87
190
imbahero
Germany
87
191
Karna Potwar
Germany
87
193
Jomppe Vesangasta
Finland
87
194
Christian Balboa
Philippines
87
195
User
China
87
196
WhiteFalcon
United Kingdom
87
197
Max Holm
Thailand
87
198
Rahul Jain
United States
87
199
User
Canada
86
200
Abhiraj
Netherlands
86
201
User
India
86
202
User
Peru
86
203
User
United States
86
204
Đăng Khôi Phùng
Vietnam
86
205
Robbe
Belgium
86
206
Lincoln Drosse
Australia
86
207
Hayk Darbinyan
Armenia
86
208
anvesh voma
India
86
209
User
Japan
86
210
Eskay
India
86
211
User
United States
86
212
RobbieBobbie
United States
86
213
Foster Honeck
United States
86
214
User
United States
86
215
RobOhMyOhMy
United States
86
216
User
Lebanon
86
218
User
India
86
219
RandomID
France
85
220
Phùng Đăng Khôi
Vietnam
85
221
Arnav Singh
Canada
85
222
User
Turkey
85
224
MATH
Germany
85
225
User
United States
85
226
User
Pakistan
85
227
Michael Sakowski
Norway
85
228
User
India
85
229
User
South Africa
85
230
Arthur Li
Canada
85
231
Chris Stockley
New Zealand
85
232
CM
Japan
85
233
Sherine Kokado
United States
85
235
User
Ireland
85
236
Rupayan Mallick
India
84
237
MartimG
Portugal
84
238
Swapnil Modani
India
84
239
Andrew W
United States
84
240
User
India
84
241
Terry Zhong
Australia
84
242
JoEp
Spain
84
243
Daniel Pedro
Canada
84
244
User
Canada
84
245
Brian Moret Ferreira da Silva
Brazil
84
246
User
Hungary
84
247
SRIJITA PAUL
India
84
248
Angel Cheikh
Lebanon
84
249
User
India
84
250
User
United States
84
251
Sandy Chung
China
84
252
User
India
84
253
juodumas
Lithuania
84
254
Ayush
India
84
255
SenorRobDof
United States
84
256
Bunk McGregor
Switzerland
84
257
Andreas Falk
Sweden
83
258
Spitzer
United States
83
259
User
India
83
260
User
Pakistan
83
261
Ankur Prakash
India
83
262
Dustin
Canada
83
263
User
Australia
83
264
Samuel honestly
United States
83
265
User
India
83
266
Tarang Desai
India
83
267
Danielle Teter
United States
83
268
User
India
83
269
Trimitra Mahesa Aditya
Indonesia
83
270
Predrag Milic
Serbia
83
271
Akash Bittu
India
83
272
User
South Africa
83
273
Tanisha Grewal
United States
83
274
pupo
Japan
83
275
User
India
83
276
User
Ireland
83
277
User
Finland
82
278
aperson
United States
82
279
Redji_Tissemsilt
Algeria
82
280
User
India
82
283
User
Tunisia
82
284
LaBoya
Mexico
82
285
User
Netherlands
82
286
Gaetan Cretton
Switzerland
82
287
listen to JJ Lin
Brunei Darussalam
82
288
Jagat
India
82
289
Anant 007
India
82
290
User
China
82
292
help me
South Korea
82
293
User
India
82
294
Brandon Williams
United States
82
295
User
Netherlands
82
296
Daregs
United States
82
297
Jacob Hess
United States
82
298
Delage David
France
82
299
Matias
Finland
82
300
Dhaval Parekh
India
82
301
MrRobDoggy
United States
82
303
KSUKan
Japan
82
304
Mario Cuomo
United States
82
305
User
India
81
307
vitor shadows
Portugal
81
308
Harsh vardhan
China
81
309
Jackie
Japan
81
310
Shrek Shrotio
United States
81
311
User
United States
81
312
Qiushi Yu
United States
81
313
leff
France
81
314
Paulo Cezar
Brazil
81
315
Matthew
United States
81
316
User
India
81
317
Akshat
India
81
318
Tyrael
France
81
320
test
United States
81
321
User
Netherlands
81
322
Priyang Tripathi
India
81
323
Jin Young Bang
Indonesia
81
324
Rodrigo Pariona Munguia
Peru
81
325
RobbieMcDoggie
United States
80
327
PapaRobDog
United States
80
328
Joonas Majuri
Finland
80
329
User
