Leaderboard
Worldwide
Rank
Level
1
Favian Ho
United States
100
2
K. Gee
United States
100
3
Ghalib Sadan
South Africa
100
4
guerdin
Russia
100
5
Matthew peters
United Kingdom
100
6
Albert
South Korea
100
7
Manup B
India
100
8
Miklós Horváth
Switzerland
100
9
Shivank Khard
India
100
10
Kane Rex
Australia
100
11
I was looking first for a Junior Trader position: snd18@ic.ac.uk
France
100
12
Giovanni Casalegno
Italy
100
13
Sawyer West
United States
100
14
M Yu
Netherlands
100
15
Algorox
Canada
100
16
Cyril Guzeev
Russia
100
17
Eddie
Sweden
100
18
Emilia san
Singapore
99
19
Yu Kai Tan
Singapore
99
20
Callan
Australia
99
21
Migs de las Llagas
Philippines
99
22
JJ Okocha
India
99
23
Leonid Didyk
Ukraine
99
24
Looking for a Junior Trader position: jl6953@stern.nyu.edu
United States
99
25
pancarte
France
99
26
Tom Scotney
United Kingdom
98
27
Gregory Eck2856
United States
98
28
Carl Letscher IV
United States
98
29
Cody Goldman
United States
98
30
ade rev
Canada
98
31
LiviuRebreanu
Romania
98
32
Gerd Wichers
Netherlands
98
33
Joonas Majuri
Finland
98
34
François Ardèche
France
98
35
isaac
Netherlands
97
36
luust
United States
97
37
Sunny Cho Arizona
Japan
97
38
The Mentalist
India
97
39
wahoud
Lebanon
97
40
Osten Loo
United States
97
41
Rahul Jain
United States
97
42
Logi SMITH
Australia
97
43
Lam Kendrick
China
97
44
James HARTLEY
Australia
97
45
Thomas Thybo
Denmark
97
46
Sanchi Wankhade
India
97
47
Owen Summers
United States
96
48
Sachin Dubey
India
96
49
Aryaseto
Indonesia
96
50
Boris Volkovich
Ukraine
96
51
Taeho Lee
South Korea
96
52
Zizi
Hungary
96
53
Marcell Szilvási
Hungary
96
54
Trym Lyssand
Norway
96
55
Panneer
India
96
56
Pavalosk
France
95
57
Brian Charles
France
95
58
Rupayan Mallick
India
95
60
Michael Ho
China
95
61
Ozymandias
Netherlands
95
62
Alberto Piva
Australia
95
63
Michelle Hosea
United States
94
64
harshit sharma
India
94
65
prankster gangster hamster damster curious furious delicious hit or miss trump is mexican obama is latino
United Arab Emirates
94
66
Rudra Parikh
India
94
67
RioXD DX
Australia
94
68
Cr7pro
Singapore
94
69
Motivation
Hungary
94
70
Giulio
Spain
93
72
Brian Caulfield
United States
93
73
Eugene Lau
China
93
75
Brian Chao
United States
93
76
Etash Guha
United States
93
77
Stefan Stankovic
United States
93
78
Kimora Williams
United States
92
79
Guo Shi Bei
United States
92
80
Ima Coder
United States
92
81
Christian Petralito
Switzerland
92
82
Mehak Dhiman
India
92
83
Lukas Johansson
Sweden
92
84
Lucas Silva
Portugal
92
86
Pi Pu
Germany
92
87
Justin Leemburg
Netherlands
92
88
Tiffany Chu
China
92
89
Dickson Wu
Canada
91
90
Jenn Jia
United States
91
91
Ariel Pedernera
Argentina
91
92
Milan van de Meeberg
Brazil
91
94
ZH Liu
Netherlands
91
95
Ajinkyasinh Rajput
India
90
96
Jorgen
Pakistan
90
97
ryan n
Australia
90
98
John Smith
United States
90
99
Joe Scarborough
United States
90
100
William Lin
China
90
101
Eduardo Mora
Mexico
90
102
Dāvis Šterns
Latvia
90
103
Michel Zou
France
89
105
Nicolas Varlot
France
89
106
