Leaderboard
Brazil
Brazil
Worldwide
Rank
Level
1
Milan van de Meeberg
Paraná
99
2
Brian Moret Ferreira da Silva
Rio de Janeiro
84
3
Paulo Cezar
Mato Grosso
81
4
Marcos Cabral
Pernambuco
77
5
ackuhl2@gmail.com
Rio de Janeiro
77
6
Filipe de Castro
Minas Gerais
76
7
HRP is the BOSS
Paraná
69
8
Jhonne Kleyton
Pernambuco
62
9
Matheus Rech
São Paulo
61
10
User
Pernambuco
60
11
Felipe de Paula
Paraná
56
12
xzy
São Paulo
55
13
Matheus Petucco
São Paulo
54
14
Fernando Henrique
Pará
53
16
Taquion
Goiás
50
17
Bruno Shirasuna
São Paulo
49
18
Michelle
Rio de Janeiro
47
19
bruno couto
Rio de Janeiro
45
20
User
Pernambuco
43
21
User
Ceará
43
22
Tiago
São Paulo
41
23
User
São Paulo
41
24
Vandré
Paraná
41
25
User
Ceará
40
27
José Freire
Sergipe
40
28
User
Paraná
39
30
Gabriel
Paraíba
39
31
User
São Paulo
39
32
User
São Paulo
39
34
User
Ceará
38
35
Renan
São Paulo
38
36
Murius
São Paulo
37
38
User
Paraná
37
42
Leonardo Almeida
Paraná
35
43
Lyfaenia
Rio Grande do Sul
35
44
User
São Paulo
34
46
User
Paraná
33
47
User
Paraná
33
48
Naiane Pontes
São Paulo
33
49
User
São Paulo
32
50
User
Pernambuco
32
52
User
São Paulo
31
53
Juliano Negri
São Paulo
31
54
Gustavo Henrik
São Paulo
30
55
User
Paraná
30
56
User
Minas Gerais
30
58
Lyfaenia
Rio Grande do Sul
29
59
User
Pernambuco
29
60
User
Paraná
29
61
Hugo Lunel
São Paulo
29
62
Milhouse Bugado
São Paulo
29
63
Roderiko
São Paulo
29
64
User
São Paulo
29
65
Tiago Teodoro
Paraná
29
67
User
São Paulo
28
68
User
Ceará
28
69
User
São Paulo
28
72
User
Paraná
27
73
User
Paraná
26
74
Nicolas Gama Coelho Rodrigues
São Paulo
26
75
Hildemar
Sergipe
26
76
Loust 055
Amapá
26
78
lucas teles
Rio Grande do Sul
26
79
User
Minas Gerais
26
80
Eduardo Freire
Distrito Federal
25
81
Kauê Godoy
São Paulo
24
82
User
Pará
24
83
frnandoh
Pará
24
86
User
Pernambuco
23
88
User
Ceará
23
89
André Shevantes
Rio Grande do Sul
23
90
User
Minas Gerais
23
91
Rosemberg Santos
Paraíba
22
92
User
Minas Gerais
22
93
Gabriel Küter
Santa Catarina
21
94
Marcos Henrique
Pernambuco
21
95
User
São Paulo
21
98
User
São Paulo
20
100
User
Ceará
19
101
User
Pernambuco
19
102
User
São Paulo
19
103
User
Amazonas
19
104
Rubia
São Paulo
18
107
User
São Paulo
17
108
User
Minas Gerais
17
110
User
São Paulo
17
111
tadeu
Rio de Janeiro
16
112
User
São Paulo
16
113
User
São Paulo
16
115
User
Minas Gerais
16
116
User
Alagoas
16
117
User
Minas Gerais
16
118
User
São Paulo
15
119
User
São Paulo
15
120
User
São Paulo
15
121
User
Bahia
15
122
User
São Paulo
15
123
Daniel
São Paulo
15
124
User
Minas Gerais
15
126
geovana p
São Paulo
14
127
User
São Paulo
14
128
User
Paraná
14
129
User
Paraná
14
131
User
São Paulo
14
132
Luis Masuko
Amapá
14
135
User
São Paulo
14
136
User
Paraná
14
138
User
Minas Gerais
13
139
Fabio Ocana Vieira
São Paulo
13
140
User
São Paulo
13
141
User
São Paulo
13
142
User
São Paulo
13
144
User
Goiás
13
145
User
São Paulo
13
146
memes
São Paulo
13
147
User
Mato Grosso
13
148
User
Goiás
13
149
beto games
São Paulo
13
150
User
São Paulo
12
152
Fierce Deity
Rio de Janeiro
12
153
User
São Paulo
12
154
User
São Paulo
12
156
Paulo Henrique dos Santos
Rio Grande do Sul
12
157
Lucas Valença
Santa Catarina
12
160
User
São Paulo
12
161
User
São Paulo
12
162
User
Minas Gerais
12
163
Caio Kaercher
São Paulo
12
165
Leandro Davila
Sergipe
11
166
João Chaloub
Rio de Janeiro
11
168
User
Minas Gerais
11
169
User
Paraná
11
170
User
São Paulo
11
171
Matheus Devincenzi
Rio Grande do Sul
11
172
User
São Paulo
11
173
User
Sergipe
11
174
User
São Paulo
11
175
User
São Paulo
11
176
User
Goiás
11
177
User
São Paulo
11
179
Talyta dos Santos Capitan Dias
São Paulo
11
180
User
São Paulo
10
181
Valter M.K.
