Leaderboard
Canada
Canada
Worldwide
Rank
Level
1
Algorox
Manitoba
100
2
ade rev
Alberta
99
3
John Smith
Ontario
95
4
Hakase
Manitoba
93
6
User
Quebec
92
7
Sabrina
Ontario
89
8
User
Ontario
86
9
Arnav Singh
Ontario
85
10
Arthur Li
Ontario
85
11
Daniel Pedro
British Columbia
84
12
User
Ontario
84
13
Dustin
Quebec
83
14
Duong,Tu
Quebec
77
15
Tyler Chiu
Ontario
77
16
Alex Wen
British Columbia
76
17
Kian Tuastomban
Alberta
75
18
User
Quebec
75
19
Quang Nguyen
New Brunswick
75
20
User
Ontario
75
22
Affan Bhandakya
British Columbia
72
23
Koyomin
Manitoba
70
24
Alex Gay
Ontario
70
25
Raiyana Shams
Alberta
69
27
User
Ontario
68
28
Ella Rasoulian
Ontario
67
29
Aegwynn
Quebec
66
30
Shahid M Salam
Quebec
65
31
User
Manitoba
64
32
User
Quebec
64
33
User
Ontario
64
34
User
Ontario
63
35
B.Arindam
British Columbia
63
36
Mr MadPanda
Ontario
63
38
Dave Coding Pro
Ontario
62
39
Ian
Alberta
61
41
Edward Lee
Ontario
60
43
Sophie Yin
Ontario
60
44
Cameron Villetard
Alberta
59
45
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
59
46
Frédérique Poitras
Alberta
58
47
User
Ontario
57
48
Mike Travis
Ontario
57
49
Andrew Hua
Ontario
57
51
Anton
Quebec
56
52
Owen Jakob
Ontario
56
53
User
Manitoba
55
54
Brielle Huang
Alberta
55
55
Zach L
Ontario
55
56
User
Ontario
55
57
User
Ontario
54
58
Balazs Somfalvi
Alberta
54
59
User
Ontario
54
60
Richard Williams
Quebec
53
61
User
Quebec
53
62
Shani Mithani
Ontario
52
63
Cornell Lee
British Columbia
52
65
User
Quebec
51
66
User
Ontario
51
67
john gay
Ontario
51
68
User
Quebec
50
69
User
Nova Scotia
50
70
Sukhminder Bath
Manitoba
50
71
User
Manitoba
49
72
Sean Bendig
Alberta
49
73
User
Quebec
49
74
User
Alberta
49
75
Steven
Quebec
48
76
Tyler Chiu
Ontario
48
77
Yatin Kapur
Ontario
48
78
Ryan Han
British Columbia
47
79
User
Quebec
47
80
User
Ontario
47
81
User
Ontario
47
82
Durraiz Ashraf
Ontario
47
83
User
Ontario
47
84
User
Ontario
47
86
User
Alberta
46
87
User
Ontario
46
88
Mattheo Boisjelo
Quebec
46
89
awesome mathman
Ontario
46
90
User
Ontario
46
91
Lisa Hines
Ontario
45
92
Jonathan Varro
Alberta
45
93
User
Quebec
45
94
User
Ontario
45
95
User
Ontario
45
96
Godmode VIII
Alberta
44
97
r/Misho
Ontario
44
98
User
Alberta
44
99
Karen Zhao
British Columbia
44
100
User
Ontario
44
101
Yichen Sun
Quebec
44
102
User
Ontario
44
103
Patrick Kaipainen
Ontario
43
104
User
Ontario
43
105
User
Alberta
43
106
User
Ontario
43
107
Jayden Kallusky
Manitoba
43
108
User
Quebec
43
109
Chester M
Ontario
43
110
User
Ontario
43
111
Jeffrey Misner
Ontario
43
112
Matteo Espinosa Ramirez
Quebec
43
113
Matthew Taylor
Ontario
42
115
User
Ontario
42
116
User
Ontario
42
118
Godmode VII
Alberta
41
119
Samuel Zheng
Ontario
41
120
Arya🙄🙂🙃😉😃😄
Quebec
41
121
Jeong Augustine
British Columbia
41
122
User
Ontario
41
123
Godmode VI
Alberta
41
124
Corbin Brooks
Newfoundland and Labrador
41
125
User
Nova Scotia
41
126
User
Ontario
41
127
robin yoo
Quebec
41
128
User
Quebec
40
129
User
Ontario
40
130
Elliot Wong
Ontario
40
131
User
Ontario
40
132
User
Ontario
40
133
User
Ontario
40
134
Chris Christensen
Ontario
39
135
Samuel Ho
Ontario
39
136
User
Alberta
39
138
Hrafn Haugr
Alberta
39
139
Cindy
Ontario
39
140
Azmy Taha
Ontario
39
142
User
Manitoba
39
143
Garv Agarwal
