Leaderboard
United States
United States
Worldwide
Rank
Level
1
Detective Pikachu
Washington
100
2
Debabi
Washington
100
3
Spondulix
Minnesota
100
4
User
Washington
99
5
User
Virginia
99
6
Timothy So
California
99
7
imo miklos
Washington
99
8
Matthew
South Carolina
98
9
User
Virginia
98
10
User
Virginia
97
11
User
Michigan
97
12
Shanon
Florida
96
13
Siva
Illinois
96
14
Sawyer West
Maryland
95
15
EASTON SINGER
Ohio
95
16
The Rarest Frog
Missouri
94
17
timmytanner21
Washington
94
18
User
Maryland
94
19
Jeffrey Huang
Texas
94
20
User
Virginia
94
21
Michalis Paizanis
Texas
94
22
Aden Smith
Texas
94
23
eixsinxicosx
New York
94
24
User
California
93
25
User
Washington
93
26
Joey
Nevada
93
27
User
Washington
93
28
User
Michigan
93
29
ravenclawstrength 7
Washington
93
30
FelixM
California
93
31
DoMo
New Jersey
93
32
User
Arkansas
93
33
User
New Jersey
93
34
Darren Chen
California
92
35
Sauske
Washington
92
36
User
Texas
92
37
Grizzly Bear
Washington
92
38
ManInJelly
Pennsylvania
92
39
Andrew Ahn
Massachusetts
92
40
Ima Coder
Massachusetts
91
41
Rich Caputo
Massachusetts
91
42
User
Virginia
91
43
User
Texas
91
44
Ben Brown
Alabama
91
45
David Keen
Indiana
91
46
User
New Jersey
91
47
User
Texas
91
48
User
Virginia
91
49
Teague Stotlar
Illinois
91
50
CEL IV
Texas
91
51
User
Pennsylvania
91
52
Spooner
Maine
91
53
Anthony McWilliams
New Jersey
91
54
Andrew Smiciklas
Nevada
91
55
Anand Bahl
Michigan
91
56
User
Texas
90
57
User
Texas
90
58
User
Texas
90
59
User
Texas
90
60
Kate Goloboy
Massachusetts
90
61
Alice
West Virginia
90
62
Kvothe
Illinois
90
63
matt dowling
South Dakota
90
64
User
Ohio
90
65
Kyle Xiao
New Jersey
90
66
User
Texas
89
67
User
Nevada
89
68
Morgan Bryant
North Carolina
89
69
User
Texas
89
70
User
Illinois
89
71
Gordon He
California
89
72
User
Texas
88
73
CEL-IV
Texas
88
74
CEL IV
Texas
88
76
Ryan Zhang
Colorado
88
77
Adam Chen
Texas
88
78
user
California
88
79
User
Illinois
88
80
User
Texas
87
81
Myra Lee
Washington
87
82
King of the Hill
Ohio
87
83
User
Texas
87
84
User
Texas
87
85
Ethan Zh
Virginia
87
86
CEL IV
Texas
87
87
Peter Andersson
Georgia
87
88
Tyler Shen
Connecticut
87
89
Andrew Stepp
California
87
90
User
New York
86
91
User
Texas
86
92
Tanishq Pauskar
Ohio
86
94
Michael Li
Washington
86
95
User
New Jersey
86
96
Andrew Couser
Texas
86
97
User
Texas
86
98
User
California
86
99
Michael Lu
Illinois
86
100
Kaden Chen
Massachusetts
86
101
Albert Caputo
Massachusetts
86
102
User
Pennsylvania
86
103
Ben Yu
Washington
86
104
FARIS
New York
86
105
User
Pennsylvania
86
106
User
New Jersey
86
107
Blaziken Boi
Georgia
86
108
User
California
86
109
Jonathan G
New York
86
110
Travis Grady
Florida
85
111
Frank Qiang
Alabama
85
112
User
Illinois
85
113
User
Texas
85
114
Luis Sanchez-Ulloa
Washington
85
115
User
California
85
116
Oleg Aprelikov
Indiana
85
117
TJ Cubing
Florida
85
119
User
New York
85
120
User
New York
85
121
Josh
Massachusetts
85
122
Sneh Patel
California
85
123
User
New York
85
124
Ellen Khat
Colorado
85
125
Angel Escobar
North Carolina
84
126
Le azien
Region
84
127
Jessica He
Georgia
84
128
User
Texas
84
129
Benjamin Kang
New York
84
130
Benjamin Kang
New York
84
131
Joseph Yu
Indiana
84
132
Alexandre Leonetti
Region
84
133
User
Kansas
84
134
Cory Huelskamp
Ohio
84
135
User
New York
84
136
UserChitti
Washington
84
137
Hui Cai
Michigan
84
138
User
Illinois
84
139
Tracy Lenahan
Vermont
83
140
User
Maine
83
141
Mike Orr
Texas
83
142
Brian Caulfield
New York
83
143
Chengjian Shi
Illinois
83
144
User
California
83
145
User
Kansas
83
146
Matias
California
83
147
User
Texas
83
148
User
Maryland
83
149
Иван Мирошниченко
California
83
150