Maranhão
10
182
User
São Paulo
10
183
Jonathan Costa
Minas Gerais
10
184
Paulo Ormonde
São Paulo
10
186
Leandro Costa
Rio de Janeiro
10
187
Bruno
Rio de Janeiro
10
188
User
Pernambuco
10
189
User
São Paulo
10
190
User
Minas Gerais
10
191
User
São Paulo
10
192
MestreGnomo
São Paulo
10
196
User
São Paulo
10
197
User
São Paulo
10
199
User
São Paulo
9
200
Leros
Goiás
9
201
User
São Paulo
9
202
Vinicius
Minas Gerais
9
203
arielnc
São Paulo
9
204
Giovana Flores
São Paulo
9
205
User
São Paulo
9
206
Veronica Cantarelli
São Paulo
9
207
User
Paraíba
9
209
Vinicius Fonseca
Rio de Janeiro
9
210
User
Paraná
9
211
Rhuan Paulo Lopes Barros
Rio Grande do Sul
9
212
User
São Paulo
9
213
User
Maranhão
9
215
User
São Paulo
9
216
User
Piauí
9
217
User
São Paulo
9
219
User
São Paulo
9
220
User
Bahia
9
221
Vinicius de Santana
São Paulo
8
222
Jessica Messias
São Paulo
8
223
User
Minas Gerais
8
226
User
São Paulo
8
228
User
Minas Gerais
8
229
Cleto
São Paulo
8
230
User
São Paulo
8
231
User
São Paulo
8
232
Alex C
São Paulo
8
233
User
São Paulo
8
234
User
Goiás
8
235
User
São Paulo
8
236
User
Paraná
8
237
User
Paraná
8
238
User
Tocantins
8
239
Flyx Hoy
Paraíba
8
240
User
Paraíba
8
241
User
São Paulo
8
242
User
São Paulo
8
243
Renan W.
São Paulo
8
244
User
São Paulo
8
245
User
São Paulo
8
246
User
São Paulo
8
247
User
São Paulo
8
250
Felipe Albarelli
São Paulo
8
251
Anna
São Paulo
8
252
User
São Paulo
8
254
User
São Paulo
8
255
User
Goiás
8
256
User
São Paulo
8
257
User
São Paulo
8
259
User
São Paulo
7
261
User
Pernambuco
7
262
Gustavo Mazzarollo
Rio Grande do Sul
7
263
User
São Paulo
7
264
User
Pernambuco
7
265
Wagner Batalha
Distrito Federal
7
266
User
São Paulo
7
267
User
São Paulo
7
268
Eder de Faria
Minas Gerais
7
270
User
Bahia
7
271
User
São Paulo
7
272
User
Paraná
7
273
User
Paraná
7
275
User
São Paulo
7
276
User
Paraná
7
277
User
São Paulo
7
278
Tadeu Moraes
Rio de Janeiro
7
279
User
São Paulo
7
280
User
Piauí
7
282
Mareval Soares Pereira
São Paulo
7
283
User
São Paulo
7
285
User
São Paulo
7
286
User
Minas Gerais
7
287
Erick Castilho
São Paulo
7
289
User
São Paulo
6
290
bruno pessoa
Minas Gerais
6
291
User
Paraná
6
292
User
São Paulo
6
293
User
Ceará
6
295
pato
Minas Gerais
6
297
User
São Paulo
6
299
User
Minas Gerais
6
301
joana wanderley
Paraná
6
303
User
São Paulo
6
304
math math
Pernambuco
6
307
filipe 2357
Paraíba
6
308
User
Pará
6
309
User
Minas Gerais
6
310
User
São Paulo
6
311
Best Botter
São Paulo
6
312
User
Bahia
6
313
User
São Paulo
6
314
User
Bahia
6
317
Vitor
São Paulo
6
319
User
São Paulo
6
320
User
Mato Grosso
6
321
User
São Paulo
6
322
User
São Paulo
6
323
User
São Paulo
6
324
User
São Paulo
6
325
User
São Paulo
6
326
User
São Paulo
6
328
User
São Paulo
6
329
User
Sergipe
6
332
User
São Paulo
6
333
User
São Paulo
6
334
Jonatas Rodrigues
Distrito Federal
6
335
User
São Paulo
5
337
User
São Paulo
5
339
User
Paraná
5
342
User
Minas Gerais
5
343
User
São Paulo
5
345
User
Paraná
5
346
User
Minas Gerais
5
347
User
São Paulo
5
348
Mario
São Paulo
