Leaderboard
United States
United States
Worldwide
Rank
Level
1
Owen Summers
Ohio
100
2
Kevin Wang
Texas
99
3
K. Gee
California
98
4
Debabi
Washington
98
5
Detective Pikachu
Washington
98
6
Brian Caulfield
New York
98
7
User
Virginia
98
8
User
California
98
9
User
Virginia
97
10
Tracy
Vermont
97
11
User
Colorado
97
12
Gianni Manginelli
New Jersey
96
13
Eugene Park
Tennessee
96
14
User
Virginia
96
15
User
Virginia
96
16
Jenn Jia
California
96
17
Hakase
Minnesota
95
18
User
Virginia
95
19
Viral Jhaveri
California
95
20
Philip Speegle
Missouri
95
21
User
Virginia
94
22
User
Virginia
94
23
Robert Hansen
Region
94
24
User
Virginia
94
25
Alan Shenkerman
New Jersey
93
26
User
Virginia
93
27
User
Texas
93
28
Cody Cao
New York
93
29
User
Virginia
93
30
User
Virginia
93
31
Michael Bosworth
New Jersey
93
32
SPONDULIX
California
93
33
User
Virginia
93
34
RobDogLovesYou
Florida
92
35
SuperRobDog
Florida
92
36
SeeRobDogRun
Florida
92
37
GoRobDogGo
Florida
92
38
RobDogRocks
Florida
92
39
RobDogSharkDoDo
Florida
92
40
User
Michigan
91
41
Hello There
Massachusetts
91
42
User
Virginia
91
43
RobCoughDog
Florida
91
44
ishaan grewal
Texas
91
45
User
Virginia
90
46
User
Virginia
90
47
Brian Chao
California
90
48
User
Virginia
90
49
User
Texas
90
50
Richard Ramirez
Region
90
51
Sharon
New Jersey
89
52
User
Virginia
89
53
User
Maine
89
54
Anthony P
New York
89
55
door glass
New Jersey
88
56
MathHead
New York
88
57
Etash Guha
North Carolina
88
58
Hannah Chu
New Jersey
88
59
Travis Grady
California
88
60
Samuel Ogunsanya
Indiana
87
61
Seb
Utah
87
62
J.D. Slattery
Illinois
87
63
Robert H
California
87
64
Rahul Jain
New Jersey
87
65
User
Michigan
86
66
User
Washington
86
67
RobbieBobbie
Florida
86
68
Foster Honeck
Missouri
86
69
User
Texas
86
70
RobOhMyOhMy
Florida
86
72
User
Texas
85
73
Sherine Kokado
California
85
74
Andrew W
California
84
75
User
Pennsylvania
84
76
SenorRobDof
Florida
84
77
Spitzer
New York
83
78
Samuel honestly
Tennessee
83
79
Danielle Teter
Louisiana
83
80
Tanisha Grewal
Texas
83
81
aperson
Maryland
82
82
KRZ
Michigan
82
83
Brandon Williams
Virginia
82
84
Daregs
Connecticut
82
85
MrRobDoggy
Florida
82
86
Mario Cuomo
New York
82
87
Shrek Shrotio
Pennsylvania
81
88
User
Virginia
81
89
Qiushi Yu
Michigan
81
90
Matthew
South Carolina
81
91
test
Texas
81
92
Jacob Hess
Massachusetts
81
93
RobbieMcDoggie
Florida
80
95
PapaRobDog
Florida
80
96
User
Washington
80
97
User
California
80
98
User
California
80
100
User
Utah
80
101
RobbieDoggie
Florida
80
102
User
Pennsylvania
80
103
Ima Coder
Massachusetts
80
104
SirRobDog
Florida
79
105
MrRobDog
Florida
79
106
Justin
Arizona
79
107
User
Kansas
79
108
CEL-IV
Texas
79
109
User
New York
79
110
sirius
California
79
111
Henry
New York
79
112
RobDog
Florida
79
113
CEL IV
Texas
79
114
Aurelie Chow
North Carolina
78
115
User
California
78
116
User
California
78
117
RobDoggy
Florida
78
118
User
Florida
78
119
clide
Connecticut
78
120
Florida