United States
80
330
Ender
Singapore
80
331
User
United States
80
332
User
India
80
333
Asish Pradhan
India
80
334
User
United States
80
335
vinu
United States
80
336
Agualva Cacem
Portugal
80
337
User
United States
80
338
Tuusdey
Australia
80
339
marcus dyson
Australia
80
340
Satya swarup Jena
India
80
341
RobbieDoggie
United States
80
342
User
United States
80
343
Ishaan Tehlan
India
80
344
Sumeet Nandi
India
80
345
Ima Coder
United States
80
346
sasidharreddy lankireddy
India
80
347
Pan Mun Jom
Spain
80
348
User
India
79
349
SirRobDog
United States
79
350
chaitanya jaolekar
India
79
351
User
India
79
352
MrRobDog
United States
79
353
User
South Africa
79
354
Justin
United States
79
355
Otot Cruz
Philippines
79
356
User
United States
79
357
CEL-IV
United States
79
358
Emmanuel Brun d'Aubignosc
Belgium
79
359
Erik M. L. Johannesen
Denmark
79
360
User
United States
79
361
sirius
United States
79
362
Michel Zou
France
79
363
User
India
79
364
Henry
United States
79
365
RobDog
United States
79
366
User
Lebanon
79
367
Petr Novák
United Kingdom
79
368
User
Australia
79
369
CEL IV
United States
79
370
Aurelie Chow
United States
78
371
User
United States
78
372
Rangga Puraditama
Indonesia
78
373
User
United States
78
374
User
India
78
375
RobDoggy
United States
78
376
User
United States
78
377
clide
United States
78
378
Florida
United States
78
379
Jagath Vytheeswaran
United States
78
381
Sanchi Wankhade
India
78
382
CEL 4
United States
78
383
User
Netherlands
78
384
User
Australia
78
385
User
United States
78
386
User
India
78
387
Mike Orr
United States
78
388
Lyn Jin
Australia
78
389
Parth Sharma
India
78
390
User
United States
78
391
User
India
78
392
User
United States
78
393
Pataki
United States
78
394
User
China
78
395
Josh Walter
United States
78
396
Fried Lohr
Ukraine
78
397
RobDogLog
United States
78
398
ongzz
Malaysia
77
399
Christopher Dick
United Kingdom
77
400
User
United States
77
401
User
United States
77
402
Justin Leemburg
Netherlands
77
403
Octavius
United States
77
404
Andrew Schirmer
Australia
77
405
Asterohnefos
Vietnam
77
406
Joonas Majuri
Finland
77
407
User
India
77
408
User
United States
77
409
Vladislav Kubík
Czechia
77
410
User
United States
77
411
User
United States
77
412
User
Belgium
77
413
User
Singapore
77
414
User
India
77
415
User
Netherlands
77
416
Duong,Tu
Canada
77
417
Hedy
Malaysia
77
418
Les Miserable
United States
77
419
Sinswien Lee
Taiwan
77
420
Gertibird
United States
77
421
icecuber
Norway
77
422
User
India
77
423
User
Germany
77
424
Sandro
Switzerland
77
425
Tyler Chiu
Canada
77
426
User
United States
77
427
User
United States
77
428
Marcos Cabral
Brazil
77
429
ackuhl2@gmail.com
Brazil
77
430
User
Philippines
77
431
User
Ukraine
77
432
User
Singapore
77
433
Nathaniel Day
United States
77
434
User
United States
77
435
hoàn phạm long
Vietnam
77
436
Samantha Hallgren
United States
77
437
Bruno
Switzerland
77
438
Yousef c t
India
77
439
User
United States
77
440
Chan Lok Hin
Malaysia
77
441
Nitish Roat
India
77
442
Vardh An
India
77
443
Wadim
Russia
77
445
Alex Wen
Canada
76
446
Ryan Burke
United States
76
447
CEL IV
United States
76
449
Cameron Moats
United States
76
450
User
Peru
76
451
User
India
76
452
User
Luxembourg
76
453
Steven Walker
United States
76
454
User
Singapore
76
455
Robin David
Netherlands
76
456
User
United States
76
457
User