5
349
User
São Paulo
5
350
User
São Paulo
5
351
User
Paraná
5
352
User
São Paulo
5
353
User
São Paulo
5
354
User
São Paulo
5
356
User
São Paulo
5
357
User
São Paulo
5
358
User
São Paulo
5
359
User
Goiás
5
361
User
São Paulo
5
362
User
Pernambuco
5
364
User
Minas Gerais
5
365
User
Mato Grosso
5
366
User
Goiás
5
367
User
São Paulo
5
368
User
São Paulo
5
369
User
Paraíba
5
370
User
Pernambuco
5
371
User
São Paulo
5
373
User
Minas Gerais
5
374
User
São Paulo
5
376
Camelzebra
São Paulo
5
378
User
Goiás
4
379
User
São Paulo
4
380
Solange Jacob
São Paulo
4
381
User
Amazonas
4
382
User
Maranhão
4
383
User
São Paulo
4
384
User
Paraná
4
385
User
Paraná
4
386
User
São Paulo
4
387
Heitor
São Paulo
4
390
User
São Paulo
4
391
User
São Paulo
4
392
User
Bahia
4
393
User
São Paulo
4
394
User
São Paulo
4
396
User
Paraíba
4
397
User
São Paulo
4
398
Riquelmy Oliveira
Bahia
4
399
User
São Paulo
4
400
User
São Paulo
4
403
User
Paraná
4
404
User
São Paulo
4
405
User
São Paulo
4
406
User
Sergipe
4
407
User
São Paulo
4
409
User
São Paulo
4
410
User
São Paulo
4
412
User
São Paulo
4
413
User
São Paulo
4
414
User
Minas Gerais
4
416
User
São Paulo
4
417
User
Alagoas
4
418
User
Minas Gerais
4
419
User
São Paulo
4
420
User
Ceará
4
421
Marcio Todeschini
São Paulo
4
422
Alisson Santos
Paraná
4
423
User
São Paulo
4
424
User
São Paulo
4
425
User
Paraíba
4
427
User
São Paulo
4
428
User
Piauí
4
429
User
Pará
4
430
User
São Paulo
4
431
Wlasmir Cavalcanti
Rio de Janeiro
4
433
User
São Paulo
4
434
User
São Paulo
3
435
User
São Paulo
3
436
User
São Paulo
3
438
User
São Paulo
3
439
User
Minas Gerais
3
440
User
São Paulo
3
441
User
São Paulo
3
443
User
São Paulo
3
444
User
São Paulo
3
446
User
São Paulo
3
447
User
São Paulo
3
448
User
São Paulo
3
450
User
Pernambuco
3
452
User
São Paulo
3
453
User
São Paulo
3
454
User
Paraíba
3
455
User
São Paulo
3
456
User
São Paulo
3
457
User
São Paulo
3
458
User
São Paulo
3
459
User
São Paulo
3
460
User
São Paulo
3
461
User
São Paulo
3
462
User
São Paulo
3
463
User
São Paulo
3
464
User
Maranhão
3
465
Giuliano Ribeiro Mezacasa
Rio Grande do Sul
3
466
User
Sergipe
3
467
Gabriel Tiberio
São Paulo
3
468
User
Minas Gerais
3
469
User
Minas Gerais
3
470
User
Tocantins
3
472
User
Paraná
3
473
User
Goiás
3
474
User
São Paulo
3
475
User
São Paulo
3
476
User
Paraná
3
478
User
São Paulo
3
481
User
Bahia
3
482
User
Pernambuco
3
483
User
São Paulo
3
484
User
Paraná
3
485
User
São Paulo
3
486
User
Bahia
3
487
User
São Paulo
3
488
User
São Paulo
3
489
User
Maranhão
3
491
User
Paraíba
3
493
User
São Paulo
3
494
User
São Paulo
3
495
User
Minas Gerais
3
496
Matthew Lewis
Rio de Janeiro
3
497
kaio costa
São Paulo
3
498
User
São Paulo
3
499
User
São Paulo
3
501
User
São Paulo
2
503
User
Paraná
2
504
User
São Paulo
2
505
User
Minas Gerais
2
506
User
Minas Gerais
2
507
User
Minas Gerais
2
508
User
Minas Gerais
2
509
User
São Paulo
2
510
User
Amazonas
2
511
User
São Paulo
2
512
User
São Paulo
2
513
User
São Paulo
2
514
User
São Paulo
2
515
User
Minas Gerais
2
516