Quebec
39
144
User
Ontario
39
145
User
Ontario
38
146
User
Ontario
38
147
User
Ontario
38
148
Muircat Chelsea
Ontario
38
149
Hugeldorf
Ontario
38
150
User
Ontario
38
151
User
Ontario
38
152
User
Ontario
38
153
User
Alberta
38
154
User
Quebec
38
155
User
Ontario
38
156
Marc-Antoine Joyal
Quebec
38
157
User
Ontario
38
158
Godmode IV
Alberta
37
159
Renee Weng
British Columbia
37
160
Shashvat Varma
Manitoba
37
162
User
Nova Scotia
37
163
User
Quebec
37
164
Lyss
Ontario
37
165
Daniel Qu
Ontario
37
166
User
Ontario
37
167
User
Nova Scotia
37
168
User
Quebec
37
169
Elliot W
Ontario
36
170
User
Manitoba
36
171
User
New Brunswick
36
172
User
Alberta
36
174
User
Alberta
36
175
Godmode II
Alberta
36
177
User
Manitoba
36
178
User
Alberta
36
179
User
Ontario
36
180
User
Ontario
36
181
User
Quebec
36
182
Pandy 2468
Quebec
35
183
User
Manitoba
35
184
User
Quebec
35
185
Godmode III
Alberta
35
186
User
Ontario
35
187
Matthew Hobbs
Ontario
35
188
User
Ontario
35
189
Khalisa Valani
Ontario
35
190
User
Ontario
35
191
User
Ontario
35
194
User
Quebec
35
195
User
Nova Scotia
34
196
User
Ontario
34
197
Connor Hogan
Saskatchewan
34
198
User
Alberta
34
199
Ricardo Chang
Ontario
34
200
User
Quebec
34
202
User
Alberta
34
203
Ryan Calvano
Ontario
34
205
User
Alberta
33
206
User
Alberta
33
207
Godmode
Alberta
33
208
User
Quebec
33
209
Joey Hotz
Ontario
33
210
Kennens
Ontario
33
211
Evan Zhou
Ontario
33
213
User
Alberta
33
214
Godmode V
Alberta
33
215
User
Ontario
33
216
Mark Gravely
Ontario
33
217
Xenia
Manitoba
33
218
Nicholas Cayla
British Columbia
33
220
User
Quebec
33
221
User
Ontario
32
222
User
Ontario
32
224
User
Ontario
32
225
User
Ontario
32
226
User
Ontario
32
229
yfradkin98@gmail.com
Ontario
32
230
Logan Peterson
British Columbia
32
231
User
Quebec
32
232
Andy Vuong
Ontario
32
233
User
Quebec
32
235
Big E
Ontario
32
236
User
Ontario
32
237
User
Ontario
32
238
User
Quebec
32
239
User
Manitoba
32
240
User
Ontario
32
241
User
Manitoba
31
243
User
Ontario
31
244
User
Quebec
31
245
User
Ontario
31
246
Prem
Ontario
31
247
Kevin Liu
British Columbia
31
248
User
Ontario
31
249
User
Quebec
31
250
User
Quebec
31
251
Renee Weng
British Columbia
31
252
Martin Tin
Ontario
31
253
Dave
Ontario
31
254
User
Quebec
31
255
User
Alberta
30
256
User
Ontario
30
258
Chocolate Chicken
Manitoba
30
259
User
Ontario
30
260
User
Nova Scotia
30
261
User
Ontario
30
262
User
Ontario
30
263
User
Alberta
30
264
User
Ontario
30
265
User
New Brunswick
30
269
User
Ontario
30
270
User
Manitoba
30
271
User
Quebec
29
273
User
Ontario
29
274
User
Ontario
29
275
User
Ontario
29
276
User
Alberta
29
277
User
Alberta
29
278
rdu
Ontario
29
279
MathVader
Ontario
29
280
ringtail doofus
Ontario
29
281
User
Alberta
29
283
Rex VS Gaming
Quebec
29
284
User
Ontario
29
285
User
Ontario
29
286
User
Ontario
29
287
User
Ontario
29
288
User
Ontario
29
289
User
Ontario
29
290
User
Ontario
28
291
Hao Hu
Ontario
28
292
User
Ontario
28
293
User
Ontario
28
294
Steven Craig
Quebec
28
295
User
Alberta
28
296
User
Ontario
28
297
Philippe
Quebec
28
298
User
Ontario
28
299
Azide
Ontario
28
300
Tyler Groves
Ontario
28
301
Don
Ontario
28
302
User
Alberta
28
303
User
Ontario
28
304
Mireille Lemoine
Manitoba
28
305
User
