Leaderboard
Spain
Spain
Worldwide
Rank
Level
1
Dani Díaz
Madrid
97
2
User
Catalunya
92
3
Gonzalo Calvar
Madrid
88
4
User
Madrid
88
5
ricardo
Madrid
87
6
User
Catalunya
86
7
User
Madrid
84
8
User
Catalunya
83
9
Jaime Benabent
Andalucía
73
10
MANUEL GONZÁLEZ
Catalunya
73
11
Dinosaureo
Madrid
71
12
Alex
Catalunya
68
13
Parke Aiken
Catalunya
67
14
Roberto
Catalunya
66
15
User
Madrid
64
16
Tim
Andalucía
62
17
User
Catalunya
61
18
User
Galicia
60
19
Bart ter Huurne
Valenciana
59
20
Eduardo Gallego
Valenciana
56
21
User
Madrid
56
22
Miguel Pérez Riera
Catalunya
54
23
Ule 17
Catalunya
53
24
Bartt
Valenciana
52
25
User
Madrid
52
26
User
Madrid
51
27
Sirdeblas
Catalunya
51
28
Pau Herrera
Catalunya
51
29
User
Andalucía
51
30
HULIO
Andalucía
51
31
Juan Vecino
Madrid
50
32
Nicholas Brauner
Catalunya
50
33
Helena Yuste
Catalunya
50
34
BIZARRE MIND
Madrid
49
35
Mònica Serra
Catalunya
49
36
Maria Corona
Catalunya
49
37
User
Madrid
49
38
Carlos Perelló
Valenciana
47
39
Poldeblas
Catalunya
46
40
User
Catalunya
46
41
GuillerLT
Madrid
45
42
User
Andalucía
45
43
José Miguel Gomis
Valenciana
44
44
Benjamin Bourasseau
Andalucía
44
45
User
Catalunya
44
46
User
Catalunya
44
47
User
Madrid
43
48
User
Madrid
43
49
User
Andalucía
43
50
User
Catalunya
42
51
User
Madrid
42
52
Arnalpi 03
Catalunya
42
53
User
Andalucía
42
54
User
Catalunya
41
55
User
Catalunya
41
56
User
Madrid
40
57
User
Canarias
39
58
Marcel Sandoval
Catalunya
38
59
User
Madrid
38
60
User
Madrid
38
61
User
Madrid
38
62
User
Andalucía
38
63
MMolero
Madrid
37
64
User
Madrid
37
65
User
Catalunya
37
66
User
Madrid
37
68
Jon Goodey
Murcia
37
69
Víctor Ballester Ribó
Catalunya
36
70
JuanJe
Andalucía
36
71
User
Catalunya
36
72
User
Madrid
36
73
User
Catalunya
36
74
User
Valenciana
36
75
User
Catalunya
35
76
User
Catalunya
35
77
Iñigo Diez
Madrid
35
78
User
Catalunya
35
79
Wayra uhlig
Catalunya
34
80
Monica Casanova
Catalunya
34
81
Damia de angulo
Catalunya
34
82
User
Catalunya
34
83
User
Madrid
34
84
User
Madrid
34
85
User
Madrid
34
86
User
Andalucía
34
87
User
Andalucía
34
88
Adrià Gasull
Catalunya
33
89
Jorge Calvar
Madrid
33
90
User
Catalunya
33
91
User
Andalucía
33
92
User
Madrid
33
93
gerard merino
Catalunya
32
94
User
Catalunya
32
95
User
Valenciana
32
96
User
Valenciana
32
97
User
Catalunya
31
98
David serra
Catalunya
31
99
User
Catalunya
31
100
User
Madrid
31
101
User
Catalunya
31
102
User
Catalunya
31
103
User
Catalunya
31
104
User
Andalucía
31
105
Joana Moya Tirado
Andalucía
30
106
Vinyet
Catalunya
30
107
Yeha Yeha
Andalucía
30
108
Martí Lahoz
Catalunya
30
109
Sergio Valencia Jaraba
Catalunya
30
110
User
Madrid
30
111
User
Madrid
30
112
User
Madrid
30
113
User
Andalucía
30
114
User
Andalucía
30
115
User
Andalucía
30
116
Sanjana Nandwani
Andalucía
30
117
User
Madrid
30
118
iarobmon
Catalunya
30
119
User
Euskal Herria
29
120
Joan Gallego Vazquez
Catalunya
29
121
User
Catalunya
29
122
User
Catalunya
29
123
User
Catalunya
29
124
User
Catalunya
29
126
Hector Sansano Parra
Catalunya
28
127
User
Asturias
28
129
Adrián Ramírez Marín
Andalucía
28
130
Inma Segura
Andalucía
28
131
User
Valenciana
27
132
User
Madrid
27
133
Àngel Oribe Godàs
Catalunya
27
134
Ayman Krikaz
Catalunya
27
135
User
Canarias
27
136
User
Andalucía
27
138
DoNotDeleteMe
Valenciana
26
139
User
Catalunya
26
140
User
Catalunya
26
141
MaxMas
Catalunya