Kevin Copeland
Texas
83
151
balsu
New Jersey
83
152
User
Illinois
83
153
User
New York
83
154
User
Tennessee
83
155
User
Illinois
83
156
Nes370
California
82
157
User
New York
82
158
Daregs
Connecticut
82
159
User
Illinois
82
160
User
Texas
82
161
User
Texas
82
162
User
New York
82
163
Jordan Katz
California
82
164
User
Texas
82
165
User
New York
82
166
User
New York
82
167
Daneo
Washington
82
168
Astik Gupta
New York
82
169
User
Massachusetts
82
170
User
Illinois
82
171
User
Illinois
82
172
John Laboe
Illinois
82
173
User
New York
81
174
User
Virginia
81
175
User
New York
81
176
Haining Zhou
Illinois
81
177
User
Illinois
81
178
MODOZ
North Carolina
81
179
User
New York
81
180
Pi
Michigan
81
181
User
Texas
81
182
User
California
81
183
User
California
81
184
User
New York
81
186
Timothy Gates
Virginia
81
187
User
New York
81
188
kwikmafs
New York
81
189
User
New York
81
190
User
Illinois
81
191
User
New York
81
192
User
New York
81
193
User
New York
81
194
User
Pennsylvania
81
196
User
New York
81
197
Shin
Illinois
81
198
User
New York
81
199
User
New Jersey
81
200
Vincent Lee
California
81
201
User
New Jersey
81
202
Davis Zhang
Tennessee
81
203
Kent A
North Carolina
81
204
User
Colorado
81
205
User
Connecticut
81
206
James Hu
Pennsylvania
81
207
User
Illinois
81
208
User
Pennsylvania
80
209
Haining Zhou
Illinois
80
210
User
Texas
80
211
Alexander Treibit
New York
80
212
User
Connecticut
80
213
User
New York
80
214
justin silver
Texas
80
215
User
New York
80
216
User
New York
80
217
User
Illinois
80
218
Jordan Wells
Texas
80
219
Shae Wang
Mississippi
80
220
III
New Jersey
79
221
User
Texas
79
222
User
New Mexico
79
223
Srikar Satluri
Georgia
79
224
rohaN 🐖
Georgia
79
225
Mohamed Tchambaz
Virginia
79
226
Parsa Novin
Georgia
79
227
camden
Illinois
79
228
Stanislav Perumov
New York
79
229
Lukas Baker
Connecticut
79
230
pianotennisman
South Dakota
79
231
User
Pennsylvania
79
232
User
Texas
79
233
Heath Vincent
Kansas
79
234
robert hou
California
79
235
Nikhar Patel
North Carolina
79
236
Jiawei Huang
Illinois
79
237
User
Virginia
79
239
User
Massachusetts
79
240
User
New York
79
241
User
California
79
242
User
New York
79
244
User
California
79
245
User
New Hampshire
79
246
Albert Samaha
Massachusetts
79
247
Caleb Strange
Texas
78
248
User
Illinois
78
249
Tim Zheng
Florida
78
250
User
New York
78
251
User
New York
78
252
User
New York
78
253
User
New York
78
254
Pi Monster
Texas
78
256
trllisbad
Massachusetts
78
257
Chris Rodriguez
Florida
78
258
User
New York
78
259
User
New York
78
260
User
New York
78
261
User
New York
78
262
Jack Wang
New York
78
263
User
California
78
264
Michelle Abigan
Pennsylvania
78
265
Jon Yoon
Ohio
78
266
User
Georgia
77
267
Dean Lee
California
77
268
User
New York
77
269
User
Pennsylvania
77
270
User
Pennsylvania
77
272
Zach George
Illinois
77
273
Shelley Chang
California
77
274
User
New York
77
275
Aiman
Illinois
77
276
User
New York
77
277
User
New York
77
278
User
New York
77
279
User
New York
77
280
User
New York
77
281
Trammell Saltzgaber
New Hampshire
77
282
Sagar Patel
Illinois
77
283
Josh Jung
Pennsylvania
77
284
Kevin Li
Massachusetts
77
285
Adrian G Casillas
Texas
77
286
User
Pennsylvania
77
287
User
Illinois
77
288
User
Indiana
77
289
User
Pennsylvania
76
290
User
Pennsylvania
76
291
User
Texas
76
292
User
Pennsylvania
76
293
User
New York
76
294
Vinu
West Virginia
76
295
User
California
76
296
Jeff Rangan
California
76
297
CellPhoneJeff
California
76
299
Daniel Cho
Texas
76
300
User
Utah
76
301
User
New York
76
302
User
New York
76
303
User
New York
76
304
User
New York
76
305
User
New York
76
306
Fred Jones
Iowa