Илья Горячев
Russia
89
107
Naël 1e6
France
89
108
dotop
Peru
89
109
Manon L
France
89
110
TheDecayedOne
Austria
89
111
Samuel Ogunsanya
United States
89
112
Justin
United States
88
113
Cameron Moats
United States
88
114
Gaetan Cretton
Switzerland
88
115
almer hutapea
Indonesia
88
116
Divanshu Anand
India
88
117
tothetop
Hungary
88
118
Will- Hole
United States
88
119
@Modoz_CR
United States
88
120
Bob Dobbs
United States
88
121
Will
United States
88
122
Arnold Smolders
Netherlands
88
123
Shampoo
Norway
88
124
Martin Emilov
Bulgaria
88
125
study
India
88
126
tony
Brazil
88
127
PA
Canada
88
128
Ihor Riabchich
Ukraine
88
129
Hedy Li
Malaysia
88
130
Andrew Smiciklas [Valley HS]
United States
88
131
Vanilla D
Russia
88
132
un deux
France
88
133
Marian Gianna Flores
Philippines
88
134
Joseph Gelb
United States
87
135
ishaan grewal
United States
87
136
chaitanya jaolekar
India
87
137
Tejas Shyam Padgelwar
India
87
138
Marcus Hodgart
United Kingdom
87
139
Antonis Peronis
United Kingdom
87
140
Joseph Hernandez
United States
87
141
Detective Pikachu
United States
87
142
Thanujeet Nanugonda
United States
87
143
Kevin Wang
United States
87
144
Sebastian Sanchez
United States
87
145
Hayley Cassandra Hay Ee Yap
Malaysia
87
146
Maayan Shav Artza
Israel
87
147
Wouter Huisman
Netherlands
86
148
Nicolas
Australia
86
149
Aarush Sahoo
China
86
150
Debabi
United States
86
151
Petralito Pet
Switzerland
86
152
Minas Alafasos
Greece
86
153
User
United States
86
154
Pepperminty
United States
86
155
first last
United States
86
156
Ahsan Barkati
India
86
157
Sphener
India
86
158
Viral Jhaveri
United States
86
159
Smith
Germany
86
160
Soren Delplace
France
86
161
Balazs
Netherlands
86
162
User
United States
86
163
ricky
Netherlands
86
164
Karam Jad
United Arab Emirates
86
165
Tracy
United States
86
166
User
United States
86
167
Sabrina Frias
United States
86
168
Miguel Juncais
Portugal
85
169
1102 Lu Hongyu
Netherlands
85
170
Stephen Cheng
United States
85
171
Diego Caligiore
Italy
85
172
Corfaitch
Indonesia
85
173
Delage David
France
85
174
Steven Goldberg
Australia
85
175
User
United States
85
177
Lawkvo
France
85
178
Youssef Aly Elsawy
Saudi Arabia
85
179
User
South Korea
85
180
Mason Lien
United States
85
181
Angel Cheikh
Lebanon
85
182
Jake Bloodsworth
United States
85
183
Jonathan
United Kingdom
85
184
User
Netherlands
85
185
User
Netherlands
85
186
Dustin
Canada
85
187
Gabba Gabba
Singapore
85
188
Gianni Manginelli
United States
84
189
Maxence Tricheur
France
84
190
Jiawei Huang
United States
84
191
Tyson Field
Australia
84
192
Ilrak Kerakeip
United States
84
193
batuhan yılmaz
Turkey
84
194
Favian Ho Four
United States
84
195
Anaik Gill
United Arab Emirates
84
196
Ethan Loftspring
United States
84
197
Eugene Park
United States
84
198
SOMASHEKAR A C
India
84
199
User
United States
84
200
User
United States
84
201
Eason Wu
Canada
84
202
Aarush
United Kingdom
84
203
abhishek sood
India
84
204
User
Lebanon
84
205
bigblackclock
India
84
206
Mário Gomes
Portugal
84