Florida
78
121
Jagath Vytheeswaran
Washington
78
122
CEL 4
Texas
78
123
User
Florida
78
124
Mike Orr
Texas
78
125
User
Texas
78
126
User
California
78
127
Pataki
New York
78
128
Josh Walter
Massachusetts
78
129
RobDogLog
Florida
78
130
User
New York
77
131
User
New York
77
132
Octavius
Florida
77
133
User
Texas
77
134
User
Florida
77
135
User
New York
77
136
Les Miserable
California
77
137
Gertibird
Texas
77
138
User
New York
77
139
User
Pennsylvania
77
140
Nathaniel Day
Ohio
77
141
User
Massachusetts
77
142
Samantha Hallgren
Minnesota
77
143
User
Illinois
77
144
CL
Texas
76
145
Ryan Burke
Illinois
76
146
CEL IV
Texas
76
147
Cameron Moats
Georgia
76
148
Steven Walker
Arkansas
76
149
User
Texas
76
150
User
Massachusetts
76
151
Bernd Pfrommer
Connecticut
76
152
TreeMint
Oregon
76
153
User
New York
76
154
User
Texas
76
155
User
Maryland
76
156
User
New York
76
157
User
Florida
76
158
Ademola Oladosu
California
76
159
User
California
76
160
MODOZ
North Carolina
76
161
Cody Goldman
Colorado
75
162
User
New York
75
163
Gurak
California
75
164
User
New York
75
165
User
Vermont
75
166
User
New York
75
167
Natalie Schnose
New York
75
168
User
New York
75
169
Jake Bloodsworth
Texas
75
170
User
Florida
75
171
User
Florida
75
172
User
Colorado
75
173
User
Utah
75
174
User
Texas
75
175
Emma Carr
North Carolina
75
176
Kaos
New York
75
177
User
Texas
75
178
Pogo
Florida
75
179
User
Florida
75
180
The Rarest Frog
Missouri
75
181
User
New York
75
182
User
Pennsylvania
74
183
User
California
74
184
Jeremy Griffin
Florida
74
185
Kara Korussi
New Jersey
74
186
User
Washington
74
187
User
Maryland
74
188
User
Pennsylvania
74
189
User
Virginia
74
190
Junan Zhang
Oregon
74
191
User
Florida
73
192
User
Massachusetts
73
193
User
California
73
194
User
Pennsylvania
73
195
User
Virginia
73
196
User
Texas
72
197
Darren Chen
California
72
198
User
Utah
72
199
User
Nevada
72
200
User
Texas
72
201
User
New Jersey
72
202
TheProdigyHenry
California
72
203
Sacheth Sathyanarayanan
District of Columbia
72
204
User
Virginia
72
205
User
New York
72
206
User
Washington
72
207
User
Virginia
72
208
Kris Patel
Arizona
72
209
FelixT
California
72
210
User
New Jersey
72
211
User
New York
72
212
Sauske
Washington
72
213
User
Florida
72
214
User
Pennsylvania
72
215
User
New York
72
216
User
New York
72
217
User
Texas
72
218
User
Florida
72
219
User
Pennsylvania
72
220
User
Texas
72
221
Frank Qiang
Alabama
71
222
Ethan Regal
Pennsylvania
71
223
User
New York
71
224
Ronit Tiwary
Michigan
71
225
User
Pennsylvania
71
226
Joseph Nguyen
Pennsylvania
71
227
User
Utah
71
228
User
Virginia
71
229
Max Anderson
Florida
71
230
User
Utah
71
231
Doug Childs
Texas
71
232
User
Colorado
71
233
Andrew Ahn
New York
71
234
Rich Akuna
Illinois
71
235
User
Florida
71
236
User
Rhode Island
71
237
Mohammed Namazi
New Jersey
71
238
User
Florida
70
239
User
New York
70
240
User
New Jersey
70
241
User
California
70
242
Chengjian Shi
Illinois