76
307
User
Illinois
76
308
User
Arizona
76
309
User
Tennessee
76
310
Alex
Nevada
75
311
ag ar
California
75
312
User
Florida
75
313
User
Missouri
75
314
James Doe
Virginia
75
315
User
California
75
316
User
Texas
75
317
Anyazelie M. Zéphyaire
Massachusetts
75
318
Robert Silvy
Texas
75
319
User
Texas
75
320
Andrew Wu
District of Columbia
75
321
User
New York
75
322
User
Washington
75
323
sourry
Texas
75
324
User
Illinois
75
325
User
Massachusetts
75
326
Long John
California
74
327
User
Georgia
74
328
Arnav Gupta
Wisconsin
74
329
User
Texas
74
330
User
California
74
331
User
Minnesota
74
332
User
Texas
74
333
Ashwin Sreevatsa
Michigan
74
334
User
Texas
74
335
User
California
74
336
User
Indiana
74
337
User
Texas
74
338
Ansun Sujoe
Texas
74
339
User
New Jersey
74
340
User
Texas
74
341
User
New York
74
342
User
Utah
74
343
User
New York
74
344
Pickle
Virginia
74
345
User
New York
73
346
User
New York
73
347
User
New York
73
348
Puddles Penguin
Ohio
73
349
Arnav Gupta
Alabama
73
350
User
California
73
351
Adub
Florida
73
352
Leafy Onion
Virginia
73
353
William Pan
Missouri
73
354
Ignacio Barela
Washington
73
355
Brandt Lawson
Missouri
73
356
Kevin Nguyen
California
73
357
User
New York
73
358
User
Maryland
73
359
Sejune Kwon
New York
73
360
User
Illinois
73
361
User
California
72
362
Daniel Benayoun
Florida
72
363
Katie Lee
Washington
72
364
User
New York
72
365
User
New York
72
366
Anitya Gupta
New Jersey
72
367
Xx_360NoscopeBigChungusDogLasagna_xX
Washington
72
368
User
Michigan
72
369
User
California
72
370
User
Texas
72
371
User
Pennsylvania
72
372
User
Texas
72
373
User
California
72
374
Mason Lien
California
72
375
Elana Greenberg
Missouri
72
376
Jiachen Xu
New York
72
377
Kailey
Washington
72
378
Xi Chen
California
72
379
User
New York
72
380
User
New York
72
381
User
New York
72
382
User
Massachusetts
72
383
User
Illinois
72
384
User
New York
72
385
John Smith
Washington
71
386
User
Texas
71
387
Doug Childs
Texas
71
388
User
Michigan
71
389
Skinny Penis
Region
71
390
Cannon Tuttle
California
71
391
Zavosh
California
71
392
User
Michigan
71
393
Anurag Gupta
Wisconsin
71
394
Varad Rasalkar
Illinois
71
395
User
California
71
396
William Qiu
New Jersey
71
397
User
Virginia
71
398
User
New York
71
399
Huidong Ren
Illinois
71
400
Cody Goldman
Colorado
71
401
User
New Jersey
71
402
User
New York
71
403
User
New York
71
404
User
California
71
405
User
New York
71
406
User
California
71
407
User
Illinois
71
408
User
Illinois
71
409
User
California
71
410
User
Illinois
71
411
Cameron Allen
Connecticut
70
412
Tom Camargo
New York
70
413
User
New York
70
414
Bernd Pfrommer
Pennsylvania
70
415
User
Georgia
70
416
User
New Jersey
70
417
User
Michigan
70
418
User
Michigan
70
419
User
New York
70
420
3.1415926535897932
Illinois
70
421
User
Texas
70
422
Theo Morrison
Hawaii
70
423
User
California
70
424
User
Texas
70
425
User
New York
70
426
Imhotep Truitt
North Carolina
70
427
sLiNky
North Carolina
70
428
User
Texas
70
429
Tushar Sharma
Illinois
70
430
User
New York
70
431
User
New York
70
432
User
Massachusetts
70
433
User
California
70
434
User
Illinois
70
435
Takahiro Fushimi
California
70
436
User
Missouri
70
437
User
Colorado
69
438
Dustin
Georgia
69
439
User
California
69
440
User
California
69
441
Aidan Lytle
North Carolina
69
442
User
California
69
443
User
New Jersey
69
444
User
Massachusetts
69
445
User
New York
69
446
Marcuschau
Washington
69
447
User
Texas
69
448
User
Michigan
69
449
j s
Hawaii
69
450
User
Alabama
69
451
User
Massachusetts
69
452
User
Texas
69
453
Sushant Vema
California
69
454
User
New York
69
455
Leina McDermott
New Hampshire
69
456
