Leaderboard
Canada
Canada
Worldwide
Rank
Level
1
Tyler Chiu
Ontario
99
2
User
Ontario
97
3
Alex Wen
British Columbia
96
5
Quang Nguyen
New Brunswick
92
6
Eggma Eggma
Ontario
88
7
User
Alberta
84
8
Raiyana Shams
Alberta
83
9
User
Ontario
83
10
Edward Lee
Ontario
82
11
Tyler Chiu
Ontario
82
12
User
Alberta
82
13
Yi Shi
Ontario
82
14
Alex Gay
Ontario
81
15
User
Alberta
81
16
User
Ontario
81
17
User
Ontario
81
19
Jeong Augustine
British Columbia
80
21
rdu
Ontario
80
24
User
Quebec
78
26
User
Ontario
78
27
i plai roblox all day bois and i dont stop and im play years haahahahhahahahahhaha and im making mac n chesse and no one can stop me
Quebec
77
28
Richard Williams
Quebec
77
29
Renee Weng
British Columbia
77
30
User
Ontario
76
31
User
Quebec
76
32
Mike Travis
Ontario
74
33
Shani Mithani
Ontario
74
34
Shahid M Salam
Quebec
73
35
Cornell Lee
British Columbia
71
36
Joey Hotz
Ontario
71
37
User
Quebec
70
38
jeong minji is better than jeong augustine
British Columbia
70
39
User
Ontario
69
40
User
Ontario
69
41
Aiko Fujita
Quebec
69
42
User
Ontario
68
43
User
Quebec
68
44
User
Ontario
68
45
User
Manitoba
67
46
Yichen Sun
Quebec
67
47
K. Miranda
Ontario
66
48
User
Ontario
66
49
Lyss
Ontario
65
50
Renee Weng
British Columbia
65
51
User
Ontario
65
52
Affan Bhandakya
British Columbia
64
53
Dave Coding Pro
Ontario
64
54
User
Ontario
64
55
Cindy
Ontario
64
56
User
Ontario
64
57
Martin Tin
Ontario
64
58
User
Ontario
64
59
john gay
Ontario
63
60
Hudson Vance
Ontario
63
61
awesome mathman
Ontario
62
62
Samuel Zheng
Ontario
62
63
robin yoo
Quebec
62
64
Chris Christensen
Ontario
62
65
User
Ontario
62
66
Ryan Han
British Columbia
61
67
User
Ontario
61
68
User
Quebec
61
69
Jayden Kallusky
Manitoba
61
70
User
Ontario
61
72
User
Ontario
61
74
User
Alberta
60
75
Lisa Hines
Ontario
60
76
Sukhminder Bath
Manitoba
60
77
User
Ontario
60
78
User
Ontario
60
79
User
Nova Scotia
60
81
User
Ontario
59
82
Azmy Taha
Ontario
59
83
User
Manitoba
59
85
Cameron Villetard
Alberta
58
86
Chester M
Ontario
58
87
User
Ontario
58
88
User
Ontario
58
89
B.Arindam
British Columbia
58
90
Kennens
Ontario
58
91
User
Ontario
58
92
User
Alberta
57
93
Matteo Espinosa Ramirez
Quebec
56
94
User
Manitoba
56
95
User
Ontario
56
96
User
Manitoba
55
97
Jeffrey Misner
Ontario
55
98
Corbin Brooks
Newfoundland and Labrador
55
99
User
Ontario
55
100
Matthew Hobbs
Ontario
55
101
User
Ontario
55
102
User
Quebec
54
103
User
Ontario
54
105
User
Ontario
54
106
User
Ontario
54
107
Khalisa Valani
Ontario
54
108
User
Ontario
54
109
User
Ontario
54
110
Marc-Antoine Joyal
Quebec
53
111
Elliot Wong
Ontario
52
112
User
Ontario
52
113
User
Ontario
52
114
Shashvat Varma
Manitoba
52
115
Yatin Kapur
Ontario
52
116
Mark Gravely
Ontario
52
118
Hao Hu
Ontario
52
119
Philippe
Quebec
52
120
User
Ontario
52
121
User
Ontario
52
122
User
Ontario
52
123
Antony Fournier
Quebec
52
124
Hugeldorf
Ontario
51
125
User
Ontario
51
126
User
Quebec
51
127
User
Ontario
51
128
Steven Craig
Quebec
51
129
User
Quebec
51
130
BenBen
Quebec
51
132
User
Ontario
51
133
Harjot Singh Parmar
Ontario
51
134
Hrafn Haugr
Alberta
50
135
User
Quebec
50
136
User
Quebec
50
137
Durraiz Ashraf
Ontario
50
138
User
Quebec
50
139
Diego
Quebec
50
140
Evan Zhu
Quebec
50
141
User