207
Deepanshu
India
84
208
RUSHI
India
84
209
Hakase
United States
84
210
TreeMint
United States
84
211
imbahero NO one
Germany
84
212
User
United States
84
213
tushar gupta
India
84
214
TGITG
India
83
215
Nolan Middendorf
United States
83
216
Woodrow Wilson
Germany
83
217
Santa Claus
United Kingdom
83
218
Leonid Didyk
Ukraine
83
219
User
Netherlands
83
220
User
India
83
221
🐧🐧🐧🐧🐧🐧🐧🐧🐧🐧🐧🐧🐧🐧🐧🐧🐧🐧🐧🐧🐧🐧🐧🐧🐧🐧🐧🐧🐧🐧🐧🐧🐧🐧🐧🐧🐧🐧🐧🐧🐧🐧🐧🐧🐧🐧🐧
Australia
83
222
Alex Aits
United Kingdom
83
223
Philip Speegle
United States
83
224
James TSANG
China
83
225
Victor
Russia
83
226
Karen Davtyan
Russia
83
227
Kartik
India
83
228
MathHead
United States
83
229
Michael Bosworth
United States
83
230
Namit Wadhwa
India
83
231
Tolga Celebi
Netherlands
83
232
Robert Hansen
United States
82
233
Gabi
Switzerland
82
234
Favian Ho Two
United States
82
235
User
United States
82
236
Alan Shenkerman
United States
82
237
User
United States
82
238
Cody Cao
United States
82
239
scjennywong
China
82
240
User
Netherlands
82
241
User
United States
82
242
Matthew Padgett
United States
82
243
User
United States
82
244
Ashu
India
82
245
User
Germany
82
246
door glass
United States
82
247
User
United States
82
248
Aditya Kumawat
India
82
249
GhulianiLakshya
India
82
250
Chris Weston
Australia
82
251
Hakase
Canada
82
252
User
Lebanon
81
253
Olga
Russia
81
254
Cucumba
Estonia
81
255
Favian Ho Three
United States
81
256
divyansh anand
India
81
257
Dekaz Turuz
Germany
81
258
User
United States
81
259
Sherwin Liu
United Kingdom
81
260
John Yin
United States
81
261
User
United States
81
262
Sean McCarthy
United Kingdom
81
263
Daregs
United States
81
264
Emerson Smith
United States
81
265
User
United States
81
266
Arpit Agarwal
India
81
267
SPONDULIX
United States
81
268
Jackydukdik
Thailand
81
269
Liam godly
Australia
81
270
Rajrishi Bhaisare
India
81
271
Sandy Chung
China
81
273
Wei Xi
Australia
81
274
Yannick Vels
Netherlands
81
275
John Smith
United States
81
276
User
Canada
81
277
RobDogLovesYou
United States
81
278
P Stevens
United States
81
279
User
Canada
81
280
SuperRobDog
United States
81
281
Jackie
Japan
80
282
Hans Olesen
Denmark
80
283
Aayush Sacheti
India
80
284
Sharon
United States
80
285
Andrew Couser
United States
80
286
Shanon
China
80
287
Richard Winterpeen
Netherlands
80
288
veeravalli ushasri
India
80
289
Kyle
Netherlands
80
290
Jeremy Stoltzfus
United States
80
291
Richard Akira Heru
Singapore
80
292
GoRobDogGo
United States
80
293
SeeRobDogRun
United States
80
294
Hello There
United States
80
295
Yohan Ninan
United States
80
296
User
United States
80
297
RobCoughDog
United States
80
298
luka riba
Croatia
80
299
RobDogSharkDoDo
United States
80
300
Steven Mai
Canada
80
301
Scott Wilton
United Kingdom
80
302
User
India
80
303
batatavada
India
80
304
RobDogRocks
United States
80
305
rishabh gupta
India
80
306
Joel Jacob Thomas
India
80
307
User
India
80
308
Karthick Jayaraman
India
80
309
User
United States
80
310
João Gomes
Portugal
80
311
Maciej Kopanski
Poland
80
312
Austin Chou