70
243
Nick Perozzi
Illinois
70
244
zhiwen xie
California
70
245
User
Pennsylvania
70
246
User
New York
70
247
User
Indiana
70
248
User
Texas
70
249
User
Illinois
69
250
Ben Brown
Alabama
69
251
Sid J
New York
69
252
Michael Li
Washington
69
253
User
Texas
69
254
robert hou
California
69
255
User
New York
69
256
Grizzly Bear
Washington
69
257
Gregor Nishino
North Carolina
69
258
Kate Goloboy
Massachusetts
69
259
David Keen
Indiana
69
260
Andrew Sheinberg
Rhode Island
69
261
User
New Jersey
69
262
User
New York
69
263
Sam Chinnery
California
69
264
User
Connecticut
69
265
Joseph Hernandez
Texas
69
266
Joey
Nevada
68
267
timmytanner21
Washington
68
268
User
Utah
68
269
Team America World Police
New York
68
270
Yury Namgung
New York
68
271
User
Washington
68
272
User
New York
68
273
William Zeng
Pennsylvania
68
274
Elana Greenberg
Illinois
68
275
User
Virginia
68
276
Una Gauper
California
68
277
Aviral Somani
Tennessee
68
278
William Wang
California
68
279
User
Virginia
67
280
User
New York
67
281
User
Florida
67
282
User
Utah
67
283
User
Maryland
67
284
User
Oklahoma
67
285
User
Virginia
67
286
User
Texas
67
287
User
New York
67
288
Haining Zhou
Illinois
67
289
User
Florida
67
290
User
Utah
67
291
Caleb Strange
Texas
67
292
User
Rhode Island
67
293
Andrew Rosenthal
Region
66
294
User
Michigan
66
295
User
Texas
66
296
User
Texas
66
297
CanterBerryBoi69
Indiana
66
298
Charles Comiter
California
66
299
User
Texas
66
300
Alice
West Virginia
66
301
User
New York
66
302
Gabe Korer
Arizona
66
303
Bob Cho
California
66
304
North man
Massachusetts
66
305
Haining Zhou
Illinois
66
306
User
Massachusetts
66
307
Bryan Vu
Texas
66
308
Ed
Texas
66
309
Ethan Zh
Virginia
66
310
Nes370
California
66
311
User
New York
66
312
Michael Lu
Illinois
66
313
User
Virginia
66
314
John L
Massachusetts
66
315
Brighton Schott
Washington
66
316
Rob McCracken
Illinois
66
317
Morgan Bryant
North Carolina
66
318
Satyrykal
Connecticut
66
319
User
Texas
65
320
User
Virginia
65
321
NEHA CHANDRASEKAR
West Virginia
65
322
User
Massachusetts
65
323
Jim Smith5614
Connecticut
65
324
Tim walsh
New York
65
325
Tanishq Pauskar
Ohio
65
326
Angel Escobar
North Carolina
65
327
User
California
65
328
User
Massachusetts
65
329
User
Michigan
65
330
Jayhawk
Indiana
65
331
urjil sanghvi
California
65
332
User
Texas
65
333
User
Pennsylvania
65
334
User
Utah
65
335
Kris Patel
Arizona
65
336
User
Maryland
65
337
User
New York
65
338
Aidan Lytle
North Carolina
65
339
Peter Luo
West Virginia
65
340
Cameron O'Brien
Maryland
65
341
User
New York
65
342
User
Illinois
65
343
User
Texas
65
344
Tom Camargo
New York
65
345
User
California
65
346
John D
Tennessee
65
347
User
Texas
64
348
User
Michigan
64
349
Oo
California
64
350
User
Texas
64
351
User
California
64
352
User
Indiana
64
353
Teague Stotlar
Illinois
64
354
CEL IV
Texas
64
355
User
Oregon
64
356
Krit
Pennsylvania
64
357
Oops Oops
New Jersey
64
358
tony huang
Virginia
