Leaderboard
United States
United States
Worldwide
Rank
Level
2
User
Michigan
97
3
Siva
Illinois
96
4
Sawyer West
Maryland
95
5
EASTON SINGER
Ohio
95
6
User
Virginia
94
7
Michalis Paizanis
Texas
94
8
Aden Smith
Texas
94
9
eixsinxicosx
New York
94
10
FelixM
California
93
11
DoMo
New Jersey
93
12
User
Arkansas
93
13
User
New Jersey
93
14
ManInJelly
Pennsylvania
92
15
Tyson Barney
Utah
92
16
Andrew Ahn
Massachusetts
92
17
Grizzly Bear
Washington
91
18
User
Pennsylvania
91
19
Spooner
Maine
91
20
Anthony McWilliams
New Jersey
91
21
Andrew Smiciklas
Nevada
91
22
Daniel Kim
New Jersey
91
23
Anand Bahl
Michigan
91
24
Rich Caputo
Massachusetts
90
25
Kvothe
Illinois
90
26
matt dowling
South Dakota
90
27
User
Ohio
90
28
Kyle Xiao
New Jersey
90
29
Sauske
Washington
89
30
User
Illinois
89
31
Osten Loo
Texas
89
32
Gordon He
California
89
33
User
Maryland
88
35
Ryan Zhang
Colorado
88
36
Adam Chen
Texas
88
37
user
California
88
38
User
Illinois
88
39
timmytanner21
Washington
87
40
Peter Andersson
Georgia
87
41
Tyler Shen
Connecticut
87
42
Andrew Stepp
California
87
43
Albert Caputo
Massachusetts
86
44
User
Pennsylvania
86
45
Ben Yu
Washington
86
46
FARIS
New York
86
47
User
Pennsylvania
86
48
User
New Jersey
86
49
Grace Chang
Hawaii
86
50
Blaziken Boi
Georgia
86
51
User
California
86
52
Jonathan G
New York
86
53
Oleg Aprelikov
Indiana
85
54
User
California
85
55
TJ Cubing
Florida
85
57
User
New York
85
58
User
New York
85
60
Sneh Patel
California
85
61
User
New York
85
62
Ellen Khat
Colorado
85
63
Joseph Yu
Indiana
84
64
Alexandre Leonetti
Region
84
65
User
Kansas
84
66
Cory Huelskamp
Ohio
84
67
User
New York
84
68
UserChitti
Washington
84
69
Hui Cai
Michigan
84
70
User
Illinois
84
71
Timothy So
California
83
72
Иван Мирошниченко
California
83
73
Kevin Copeland
Texas
83
74
balsu
New Jersey
83
75
User
Illinois
83
76
User
New York
83
77
User
Tennessee
83
78
User
Illinois
83
79
User
Illinois
82
80
camden
Georgia
82
81
User
Texas
82
82
Mark Wo
Hawaii
82
83
User
New York
82
84
User
New York
82
85
Daneo
Washington
82
86
User
Florida
82
87
Astik Gupta
New York
82
89
User
Illinois
82
90
User
Illinois
82
91
John Laboe
Illinois
82
92
Sora Ngin
California
82
93
User
New York
81
95
Timothy Gates
Virginia
81
96
User
New York
81
97
kwikmafs
New York
81
98
User
New York
81
99
User
Illinois
81
100
User
New York
81
101
User
New York
81
102
User
New York
81
103
User
Pennsylvania
81
105
User
New York
81
106
Shin
Illinois
81
107
User
New York
81
108
User
New Jersey
81
109
Vincent Lee
California
81
110
User
New Jersey
81
111
Davis Zhang
Tennessee
81
112
Kent A
North Carolina
81
113
User
Colorado
81
114
User
Connecticut
81
115
James Hu
Pennsylvania
81
116
User
Illinois
81
117
Michael Cognata
Texas
80
118
Alexander Treibit
New York
80
119
User
Connecticut
80
120
User
New York
80
121
justin silver
Texas
80
122
User
New York
80
123
User
New York
80
124
LIGHTSTARE
Ohio
80
125
User
Kansas
80
126
User
Illinois
80
127
Jordan Wells
Texas
80
128
Shae Wang
Mississippi
80
129
Le azien
New York
79
130
Jiawei Huang
Illinois
79
131
User
Virginia
79
133
User
Massachusetts
79
134
User
New York
79
135
User
California
79
136
User
New York
79
138
User
California
79
139
User
New Hampshire
79
140
Albert Samaha
Massachusetts
79
141
Matthew
South Carolina
78
143
trllisbad
Massachusetts
78
144
Chris Rodriguez
Florida
78
145
User
New York
78
146
User
New York
78
147
User
New York
78
148
User
New York
78
149
Jack Wang
New York
78
150
User
California
78
151
Michelle Abigan
Pennsylvania
78
152
Jon Yoon
Ohio
78
153
User
Pennsylvania
77
155
Zach George
Illinois
77
156
Shelley Chang
California
77
157
User
New York
77
158
Aiman
Illinois
77
159
User