United States
76
458
Bernd Pfrommer
United States
76
459
TreeMint
United States
76
460
User
South Africa
76
462
Palfi Csaba Sandor
Romania
76
463
User
United States
76
464
Skyer
Lithuania
76
465
User
United States
76
466
User
France
76
467
User
United States
76
468
User
India
76
469
metalzt
Norway
76
470
San Antonik
Ukraine
76
471
Justin Lau
Singapore
76
472
Aleksandra Strzelecka
Poland
76
473
User
United States
76
474
User
United States
76
475
User
India
76
476
Ademola Oladosu
United States
76
477
User
Georgia
76
478
User
India
76
479
Filipe de Castro
Brazil
76
480
User
India
76
481
User
United States
76
482
User
Italy
76
483
Mo
India
76
484
MODOZ
United States
76
485
Luca HOSKING-HOARE
Australia
76
486
Psyl Monst
France
76
487
Cilla
Hungary
76
488
Tanmay Singh
India
76
489
Bruno Jahn
Germany
75
490
User
India
75
491
Cody Goldman
United States
75
492
User
United States
75
493
Kian Tuastomban
Canada
75
494
Gurak
United States
75
495
User
Colombia
75
496
User
United States
75
497
User
Malaysia
75
498
User
United States
75
499
Sanjay Reddy
India
75
500
User
United States
75
501
nurik bissenbay
Kazakhstan
75
502
User
South Africa
75
503
User
Philippines
75
504
User
India
75
505
User
India
75
506
Tim
Germany
75
507
Realrova
Mexico
75
508
dn02
India
75
509
Natalie Schnose
United States
75
510
User
United States
75
511
ChainLink
United Kingdom
75
512
David Guo
Australia
75
513
Lars Holtappels
Netherlands
75
514
User
Philippines
75
515
Jake Bloodsworth
United States
75
516
User
China
75
517
User
United States
75
518
Peter Gibson
Australia
75
519
User
United States
75
520
Daniele Transerici
Italy
75
521
ROBIN KUMAR
India
75
522
User
India
75
523
User
India
75
524
User
South Africa
75
525
Andy Wang
Australia
75
526
AmauryM
France
75
527
User
India
75
528
Siany
Croatia
75
529
User
United States
75
530
Joonas Majuri, Vesanka
Finland
75
531
User
Philippines
75
532
User
India
75
533
User
United States
75
534
User
Canada
75
535
User
United States
75
536
Emma Carr
United States
75
537
User
Peru
75
538
Quang Nguyen
Canada
75
539
Kaos
United States
75
540
Sachit Mehrotra
India
75
541
User
India
75
542
User
Canada
75
543
User
United States
75
545
User
Netherlands
75
546
Pak Bun Ko
China
75
547
Frank Burton
United Kingdom
75
548
Pogo
United States
75
549
User
India
75
550
User
India
75
551
User
United States
75
552
The Rarest Frog
United States
75
553
User
India
75
554
User
India
75
555
User
United States
75
556
User
Singapore
75
557
Mani Gidla
India
75
558
User
Malaysia
75
559
Kane Rex
Australia
75
560
Himanshu Raj Khandelwal
India
75
561
Epycle
Philippines
75
562
User
United States
74
563
rahma safraoui
China
74
564
User
Portugal
74
565
User
Qatar
74
566
User
Germany
74
567
User
Indonesia
74
568
User
United States
74
569
Jonathan
United Kingdom
74
570
Jeremy Griffin
United States
74
571
Kara Korussi
United States
74
572
Alexander Zhang
United Kingdom
74
573
User
United States
74
574
User
Canada
74
575
User
United States
74
576
User
Australia
74
577
User
United States
74
578
User
United States
74
580
User
Philippines
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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 math 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 math, 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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