64
359
Jeffrey Huang
Texas
64
360
User
Pennsylvania
64
361
J Hulme
Texas
64
362
User
Hawaii
64
363
III
New Jersey
64
364
User
New York
64
365
User
New Jersey
64
366
User
Texas
64
367
User
New York
64
368
Cory Sternin
New Jersey
64
369
Jessica He
Georgia
63
370
User
New Mexico
63
371
User
Illinois
63
372
User
California
63
373
User
Washington
63
374
User
New York
63
375
User
California
63
376
Maaz Ahmad
Arkansas
63
377
User
Iowa
63
378
User
Texas
63
379
User
Michigan
63
380
User
California
63
381
Sawyer West
Maryland
63
382
User
Texas
63
383
User
Texas
63
384
User
California
63
385
Andreas Louskos
New Hampshire
63
386
King of the Hill
Ohio
63
387
User
Texas
63
388
Myra Lee
Washington
63
389
Sanchit Dhiman
Illinois
63
390
joe smith
Massachusetts
63
391
User
Illinois
62
392
Jonathan Davis
Virginia
62
393
CEL-IV
Texas
62
394
User
Virginia
62
395
User
Texas
62
396
User
Florida
62
397
User
Texas
62
398
Will St. Pierre
Illinois
62
399
User
Florida
62
400
Carina
New York
62
401
User
Oregon
62
402
SOPHIE LI
Florida
62
403
User
California
62
404
bobopop
Connecticut
62
405
Dirk Noom
New Jersey
62
406
Cameron Allen
Florida
62
407
Jared Haertel
California
62
408
User
Illinois
62
409
User
Connecticut
62
410
xXbithcesXx
Kentucky
62
411
User
California
62
412
User
Kansas
62
413
George Washington
New York
62
414
ravenclawstrength 7
Washington
62
415
User
Pennsylvania
62
416
Adub
Florida
62
417
Thomas Tran
New York
62
418
Brandon Pozil
North Carolina
62
419
Kailey Johnson
Washington
62
420
User
New York
61
421
User
New York
61
422
User
California
61
423
User
New York
61
424
SPicy
California
61
425
Luis Sanchez-Ulloa
Washington
61
426
User
Pennsylvania
61
427
Phong Lee
Massachusetts
61
428
Srikar Satluri
Georgia
61
429
Andrew Couser
Texas
61
430
Joe Scarborough
New York
61
431
User
Region
61
432
User
California
61
433
User
Texas
61
434
Tim Zheng
Florida
61
435
David Taylor
South Carolina
61
436
midnightcoffee
Mississippi
61
437
Jack Li
Massachusetts
61
438
CEL IV
Texas
61
439
rohaN 🐖
Georgia
61
440
User
Texas
61
441
User
Utah
61
442
User
Illinois
61
443
Svante
Texas
61
444
User
Illinois
61
445
User
Texas
61
446
Mohamed Tchambaz
Virginia
60
447
Parsa Novin
Georgia
60
448
User
Ohio
60
449
User
New York
60
450
camden
Illinois
60
451
Need Quant
New York
60
452
User
Massachusetts
60
453
User
New York
60
454
David W
Washington
60
455
User
Michigan
60
456
Stanislav Perumov
New York
60
457
User
New York
60
458
User
Florida
60
459
Kyle Ward
Tennessee
60
460
Lukas Baker
Connecticut
60
461
User
Florida
59
462
CEL IV
Texas
59
463
User
Pennsylvania
59
464
Jingyang Guo
Washington
59
465
User
New York
59
466
User
Kansas
59
467
User
Pennsylvania
59
468
Chris Hansen
California
59
469
User
Pennsylvania
59
470
User
Pennsylvania
59
471
Sebastian Degner
Massachusetts
59
472
pianotennisman
South Dakota
59
473
User
Texas
59
474
User
New York
59
475
User
New York
59
476
User
Massachusetts
59
477
User
California