United States
80
313
Lin Bing Han
Singapore
80
314
Max Holm
Thailand
80
315
Calculator
France
80
316
Dikshant Chawla
India
80
317
Chris
Indonesia
80
319
Andrew W
United States
80
320
Richard Ramirez
United States
80
321
Joonas Majuri
Finland
80
322
JoEp
Czechia
80
323
Joonas Jõeleht
Estonia
80
324
Jiunhao Lau
Singapore
80
325
User
United States
80
326
Eduardo Escamilla
Mexico
80
327
Xuan Yu
Australia
80
328
Sanjana Pramanik
India
80
329
Pipi Zhao
Australia
80
330
Nikhil Nehra
India
79
331
Abhishek Kaushal
India
79
332
Shields
Hungary
79
333
User
Mexico
79
334
Gilbert Zenner
United States
79
335
Jeff
Netherlands
79
336
User
United States
79
337
SuperMarioGalaxyPRO👌✌👍
Australia
79
338
HiepQH
Vietnam
79
339
Otot Cruz
Philippines
79
340
Alexander Belushkin
Switzerland
79
341
Anubhav
India
79
342
User
United States
79
343
User
United States
79
344
imbahero NO one
Germany
79
345
Aidan Lytle
United States
79
346
Guttacaro
Indonesia
79
347
Math McMathface
Germany
79
348
Kashyap Nepaliya
India
79
349
Anthony Pei
United States
79
350
Sherine Kokado
United States
79
351
Teow Sin Wei
Malaysia
79
352
siddhu salvi
India
79
353
Severnoe Siyanie
Russia
79
354
ประยุทธ์ จันทร์อังคารพุธ
Thailand
79
355
Ashish Ranjan
India
79
356
User
India
79
357
User
Ireland
79
358
User
India
79
359
ADITYA KUMAR Singh
India
79
360
Michael Sakowski
Norway
79
361
TBTR
Philippines
79
362
Alex Musslewhite
United States
79
363
Kushagra Kacholia
India
78
364
TARAN KATTA
United States
78
365
Kaushik Samantaray
India
78
366
Phung Dang Khoi
Vietnam
78
367
Travis Grady
United States
78
368
HoanNguyen
Vietnam
78
369
J.D. Slattery
United States
78
370
Hayk Darbinyan
Armenia
78
371
User
India
78
372
Janez Novak
Slovenia
78
373
User
United States
78
374
User
Netherlands
78
375
Busy Bee
Australia
78
376
User
Greece
78
377
Nils Asklöv
Sweden
78
378
黃兆宏
Taiwan
78
379
User
Russia
78
380
Robert H
United States
78
381
Gregor Beni
Hungary
78
382
Saptarshi Kumar
India
78
383
asdada
Portugal
78
384
Julian Vidal Polania
Russia
78
385
hellenjoanna
Brazil
78
386
User
China
78
387
getSmart
Australia
78
389
Shrootz
United Kingdom
78
390
Hannah Chu
United States
78
391
ReynaertPJ
Netherlands
78
392
Lyn Jin
Australia
78
393
Vlad Sitaru
United Kingdom
78
394
LaBoya
Mexico
78
395
Jason Xie
United States
78
396
Hans Gilbert
Philippines
78
397
Francesco Perniciaro
Italy
78
398
Vladimir Sagan
United States
78
399
SeptemberDU3
United Kingdom
78
400
imbahero
Germany
77
401
Jagat
India
77
402
Nvc Nvc
Poland
77
403
Karna Potwar
Germany
77
404
ID ZHAO
Australia
77
405
Aidan
United States
77
406
Justin Yan
United States
77
407
shield
Hungary
77
408
Ronit Chidara
India
77
409
Longinus torwaldzki
Poland
77
410
fox
China
77
411
Tony Grippando
United States
77
412
Ayush
India
77
413
MATH
Germany
77
414
Jomppe Vesangasta
Finland
77
415
Elvis nguyen
Australia
77
416
James Krano
Germany
77
417
Tanay Batra
India
77
418
jplt
South Africa
77
419
Akshat Kumar
India
77
420
Ashutosh Gupta
India
77
421
kashmiriMirch
Netherlands