Quebec
28
306
Annie
Ontario
28
307
Diego
Quebec
28
308
User
Region
28
309
User
Alberta
28
310
User
Ontario
28
314
User
Ontario
28
315
User
Ontario
28
316
User
Manitoba
28
317
User
Ontario
28
318
Massimo Mazza
Quebec
28
319
User
Quebec
28
320
User
Ontario
28
321
User
Ontario
28
322
User
Ontario
28
323
User
Ontario
28
324
Evan Zhu
Quebec
27
325
Donald Moodger
Quebec
27
326
Corey
Quebec
27
327
Alec van
Ontario
27
328
User
Alberta
27
329
Katherine Germatenko
Ontario
27
330
User
Alberta
27
332
BenBen
Quebec
27
333
User
Ontario
27
334
User
Quebec
27
335
User
Alberta
27
337
Jack Sunhaoya
Quebec
27
338
User
Ontario
27
339
User
Ontario
27
340
Reese Jolicoeur
Ontario
26
341
Sara Blais
British Columbia
26
342
User
Ontario
26
343
SteveB
British Columbia
26
344
User
Ontario
26
345
K. Miranda
Ontario
26
346
Joshua Lawrence
Quebec
26
347
Stefan Pizzinato
Ontario
26
349
User
Ontario
26
350
User
Manitoba
26
351
Balazs Somfalvi
Alberta
26
352
Michael Gray
Saskatchewan
26
353
Mark
Ontario
26
354
Noah Lavarias
Quebec
26
355
User
Ontario
26
356
Hudson Vance
Ontario
26
358
User
Quebec
26
359
User
Manitoba
26
360
User
Quebec
26
361
User
Ontario
26
362
User
Ontario
26
363
User
Quebec
26
365
Michael Baldeo
Ontario
25
366
Joel_
Alberta
25
367
User
Ontario
25
368
User
Ontario
25
369
User
Ontario
25
370
JiNo Yoon
Quebec
25
372
eric
Ontario
25
373
User
Alberta
25
374
Paula San Jose Vasquez
New Brunswick
25
375
Darwin To
British Columbia
25
376
User
Ontario
25
377
User
Saskatchewan
25
378
Yi Shi
Ontario
25
379
Renee Weng
British Columbia
25
380
Josh Craig
Quebec
25
381
User
Region
24
382
User
Ontario
24
383
User
Ontario
24
384
User
Nova Scotia
24
385
User
Alberta
24
386
fortyfivewidthmarines
British Columbia
24
387
User
Ontario
24
388
User
Quebec
24
389
User
Ontario
24
390
User
Ontario
24
391
User
Ontario
24
393
Cobra
Alberta
24
394
User
Ontario
24
395
User
Ontario
24
396
User
Ontario
24
397
Naz
Alberta
24
398
User
Ontario
24
399
User
Ontario
24
400
User
Quebec
24
401
User
Ontario
24
402
User
Ontario
24
403
User
Ontario
24
404
Jacob Fitzgerald
Nova Scotia
24
405
Tony
Alberta
24
406
ANTON SHAKOV
Ontario
24
407
User
Ontario
24
408
User
Quebec
24
410
UnKnOwN
Quebec
23
414
User
Ontario
23
416
User
Quebec
23
417
User
Alberta
23
418
User
Alberta
23
420
User
Ontario
23
421
Bryan C
Ontario
23
422
User
Ontario
23
424
User
Alberta
23
425
User
Ontario
23
426
User
Ontario
23
427
User
New Brunswick
23
428
User
Alberta
23
430
User
Quebec
23
431
Shania
Ontario
23
432
User
Ontario
23
433
Cynthia
Ontario
23
434
User
Ontario
23
435
User
Quebec
23
436
User
Ontario
23
437
User
Ontario
22
438
User
Alberta
22
439
User
Ontario
22
440
zakria ismail
Alberta
22
441
User
Ontario
22
442
User
Ontario
22
443
User
Manitoba
22
444
User
Ontario
22
445
User
Alberta
22
446
Ismaël
Ontario
22
448
Mireille Lemoine
Manitoba
22
449
User
Ontario
22
450
User
Alberta
22
451
User
Ontario
22
452
User
Ontario
22
453
Brady Metherall
Ontario
22
454
Angus Ng
Ontario
22
455
User
Ontario
22
456
User
Alberta
22
457
User
Ontario
22
458
Stephen She
Alberta
22
459
User
Quebec
22
460
Jrboyz
British Columbia
22
461
User
Alberta
22
463
User
Alberta
22
465
User
New Brunswick
22
466
Miranda
Ontario
22
468
User
Ontario
22
469