Ontario
50
142
User
Ontario
50
143
Ismaël
Ontario
50
144
Miniapple Emily
Quebec
50
145
User
Ontario
50
146
Tuong,Hoang
Quebec
50
148
User
Ontario
50
149
User
Alberta
50
150
User
Quebec
49
152
User
Ontario
49
153
User
Quebec
48
154
User
Alberta
48
155
User
Nova Scotia
48
157
User
Ontario
48
158
User
Ontario
48
159
User
Ontario
48
160
Joyce
Ontario
48
161
Kian Tuastomban
Alberta
47
162
Samuel Ho
Ontario
47
163
User
Manitoba
47
166
User
Ontario
47
167
User
Quebec
47
168
JiNo Yoon
Quebec
47
169
User
Ontario
47
170
User
New Brunswick
47
171
User
Ontario
47
172
Matthew Taylor
Ontario
46
173
User
Alberta
46
174
Nicholas Cayla
British Columbia
46
175
User
Ontario
46
176
Mark
Ontario
46
177
User
Ontario
46
178
User
Ontario
46
179
User
Ontario
46
180
Kevin Kyle
Ontario
46
181
User
Ontario
46
182
Dominic Gapero
Alberta
46
183
User
Ontario
46
184
User
Ontario
46
185
User
Ontario
46
186
User
Ontario
46
187
User
Ontario
46
188
Jagadish Gowtham Gokulaparanthaman
Ontario
45
189
User
Quebec
45
190
User
Ontario
45
191
User
Ontario
45
192
Connor Hogan
Saskatchewan
45
193
Ricardo Chang
Ontario
45
194
User
Alberta
45
196
User
Manitoba
45
197
User
Ontario
45
198
Prem
Ontario
45
199
User
Ontario
45
200
User
Quebec
45
201
Stefan Pizzinato
Ontario
45
202
Darwin To
British Columbia
45
203
User
Quebec
45
204
User
Ontario
45
205
User
Ontario
45
206
User
Quebec
45
207
Jordan Tarasoff
Alberta
45
209
User
Ontario
44
210
ringtail doofus
Ontario
44
211
User
Ontario
44
212
Jacob Fitzgerald
Nova Scotia
44
215
Meriem Saadaoui
Quebec
44
217
User
Ontario
44
218
User
Manitoba
44
219
jeong minji
British Columbia
44
220
Mohammed
Ontario
44
221
Heather M
British Columbia
44
222
User
Quebec
43
223
Arya holmukhe
Quebec
43
224
User
Ontario
43
226
yfradkin98@gmail.com
Ontario
43
227
Cobra
Alberta
43
228
User
New Brunswick
43
229
User
Quebec
43
230
User
Ontario
43
231
User
Ontario
43
232
User
Ontario
43
233
User
Ontario
42
234
Logan Peterson
British Columbia
42
235
User
New Brunswick
42
236
Annie
Ontario
42
237
User
Ontario
42
238
Donald Moodger
Quebec
42
239
Corey
Quebec
42
240
User
Ontario
42
241
User
Alberta
42
243
User
Ontario
42
244
User
Quebec
42
246
tender
Ontario
42
247
User
Ontario
42
248
Big E
Ontario
41
249
User
Quebec
41
251
Chocolate Chicken
Manitoba
41
252
Tyler Groves
Ontario
41
254
Jack Sunhaoya
Quebec
41
256
User
Alberta
41
257
User
Ontario
41
258
User
Quebec
41
259
User
Ontario
41
260
User
Manitoba
41
261
User
Alberta
41
262
Ethan
Ontario
41
263
User
Ontario
41
264
User
Ontario
41
265
User
Ontario
41
266
User
Quebec
40
267
Kevin Liu
British Columbia
40
268
User
Manitoba
40
270
User
Alberta
40
271
Joshua Lawrence
Quebec
40
272
Balazs Somfalvi
Alberta
40
273
User
Ontario
40
274
Vanessa Sementilli
Quebec
40
275
Sanskriti Sharma
British Columbia
40
276
User
Quebec
40
277
User
Ontario
40
278
User
Alberta
40
279
User
Ontario
40
280
User
Ontario
40
281
User
Ontario
40
282
Balazs Somfalvi
Alberta
40
283
User
Quebec
40
284
User
Ontario
39
285
User
Ontario
39
286
User
Ontario
39
287
Brady Metherall
Ontario
39
288
User
Ontario
39
289
User
Ontario
39
290
User
Ontario
39
291
User
Ontario
39
292
Nolan MacLean
Alberta
39
293
User
Ontario
39
294
User
Ontario
39
295
User
Ontario
39
296
Sam C
Ontario