Julia Campiolo Porto
São Paulo
2
518
Pedro Cabral
Ceará
2
519
User
São Paulo
2
520
User
Pará
2
521
User
São Paulo
2
522
User
São Paulo
2
523
User
Tocantins
2
524
User
São Paulo
2
525
User
São Paulo
2
526
User
São Paulo
2
527
User
São Paulo
2
528
User
São Paulo
2
529
User
São Paulo
2
531
User
São Paulo
2
534
User
São Paulo
2
535
User
São Paulo
2
537
User
São Paulo
2
538
User
Paraíba
2
539
User
Minas Gerais
2
544
User
São Paulo
2
545
User
São Paulo
2
546
User
Paraná
2
547
User
Paraná
2
549
User
Mato Grosso
2
550
User
Ceará
2
551
User
São Paulo
2
552
User
São Paulo
2
553
User
Tocantins
2
554
User
Minas Gerais
2
555
User
São Paulo
2
556
User
São Paulo
2
557
User
Pernambuco
2
558
User
São Paulo
2
560
User
São Paulo
2
563
User
São Paulo
2
564
User
São Paulo
2
565
Luana Castro
Minas Gerais
1
566
User
São Paulo
1
569
User
São Paulo
1
570
User
São Paulo
1
571
User
São Paulo
1
572
User
São Paulo
1
573
User
Ceará
1
574
User
Bahia
1
575
Vandre
Paraná
1
576
User
São Paulo
1
577
Camila Vecchi
São Paulo
1
578
User
Paraíba
1
579
User
São Paulo
1
580
Milan “youngscience” v.de Meeberg
Paraná
1
581
User
Sergipe
1
582
Patricia de Souza Machado
São Paulo
1
584
User
São Paulo
1
586
Ruan Silva
Ceará
1
587
User
São Paulo
1
588
User
Minas Gerais
1
589
User
Paraná
1
590
User
São Paulo
1
592
User
São Paulo
1
593
User
Amazonas
1
595
User
São Paulo
1
596
User
Paraná
1
597
User
São Paulo
1
598
Thiago Ramos
Bahia
1
599
User
Bahia
1
602
User
Maranhão
1
603
User
Bahia
1
604
User
São Paulo
1
605
User
Pernambuco
1
606
Fabiele Fritzen
Distrito Federal
1
607
User
São Paulo
1
608
User
Goiás
1
609
User
São Paulo
1
610
User
São Paulo
1
612
User
Minas Gerais
1
613
User
Amazonas
1
614
User
Amapá
1
620
User
Sergipe
1
621
User
Sergipe
1
622
User
Paraná
1
623
User
Paraná
1
624
Eduardo Marchetti
Santa Catarina
1
625
User
São Paulo
1
626
User
Paraná
1
627
Jeanne Brun
Paraná
1
628
User
São Paulo
1
629
User
São Paulo
1
630
User
Ceará
1
631
User
São Paulo
1
632
User
São Paulo
1
633
User
Pernambuco
1
634
User
São Paulo
1
635
User
Bahia
1
636
User
Bahia
1
638
User
São Paulo
1
641
User
Paraná
1
643
User
São Paulo
1
644
User
São Paulo
1
645
User
São Paulo
1
646
User
São Paulo
1
647
User
São Paulo
1
648
User
São Paulo
1
649
User
São Paulo
1
650
User
São Paulo
1
651
User
São Paulo
1
653
User
São Paulo
1
654
User
São Paulo
1
655
User
São Paulo
1
656
User
São Paulo
1
657
User
Pará
1
658
User
Pernambuco
1
659
User
São Paulo
1
660
User
Minas Gerais
1
661
User
São Paulo
1
662
User
Minas Gerais
1
663
User
São Paulo
1
664
User
Paraná
1
665
User
São Paulo
1
666
User
São Paulo
1
668
User
São Paulo
1
669
User
São Paulo
1
670
rAPHAEL
Goiás
1
672
Vinicius Willer Alencar Haas
Distrito Federal
1
673
User
Pará
1
675
User
Paraná
1
676
User
Minas Gerais
1
678
User
São Paulo
1
679
User
São Paulo
1
680
User
São Paulo
1
681
User
São Paulo
1
682
User
São Paulo
1
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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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