sashimi x soysauce
Alberta
22
471
User
Ontario
21
473
User
Quebec
21
474
User
Alberta
21
475
User
Alberta
21
476
User
Ontario
21
477
User
Ontario
21
478
User
Ontario
21
479
User
Ontario
21
481
User
Quebec
21
482
User
Quebec
21
483
User
Ontario
21
484
User
Ontario
21
485
User
Alberta
21
486
User
Ontario
21
487
User
Ontario
21
488
Omar Abukhodair
Alberta
21
489
User
Alberta
21
490
User
Ontario
21
491
User
Ontario
21
492
User
Ontario
21
493
User
Ontario
21
494
Samson Gao
British Columbia
21
495
User
Ontario
21
496
User
Quebec
21
497
User
Alberta
21
498
User
Ontario
21
499
User
Ontario
21
500
User
Saskatchewan
21
501
Jagadish Gowtham Gokulaparanthaman
Ontario
21
502
Denny Jiang
Manitoba
20
504
Vanessa Sementilli
Quebec
20
505
User
Ontario
20
506
User
Alberta
20
507
User
Ontario
20
508
Sanskriti Sharma
British Columbia
20
509
User
Ontario
20
510
User
Ontario
20
512
Kevin-Joseph Sioquin
Quebec
20
513
User
Ontario
20
514
Nicholas Plamondon
Ontario
20
515
User
Quebec
20
516
User
Alberta
20
517
User
Ontario
20
518
User
Alberta
20
519
ibuats
Ontario
20
520
User
Alberta
20
521
User
Alberta
20
522
User
Ontario
20
523
User
New Brunswick
20
524
User
Ontario
20
525
User
Alberta
20
526
User
Quebec
20
527
User
Alberta
20
529
User
Ontario
20
532
User
Ontario
20
533
User
Ontario
20
534
User
Ontario
20
535
User
Alberta
20
536
User
Ontario
20
538
User
Manitoba
20
539
User
Ontario
20
541
User
Ontario
20
543
Aiko Fujita
Quebec
20
544
User
Ontario
20
545
User
Ontario
20
546
User
Ontario
20
547
User
Ontario
20
548
User
Region
20
549
User
Ontario
20
550
User
Ontario
20
551
User
Ontario
20
553
Kevin Kyle
Ontario
20
554
User
Ontario
20
555
Leo Alvarino
Ontario
20
556
User
Alberta
20
557
User
Ontario
20
558
Marisse Roco
Ontario
19
559
User
Ontario
19
560
User
Quebec
19
561
User
Alberta
19
562
User
Region
19
563
User
Manitoba
19
564
Brencis
British Columbia
19
565
User
Alberta
19
567
Justin Belcourt
Alberta
19
568
User
Alberta
19
569
User
Ontario
19
570
User
Ontario
19
571
User
Quebec
19
572
Tony Bui
Ontario
19
573
User
Ontario
19
574
User
Ontario
19
575
Thomas
British Columbia
19
576
Martin Lai
Ontario
19
577
User
Ontario
19
578
User
Ontario
19
579
User
Alberta
19
580
User
Alberta
19
581
User
Alberta
19
582
User
Ontario
19
583
Colin Fairburn
British Columbia
19
584
User
Alberta
19
586
User
Ontario
19
587
User
Quebec
19
588
User
Quebec
19
589
User
Ontario
19
590
User
Ontario
19
591
User
Ontario
18
592
User
Ontario
18
593
User
Ontario
18
594
User
Saskatchewan
18
595
User
Alberta
18
596
Miniapple Emily
Quebec
18
598
User
Ontario
18
599
User
Ontario
18
600
User
Ontario
18
601
Why you bully me
Quebec
18
602
User
Ontario
18
603
User
Region
18
604
User
Ontario
18
605
User
Ontario
18
606
User
Alberta
18
607
Hakase
Manitoba
18
608
User
Ontario
18
609
User
Quebec
18
610
User
Ontario
18
611
User
Ontario
18
613
User
Quebec
18
614
eduardo immigrant
Alberta
18
615
User
Alberta
18
616
Gary Ikez
Ontario
18
617
User
Ontario
18
618
User
Quebec
18
620
Ramya Venkataraman
Ontario
18
621
User
Ontario
18
622
User
Ontario
18
623
User
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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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