39
297
William Zhang
British Columbia
39
298
User
Ontario
39
299
shezshade
Ontario
39
302
User
Ontario
38
303
Don
Ontario
38
306
User
Quebec
38
307
Stephane
Quebec
38
308
User
Ontario
38
309
Edward Ji - Ellengale PS (1376)
Ontario
38
310
User
Ontario
38
311
User
Ontario
38
312
User
Alberta
38
313
Madison Buchberger
Alberta
38
314
Greg
Quebec
38
315
User
Ontario
38
316
User
Alberta
37
317
Josh Craig
Quebec
37
318
User
Quebec
37
319
Hoang Khanh Tu Duong
Quebec
37
320
User
Ontario
37
321
User
Ontario
37
322
User
Ontario
37
324
User
Quebec
37
325
David Bedayo
Alberta
37
328
Katherine Germatenko
Ontario
36
329
Noah Lavarias
Quebec
36
330
Joel_
Alberta
36
331
eric
Ontario
36
332
User
Alberta
36
334
Miranda
Ontario
36
335
Nicholas Plamondon
Ontario
36
337
User
Quebec
36
338
Cindy Zou
Ontario
36
339
User
Ontario
36
340
User
Ontario
36
342
User
Ontario
36
343
User
Quebec
36
344
User
Ontario
36
345
User
Ontario
36
346
User
Quebec
36
347
User
Ontario
36
348
User
Ontario
36
349
User
Region
35
350
Andy Vuong
Ontario
35
351
User
Ontario
35
352
ANTON SHAKOV
Ontario
35
353
User
Quebec
35
354
User
Ontario
35
355
User
Ontario
35
356
Thomas
British Columbia
35
357
User
Ontario
35
358
User
Ontario
35
359
User
Ontario
35
360
User
Ontario
35
361
Mike Kusyk
Manitoba
35
362
User
Ontario
35
363
User
Ontario
35
365
Nathan Dyck
Manitoba
35
366
User
Ontario
35
368
Art Yu
British Columbia
35
369
User
Quebec
35
371
User
Quebec
34
372
Reese Jolicoeur
Ontario
34
375
User
Ontario
34
376
Tian Lai Li
Quebec
34
377
User
Ontario
34
378
User
Ontario
34
379
User
Ontario
34
381
User
Manitoba
33
382
User
Quebec
33
383
Naz
Alberta
33
384
User
Ontario
33
385
User
Alberta
33
386
User
Ontario
33
387
User
Ontario
33
388
User
Alberta
33
389
Rex VS Gaming
Quebec
33
390
User
Ontario
33
391
User
Ontario
33
393
User
Ontario
33
394
User
Alberta
33
395
User
Ontario
33
397
User
Quebec
33
398
Eyes Mojave
Ontario
33
399
User
Ontario
33
400
Mireille Lemoine
Manitoba
32
402
Renee Weng
British Columbia
32
403
User
Alberta
32
405
User
Alberta
32
406
User
Ontario
32
407
User
Alberta
32
408
User
Ontario
32
409
iPad
Quebec
32
410
User
Ontario
32
411
User
Alberta
32
412
User
Ontario
32
413
Ramtin Shekari
Ontario
32
414
User
Ontario
32
416
User
Alberta
32
417
katherineee
Ontario
32
418
User
Ontario
32
419
User
Region
31
420
User
Ontario
31
421
User
Ontario
31
423
UnKnOwN
Quebec
31
425
User
Ontario
31
426
User
Ontario
31
427
User
Ontario
31
428
User
Ontario
31
429
User
Ontario
31
430
User
Ontario
31
431
User
Ontario
31
432
User
Ontario
31
433
User
Alberta
31
434
User
Ontario
31
435
User
Ontario
31
436
User
Ontario
31
437
User
Ontario
31
438
User
New Brunswick
31
439
Marcella
Ontario
31
440
User
Ontario
31
441
User
Manitoba
31
442
User
Ontario
31
443
User
Ontario
31
444
User
Ontario
31
445
User
Ontario
31
446
User
Ontario
31
447
User
Quebec
31
449
User
Ontario
31
450
fortyfivewidthmarines
British Columbia
30
451
User
Alberta
30
453
Cynthia
Ontario
30
454
User
Alberta
30
455
User
Ontario
30
457
Kevin-Joseph Sioquin
Quebec
30
459
Ramya Venkataraman
Ontario
30
460
User
Ontario
30
461
User
Ontario
30
462
User
Ontario
30
463
User
Quebec
30
464
ahmedshaikh626
Newfoundland and Labrador
30
465
User
Ontario
30
466
User
Ontario
30