User
New York
69
457
a p
Rhode Island
69
458
kp
New York
69
459
User
New York
69
460
MelGibson
Illinois
69
461
User
Virginia
68
462
tony huang
Virginia
68
463
User
Massachusetts
68
464
User
New York
68
465
Aditya Gupta
Alabama
68
466
User
New York
68
467
Andrea Lim
Washington
68
468
User
Missouri
68
469
User
New York
68
470
User
Texas
68
471
User
Michigan
68
472
Joy Kim
Washington
68
473
SOPHIE LI
Florida
68
474
Zhangyi Chen
New York
68
475
Nate Kinast
Arkansas
68
476
User
California
68
477
Joseph Holbert
West Virginia
68
478
User
Tennessee
68
479
User
Florida
68
480
User
New Jersey
68
481
Bryan Vu
Texas
67
482
Brandon Pozil
North Carolina
67
483
User
New York
67
484
User
Pennsylvania
67
485
User
New York
67
486
User
New York
67
487
User
New York
67
488
User
New York
67
489
User
Ohio
67
490
User
New York
67
492
User
Texas
67
493
Estra
Washington
67
494
Will St. Pierre
Colorado
67
495
Bob Basmaji
District of Columbia
67
496
Justin Blank
North Carolina
67
497
Derek Cheung
Texas
67
498
User
Michigan
67
500
User
California
67
501
Naeem Semerkant
New York
67
502
User
New York
67
503
User
Illinois
67
504
Jeff Paris
Nevada
67
505
User
Texas
66
506
Nate
Ohio
66
507
User
Texas
66
508
Joseph Cano
Texas
66
509
Dennisson Sherrell
Tennessee
66
510
Traveon Parker [STUDENT]
Nevada
66
511
User
Illinois
66
512
Ryan H
Florida
66
513
User
Illinois
66
514
Collin Crader
Connecticut
66
515
User
Texas
66
516
Jeu
Michigan
66
517
User
Colorado
66
518
JarredHodges
Alabama
66
519
User
Washington
66
520
User
New Jersey
66
521
User
California
66
522
User
Massachusetts
66
523
ROBERT BATCHLET
Ohio
66
524
User
Illinois
66
525
User
Washington
65
526
User
California
65
527
User
California
65
528
User
California
65
529
User
Missouri
65
530
KK SS
Texas
65
531
User
Texas
65
532
User
Virginia
65
533
User
Texas
65
534
User
Illinois
65
535
James Milgram
Wisconsin
65
536
User
California
65
537
User
New York
65
538
User
Colorado
65
539
I got many skills but failing is not one
California
65
540
User
California
65
541
User
California
65
542
User
California
65
543
User
California
65
544
Lincoln
Kentucky
65
545
User
Connecticut
65
546
Sunil Devaraj
Arizona
65
547
User
California
65
548
Chrissy Day
Virginia
65
549
User
New York
65
550
User
Illinois
65
551
User
Pennsylvania
65
552
User
Texas
65
554
User
Virginia
64
555
User
Pennsylvania
64
556
User
Texas
64
557
Math god
Texas
64
558
User
Texas
64
559
User
New York
64
560
User
Indiana
64
561
User
Texas
64
562
Sheikh Selim
Maryland
64
563
User
Texas
64
564
User
Michigan
64
565
User
Texas
64
566
Mickey
Maine
64
567
User
Massachusetts
64
568
LM
New York
64
569
User
South Dakota
64
570
Joel Smith
California
64
571
Daniel Bela Szerbak
Arkansas
64
572
User
California
64
573
User
Georgia
64
574
Gabriel Diraviam
Florida
64
575
User
Texas
63
576
J Hulme
Texas
63
577
User
Arizona
63
578
User
New York
63
579
Amineh Beltran
California
63
580
User
New York
63
581
User
New York
63
582
Steven Johnson
Tennessee
63
583
George Culver
New York
63
584
Joseph Churilla
Ohio
63
585
User
Utah
63
586
User
New York
63
587
Akash Marakath
Maryland
63
588
User
Texas
63
589
caden tan
Washington
63
590
Alla Shved
Connecticut
63
591
Arjun Arun
Illinois
63
592
User
Texas
63
593
User
Texas
63
594
User
Florida
63
595
ChessWhiz
Oregon
63
596
Andrew James Smiciklas
Nevada
63
597
User
Texas
63
598
User
New York
63
599
User
Pennsylvania
63
600
User
California
63
601
User
Texas
63
602
Abe Lusk
New York
63
603
User
Connecticut
63
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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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