New York
77
160
User
New York
77
161
User
New York
77
162
User
New York
77
163
User
New York
77
164
Trammell Saltzgaber
New Hampshire
77
165
Sagar Patel
Illinois
77
166
Josh Jung
Pennsylvania
77
167
Kevin Li
Massachusetts
77
168
Adrian G Casillas
Texas
77
169
User
Pennsylvania
77
170
User
Illinois
77
171
User
Indiana
77
172
CellPhoneJeff
California
76
173
User
Indiana
76
175
Daniel Cho
Texas
76
176
User
Utah
76
177
User
New York
76
178
User
New York
76
179
User
New York
76
180
User
New York
76
181
User
New York
76
182
Fred Jones
Iowa
76
183
Mohamed Tchambaz
Virginia
76
184
User
Illinois
76
185
User
Arizona
76
186
User
Tennessee
76
187
User
Maine
75
188
Vinu
West Virginia
75
189
Chengjian Shi
Illinois
75
190
Robert Silvy
Texas
75
191
User
Texas
75
192
Andrew Wu
District of Columbia
75
193
User
New York
75
194
User
Washington
75
195
sourry
Texas
75
196
User
Illinois
75
197
User
Massachusetts
75
198
User
Texas
74
199
Ansun Sujoe
Texas
74
200
User
New Jersey
74
201
User
Texas
74
202
User
New York
74
203
User
Utah
74
204
User
New York
74
205
Pickle
Virginia
74
206
Leafy Onion
Virginia
73
207
William Pan
Missouri
73
208
Ignacio Barela
Washington
73
209
Brandt Lawson
Missouri
73
210
Kevin Nguyen
California
73
211
User
New York
73
212
User
Maryland
73
213
Sejune Kwon
New York
73
214
User
Illinois
73
215
Adub
Florida
72
216
Mason Lien
California
72
217
Elana Greenberg
Missouri
72
218
Jiachen Xu
New York
72
219
Kailey
Washington
72
220
Xi Chen
California
72
221
User
New York
72
222
User
New York
72
223
User
New York
72
224
User
Massachusetts
72
225
User
Illinois
72
226
User
New York
72
227
User
New York
71
228
Matias
California
71
229
Huidong Ren
Illinois
71
230
Cody Goldman
Colorado
71
231
User
New Jersey
71
232
User
New York
71
233
User
New York
71
234
User
California
71
235
User
New York
71
236
User
California
71
237
Paige
Arkansas
71
238
User
Illinois
71
239
User
Illinois
71
240
User
California
71
241
User
Illinois
71
242
Pi Monster
Texas
70
243
Theo Morrison
Hawaii
70
244
User
California
70
245
User
Texas
70
246
User
New York
70
247
Imhotep Truitt
North Carolina
70
248
sLiNky
North Carolina
70
249
User
Texas
70
250
Tushar Sharma
Illinois
70
251
User
New York
70
252
User
New York
70
253
User
Massachusetts
70
254
User
California
70
255
User
Illinois
70
256
Takahiro Fushimi
California
70
257
User
Missouri
70
258
User
Florida
70
259
HONGKONGNEES
Virginia
69
260
User
Massachusetts
69
261
User
Texas
69
262
Ima Coder
Massachusetts
69
263
Sushant Vema
California
69
264
Angel Escobar
North Carolina
69
265
User
New York
69
266
Leina McDermott
New Hampshire
69
267
User
New York
69
268
a p
Rhode Island
69
269
kp
New York
69
270
User
New York
69
271
MelGibson
Illinois
69
272
Heath Vincent
Kansas
68
273
Zhangyi Chen
New York
68
274
Nate Kinast
Arkansas
68
275
User
California
68
276
Joseph Holbert
West Virginia
68
277
User
Tennessee
68
278
User
Florida
68
279
User
New Jersey
68
280
User
California
67
281
King of the Hill
Ohio
67
282
Bob Basmaji
District of Columbia
67
283
Justin Blank
North Carolina
67
284
Derek Cheung
Texas
67
285
User
Michigan
67
287
User
California
67
288
Yohan Ninan
New Jersey
67
289
Naeem Semerkant
New York
67
290
User
New York
67
291
Justin D'Souza
California
67
292
User
Illinois
67
293
User
Nevada
67
294
Jeff Paris
Nevada
67
295
User
Colorado
66
296
JarredHodges
Alabama
66
297
User
Washington
66
298
User
New Jersey
66
299
User
California
66
300
User
Massachusetts
66
301
ROBERT BATCHLET
Ohio
66
302
Ryan H
Massachusetts
66
303
User
Illinois
66
304
Krit
Pennsylvania
66
305
User
California
65
306
User
California
65
307
User
California
65
308
Brian Caulfield
New York
65
309
User