59
478
User
Georgia
59
479
User
Texas
59
480
User
New York
59
481
Kaden Chen
Massachusetts
59
482
Alex
Nevada
59
483
Dede W
California
58
484
ag ar
California
58
485
Edna Castellanos
Pennsylvania
58
486
Joseph Kijewski
Massachusetts
58
487
Julia Levcovets
New York
58
488
User
California
58
489
User
California
58
490
Tom Solomon
California
58
491
Mason Sosa
Nevada
58
492
User
Pennsylvania
58
493
User
Texas
58
494
Daniel Benayoun
Florida
58
495
User
Massachusetts
58
496
Epic Bhangra
New Jersey
58
497
User
Texas
58
498
User
Georgia
58
499
Rawlison Zhang
New Jersey
58
501
User
Utah
57
502
User
Texas
57
503
User
Texas
57
504
User
New York
57
505
Yi L Lin
Minnesota
57
506
Gabe B
Arizona
57
507
Stlove24@gmail.com
Florida
57
508
Pi
Michigan
57
509
User
New York
57
510
User
New York
57
511
User
California
57
512
Dean Lee
California
57
513
User
Texas
57
514
Osten Loo
Texas
57
515
Benjamin Kang
New York
57
516
User
Florida
57
517
Benjamin Kang
New York
57
518
User
New Jersey
57
519
sakshi agte
California
57
520
User
New York
57
521
User
Illinois
57
522
User
Georgia
57
523
User
New York
57
524
User
New York
57
525
User
Maryland
57
526
Epic Gamer
New York
57
527
User
New Jersey
57
529
Melinda Zhang
California
57
530
User
Illinois
56
531
User
California
56
532
Nathan Johnson
Ohio
56
533
Syed Mohammad
Pennsylvania
56
534
Albert Caputo
Massachusetts
56
535
Greck
North Carolina
56
536
noob
California
56
537
User
New York
56
538
Christopher D'Onofrio
New York
56
539
User
Ohio
56
540
User
Indiana
56
541
User
Indiana
56
542
User
Georgia
56
543
User
Texas
56
544
Matthew Nolan-Cintron
Virginia
56
545
User
California
56
546
User
New York
56
547
Pranay Ranjan
New York
56
548
Sandy M.
Wisconsin
56
549
Long John
California
56
550
User
New York
56
551
User
New York
56
552
Eric Mozeika
North Carolina
56
553
User
Washington
56
554
John Appleseed
New York
56
555
User
California
56
556
User
New York
56
557
User
Pennsylvania
56
558
Oleg Aprelikov
Indiana
56
559
User
New York
56
560
Reva Sobti
California
56
561
User
New York
56
562
User
Region
56
563
User
Pennsylvania
55
564
User
Vermont
55
565
User
New York
55
566
Katie Lee
Washington
55
567
User
Massachusetts
55
568
Joseph Buzzi
Illinois
55
569
User
New Mexico
55
570
User
Texas
55
571
FelixM
California
55
572
User
Texas
55
573
Sheikh Selim
Maryland
55
574
Cannon Tuttle
California
55
575
Paula Paintsil
Illinois
55
576
Rebecca MacDonald
Washington
55
577
User
Missouri
55
578
User
New York
55
579
Tyler Stetson
New York
55
580
User
Texas
55
581
User
Virginia
55
582
User
Region
55
583
Dan Stewart
New York
55
584
Can Yavas
Rhode Island
55
585
Yamen Al Masalmeh
Illinois
55
586
User
New York
55
587
User
California
55
588
User
California
55
589
User
Pennsylvania
55
590
Puddles Penguin
Ohio
55
591
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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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