77
422
Free Rider
United States
77
423
vasty
Turkey
76
424
Harryyyy
Taiwan
76
425
compte gmail
France
76
426
WhiteFalcon
United Kingdom
76
427
Trent
United States
76
428
User
India
76
429
zipsy pony
France
76
430
User
United States
76
431
Đăng Khôi Phùng
Vietnam
76
432
Arnav Singh
Canada
76
433
User
China
76
434
Nitin Shukla
India
76
435
Abhiraj
Netherlands
76
437
User
Peru
76
438
anvesh voma
India
76
439
Stanislav Kenzie Handoyo
Indonesia
76
440
Aarij Rehman
United States
76
441
Simon Lin
United States
76
442
Christian Balboa
Philippines
76
443
Phùng Đăng Khôi
Vietnam
76
444
User
Japan
76
445
User
Lebanon
76
446
User
Turkey
75
447
Anais Deere
United Kingdom
75
448
User
United States
75
449
RobOhMyOhMy
United States
75
450
RobbieBobbie
United States
75
452
User
India
75
453
Dhaval Parekh
India
75
454
J L
Costa Rica
75
455
User
United States
75
456
Aiden Wong
Australia
75
457
Bublina
Czechia
75
458
Nikhil Mudholkar
India
75
459
Prakash Milan
India
75
460
Andy Chen
United States
75
462
Chris Stockley
New Zealand
75
463
Eskay
India
75
464
Kuvam Shahane
United States
75
465
Ryan Burke
United States
75
466
Swapnil Modani
India
75
467
User
Ireland
75
468
Robbe
Belgium
75
469
Izaan A
Canada
75
470
Arthur Li
Canada
75
471
Foster Honeck
United States
75
472
Agrim
India
75
473
Sid
India
75
474
User
South Africa
75
475
Bunk McGregor
Switzerland
75
476
CM
Japan
75
477
User
India
75
478
User
India
75
479
Helyey
Hungary
75
480
RP Gaur
India
75
481
Kornel Rakowski
Poland
75
482
Albert Chefen
Sweden
75
483
Brian Moret Ferreira da Silva
Brazil
75
484
BichodoMato
Portugal
75
485
Shahid M Salam
Canada
75
486
Pan Mun Jom
Spain
75
487
Siany
Croatia
74
488
User
United States
74
489
User
Hungary
74
490
User
India
74
491
Anvi
India
74
492
juodumas
Lithuania
74
493
Brandon Williams
United States
74
494
Ankur Prakash
India
74
495
Andreas Falk
Sweden
74
496
Jack
Switzerland
74
497
tran nhat minh
Vietnam
74
498
Rahul Chanduka
India
74
499
Jana
Switzerland
74
501
Fabrício Santos
Brazil
74
502
User
Pakistan
74
503
iamgermanshepherd
India
74
504
User
Australia
74
505
coffee or vodka
Turkey
74
506
User
India
74
507
ANDREW SCHIRMER
Australia
74
508
Maarten van den Akker
Belgium
74
509
Jeffrey Hu
United States
74
510
Kalen Meine
United States
74
511
commandotaco
United States
74
512
Samuel honestly
United States
74
513
Spitzer
United States
74
514
User
India
74
515
Predrag Milic
Serbia
74
516
User
Canada
74
517
MG123
Australia
74
518
Hoang Khanh Tu Duong
Canada
74
519
San Antonik
Ukraine
74
520
Tyson Barney
United States
74
521
Robert Dole
United States
74
522
Utkarsh Saxena
India
74
523
User
India
74
524
SenorRobDof
United States
74
525
MartimG
Portugal
74
526
Trimitra Mahesa Aditya
Indonesia
74
527
Tarang Desai
India
74
528
Luca HOSKING-HOARE
Australia
74
529
Cumsa Treasurer
United Kingdom
74
530
Kaustubh Bhoir
India
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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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