California
65
310
Lincoln
Kentucky
65
311
User
Connecticut
65
312
Sunil Devaraj
Arizona
65
313
User
California
65
314
Chrissy Day
Virginia
65
315
User
New York
65
316
User
Illinois
65
317
User
Pennsylvania
65
318
User
Texas
65
319
User
Tennessee
64
320
Myra Lee
Washington
64
321
3.1415926535897932
Illinois
64
322
Mickey
Maine
64
323
Anyazelie M. Zéphyaire
Massachusetts
64
324
Nikhar Patel
North Carolina
64
325
User
Massachusetts
64
326
LM
New York
64
327
User
South Dakota
64
328
Joel Smith
California
64
329
Daniel Bela Szerbak
Arkansas
64
330
User
California
64
331
User
Georgia
64
332
Gabriel Diraviam
Florida
64
333
ChessWhiz
Oregon
63
334
Andrew James Smiciklas
Nevada
63
335
User
Texas
63
336
User
New York
63
337
User
Pennsylvania
63
338
User
Texas
63
339
User
California
63
340
User
Texas
63
341
Abe Lusk
New York
63
342
User
Connecticut
63
343
User
New York
63
344
Carl Svahn
Texas
63
345
User
Pennsylvania
63
346
User
Florida
63
347
User
Illinois
63
348
User
California
63
349
User
Hawaii
62
350
User
Indiana
62
351
User
California
62
352
User
Texas
62
353
User
Georgia
62
354
Lou
Florida
62
355
Jingyi Wang
Region
62
356
Kenny Huang
Texas
62
357
Evgenia Chubata
Illinois
62
358
User
California
62
359
frances
Pennsylvania
62
360
User
California
62
361
User
Pennsylvania
62
362
User
California
62
363
User
South Dakota
62
364
User
Illinois
62
365
Cian C
New York
62
366
User
Maryland
62
367
User
Tennessee
62
368
User
Texas
61
369
User
Texas
61
370
User
California
61
371
User
New York
61
372
User
California
61
373
Takanori Tojikubo
Virginia
61
374
User
California
61
375
User
Texas
61
376
User
Illinois
61
377
Christopher Ortiz
North Carolina
61
378
Matteo M
Region
61
379
User
Maryland
61
380
User
New Jersey
61
381
User
New York
61
382
Sabrina Noborikawa
Florida
61
383
User
Pennsylvania
61
384
Yeonwoo Jang
New York
61
385
Raphael Zerah
New York
61
386
User
Washington
61
387
User
California
61
388
Anna Claire Kinsey
Alabama
61
389
Nolan Raimo
Texas
61
390
User
California
61
392
Katherine Tran
California
61
393
Joelle Young
Mississippi
61
394
User
New Jersey
61
395
User
New Jersey
61
396
User
Virginia
61
397
Xx_360NoscopeBigChungusDogLasagna_xX
Washington
60
398
Bob Luo
Texas
60
399
Stanislav Perumov
Massachusetts
60
400
User
Missouri
60
401
Abhinav Goel
Ohio
60
402
PatrickS
Virginia
60
403
User
Texas
60
404
User
Illinois
60
405
pizza
California
60
406
User
Texas
60
407
Ted Silver
New York
60
408
User
California
60
409
Yiteng Zhang
Illinois
60
410
User
Pennsylvania
60
411
Teddy
New York
60
412
User
Pennsylvania
60
413
Arnav Gupta
Alabama
59
414
Neil Chriss
New York
59
415
1. Thou Shant Lewd Kaori
Pennsylvania
59
416
User
New York
59
417
User
California
59
418
User
Texas
59
419
Eric
Illinois
59
420
User
Texas
59
421
User
Connecticut
59
422
EVAN SOMMER - STUDENT
Missouri
59
423
dava
West Virginia
59
424
User
Texas
59
425
Mark Hernandez
Oregon
59
426
bhaskar mishra
Florida
59
427
Timothy Tan
Washington
59
428
User
California
59
429
User
Pennsylvania
59
430
User
Michigan
59
431
User
Utah
59
432
User
Texas
59
433
User
Michigan
59
434
User
New Jersey
59
435
User
California
59
436
User
New York
59
437
Bernd Pfrommer
Missouri
58
438
User
Delaware
58
439
Amandine Roche
New York
58
440
Amineh Beltran
California
58
441
Danielle Teter
Louisiana
58
442
User
California
58
443
User
New York
58
444
User
Texas
58
445
User
California
58
446
Hector Orea
Florida
58
447
User
New York
58
448
User
New York
58
449
Prachet Trivedi
New Jersey
58
450
User
New York
58
451
Gianni Crivello
North Carolina
58
452
Reed Tubbs
Texas
58
453
User
Colorado
58
454
Andrew
California
58
455
al s
Nevada
58
456
User
Illinois
58
457