467
User
Ontario
30
470
Froylanne Paril
Alberta
30
471
Rene Tung
Quebec
30
472
SSR RAO
Manitoba
30
473
Mahendirakumar Krishnakumar
British Columbia
30
474
User
Alberta
30
475
User
Ontario
30
476
User
Ontario
30
477
User
Alberta
29
478
User
Ontario
29
479
User
Ontario
29
480
User
Ontario
29
481
User
Ontario
29
482
User
Alberta
29
483
Omar Abukhodair
Alberta
29
484
User
Alberta
29
485
User
Quebec
29
486
User
Ontario
29
487
User
Ontario
29
488
User
Ontario
29
489
User
Ontario
29
490
Jim Kerry
Ontario
29
492
User
Ontario
29
493
User
Ontario
29
494
User
Quebec
29
495
User
Ontario
29
496
User
Quebec
29
497
John Walsh
Ontario
29
498
User
Ontario
29
499
User
Ontario
29
500
User
Quebec
29
501
User
Alberta
29
502
User
Manitoba
29
504
User
Ontario
29
505
User
Ontario
29
507
User
Quebec
29
508
User
Ontario
29
509
User
Ontario
29
510
User
Alberta
29
511
zakria ismail
Alberta
28
512
User
Ontario
28
513
User
Ontario
28
514
User
Ontario
28
516
User
Alberta
28
517
Denny Jiang
Manitoba
28
518
User
Quebec
28
519
User
Ontario
28
520
User
Quebec
28
521
User
Ontario
28
522
User
Quebec
28
524
User
Alberta
28
525
User
Alberta
28
526
User
Manitoba
28
527
User
Ontario
28
528
User
Ontario
28
529
User
Ontario
28
530
User
Ontario
28
531
User
Ontario
28
532
User
Quebec
28
533
User
Ontario
28
534
User
Ontario
28
535
User
Ontario
28
536
User
Ontario
28
538
Ryan dai
Ontario
28
539
User
Ontario
28
540
User
Ontario
28
541
User
Ontario
28
542
User
Quebec
28
543
User
Manitoba
28
544
User
Quebec
28
545
Alex Zed
Quebec
28
546
User
Quebec
28
547
Mini Store
Ontario
28
548
User
Ontario
28
549
User
Ontario
28
551
User
Alberta
28
552
User
Ontario
28
553
User
Ontario
28
554
User
Ontario
28
555
User
Manitoba
28
557
User
Manitoba
28
559
Jrboyz
British Columbia
27
561
User
Ontario
27
562
User
Alberta
27
563
User
Alberta
27
564
User
Ontario
27
566
Leo Alvarino
Ontario
27
567
Brencis
British Columbia
27
568
User
Alberta
27
569
MathVader
Quebec
27
570
Sahil Jamani
Alberta
27
571
User
Ontario
27
572
User
Manitoba
27
573
User
Nova Scotia
27
574
Tori J
Saskatchewan
27
575
User
Ontario
27
576
User
Ontario
27
577
User
Ontario
27
579
Rodela Afnam
Ontario
27
580
User
Ontario
27
581
User
Ontario
27
582
User
Ontario
27
583
User
Alberta
27
584
User
Nova Scotia
27
585
User
Alberta
27
586
User
Ontario
27
587
User
Nova Scotia
27
588
User
Ontario
27
589
User
Ontario
27
590
User
Ontario
27
591
User
Alberta
27
592
User
New Brunswick
27
593
User
Ontario
27
594
User
Manitoba
27
595
User
Quebec
27
596
User
Manitoba
27
597
User
Alberta
27
598
User
Ontario
26
599
User
Ontario
26
600
User
Ontario
26
601
User
Ontario
26
604
User
Quebec
26
605
User
Quebec
26
606
User
Alberta
26
607
Ibrahima Diallo
Ontario
26
608
Ethan C3285
Alberta
26
609
Chuck Duford
Quebec
26
610
Alec van
Quebec
26
611
User
Nova Scotia
26
612
User
Ontario
26
613
youm
Quebec
26
614
User
Quebec
26
615
User
Ontario
26
617
User
Ontario
26
618
User
Ontario
26
619
User
Ontario
26
621
User
Ontario
26
623
User
Ontario
26
624
Quantum
Ontario
26
625
User
Ontario
26
626
User
Manitoba
26
627
User
Saskatchewan
26
628
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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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