SemperJon B82
Illinois
58
458
Andy Lee
California
58
459
User
Massachusetts
58
460
mishmash
Georgia
58
461
User
New York
58
462
User
Illinois
58
463
Pinchen Fan
Minnesota
58
464
AresTheMars
New York
58
465
User
California
58
466
Sheikh Selim
Maryland
57
467
User
California
57
468
User
New York
57
469
User
California
57
470
I got many skills but failing is not one
California
57
471
Danny Dark
New York
57
472
L. J.
Pennsylvania
57
473
User
California
57
475
Joseph Holbert
West Virginia
57
476
User
New Jersey
57
477
Benjamin Ward
Illinois
57
478
User
New Jersey
57
479
User
West Virginia
57
480
Michael B
New Jersey
57
481
User
Georgia
57
482
User
Missouri
57
483
User
Connecticut
57
484
User
California
57
485
User
New York
57
486
User
California
57
487
User
California
57
488
User
Illinois
57
489
Jennifer Guerra
North Carolina
57
491
User
Minnesota
57
492
User
Utah
57
493
User
Minnesota
57
494
User
New York
57
495
pigeonrepublic
New York
57
496
User
New York
56
497
Daregs
Connecticut
56
498
Jacob Yuruki
Hawaii
56
499
Akash Marakath
Maryland
56
500
User
Louisiana
56
501
User
Colorado
56
502
Chris Davis
Oregon
56
503
User
Florida
56
504
David Barr
Washington
56
505
Barrett Block
Kentucky
56
506
User
Virginia
56
507
User
New Hampshire
56
508
User
Arizona
56
509
User
Washington
56
510
User
Illinois
56
511
Alice
California
56
512
Al s
Nevada
56
513
User
California
56
514
User
Georgia
56
515
User
Connecticut
56
516
User
Georgia
56
517
User
Tennessee
56
518
Gabe Korer
Arizona
56
519
User
Connecticut
56
520
Xander Bystrom
California
56
521
User
Tennessee
56
522
User
Virginia
55
523
D G
Connecticut
55
524
User
Missouri
55
525
Lukas Baker
Connecticut
55
526
Zack Lee
Pennsylvania
55
527
User
Michigan
55
528
User
Michigan
55
529
Mark Durbin
California
55
530
User
Ohio
55
531
User
Illinois
55
532
User
California
55
533
User
California
55
534
User
Tennessee
55
535
Zhiwei Jing
California
55
536
User
Utah
55
537
Steven Maradiaga
Florida
55
538
Alexander Paskal
Maryland
55
539
User
New York
55
540
cool guy
Texas
55
541
User
New Jersey
55
542
Karol Rychlik
New York
55
543
User
Florida
55
544
User
New Hampshire
55
545
User
Pennsylvania
55
546
Megan Le
New York
55
547
User
Illinois
55
548
AruG
Alabama
54
549
Arjun Arun
Illinois
54
550
Victoria Phillips
Missouri
54
551
Joseph Cano
Texas
54
552
Travis Grady
California
54
553
User
Indiana
54
554
User
Minnesota
54
555
User
New York
54
556
User
Virginia
54
557
User
California
54
558
User
Florida
54
559
User
Tennessee
54
560
jtv atheist
Kentucky
54
561
da baby
Rhode Island
54
562
AT
Illinois
54
563
User
New York
54
564
Dee Sullivan
Illinois
54
565
Connor Stanton
Kansas
54
566
User
Illinois
54
567
Emanuel Melaku
Ohio
54
568
Raj Amarnani
New York
54
569
User
New Jersey
54
570
User
New York
54
571
User
Tennessee
54
572
jkuzi
Massachusetts
54
573
User
New York
54
574
karanbir Singh
California
54
575
Owen Smith
North Carolina
54
576
Bosssssssss
Colorado
54
577
Rohan Jasani
California
54
578
User
Pennsylvania
54
579
User
Illinois
53
580
chemita
Wisconsin
53
581
Ricky Kirk
Texas
53
582
Will St. Pierre
Texas
53
583
User
California
53
584
User
Texas
53
585
List
New York
53
586
User
California
53
587
User
Michigan
53
588
NOLA Math
Louisiana
53
589
Christian Tang
Florida
53
590
Anton Yakunin
Iowa
53
591
Latissimus65
Pennsylvania
53
592
User
New York
53
593
User
New York
53
594
User
Illinois
53
595
User
Mississippi
53
596
User
Wisconsin
53
597
Euclid1023
Connecticut
53
598
User
Florida
53
600
akansha sehgal
Maryland
53
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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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