26
142
CarlosNT
Madrid
26
143
User
Galicia
26
144
User
Madrid
26
145
Javier Fernandez
Madrid
26
146
User
Catalunya
26
147
User
Illes Balears
26
148
User
Madrid
26
149
User
Catalunya
26
150
User
Valenciana
26
151
Juanjo
Madrid
25
152
User
Catalunya
25
153
User
Catalunya
25
154
User
Catalunya
25
155
User
Catalunya
25
156
marti serra
Catalunya
25
157
User
Catalunya
25
158
User
Catalunya
25
159
User
Euskal Herria
25
160
User
Catalunya
25
162
User
Catalunya
25
163
User
Madrid
24
164
User
Cantabria
24
165
User
Catalunya
24
166
User
Catalunya
24
167
User
Catalunya
24
168
Ana Núñez
Catalunya
24
169
User
Aragón
24
170
User
Madrid
24
171
User
Madrid
24
172
User
Madrid
24
173
User
Catalunya
24
174
José Joaquín Ruzafa
Murcia
23
175
User
Catalunya
23
176
User
Madrid
23
177
Carlos Torrecillas Vidal
Valenciana
23
178
User
Andalucía
23
179
User
Andalucía
23
180
User
Catalunya
23
181
User
Catalunya
23
182
User
Andalucía
23
183
User
Andalucía
23
184
Jaime
Andalucía
22
185
Japheth
Asturias
22
186
User
Catalunya
22
187
User
Catalunya
22
188
Jep Lezaun Sola
Catalunya
22
189
roller 11
Catalunya
22
190
Arnau Estella Baeza
Catalunya
22
191
Arnau Beulas Vinaixa
Catalunya
22
192
User
Catalunya
22
193
User
Valenciana
22
194
User
Madrid
22
195
User
Madrid
22
196
User
Catalunya
22
197
User
Andalucía
22
198
User
Catalunya
22
199
User
Andalucía
22
201
User
Catalunya
21
202
laia
Catalunya
21
203
William Eddleston
Nafarroako Foru Komunitatea*
21
204
User
Valenciana
21
205
User
Catalunya
21
206
LUNA ELIGIO CALZADO
Catalunya
21
208
User
Catalunya
21
209
User
Catalunya
21
210
User
Catalunya
21
211
User
Galicia
21
212
User
Catalunya
21
213
User
Catalunya
21
214
User
Catalunya
21
215
User
Catalunya
21
216
User
Catalunya
21
217
User
Valenciana
21
218
User
Valenciana
21
219
User
Madrid
21
220
User
Aragón
21
221
User
Catalunya
21
222
User
Catalunya
21
223
User
Catalunya
21
225
User
Andalucía
21
226
User
Aragón
21
227
User
Catalunya
21
228
User
Catalunya
20
229
User
Galicia
20
230
Wayra
Catalunya
20
231
User
Catalunya
20
232
User
Galicia
20
233
Angel Sanz Barragan
Catalunya
20
234
User
Madrid
20
235
User
Catalunya
20
236
User
Catalunya
20
237
User
Catalunya
20
238
User
Catalunya
20
239
User
Catalunya
20
240
User
Madrid
20
241
User
Andalucía
20
242
User
Catalunya
20
243
User
Andalucía
20
244
Miguel López Alcalde
Murcia
20
245
User
Catalunya
20
246
User
Catalunya
20
247
User
Madrid
20
248
User
Catalunya
20
249
Jorge ruiz
Asturias
20
250
User
Valenciana
20
251
User
Madrid
20
252
Elena Carulla
Cantabria
20
253
Javier Cuerva Heredia
Madrid
19
254
Guillem Parra
Catalunya
19
255
Alejo Soriano
Catalunya
19
256
Alex
Catalunya
19
257
User
Catalunya
19
258
MARC ORIOL YBERN
Catalunya
19
259
Ignasi Valls
Catalunya
19
260
User
Catalunya
19
261
User
Andalucía
19
262
User
Andalucía
19
263
User
Catalunya
19
264
User
Madrid
19
265
User
Madrid
19
266
User
Catalunya
19
267
User
Catalunya
19
268
User
Catalunya
19
269
Juan Del Guayo
Madrid
18
270
User
Madrid
18
271
User
Madrid
18
272
David Peralta
Catalunya
18
273
User
Catalunya
18
274
User
Catalunya
18
275
User
Madrid
18
276
User
Catalunya
18
277
User
Catalunya
18
278
User
Catalunya
18
279
User
Catalunya
18
280
Quim Talamas
Madrid
18
281
User
Catalunya
18
282
User
Andalucía
18
283
User
Andalucía
18
284
User
Andalucía
18
285
User
Andalucía
18
286
User
Catalunya
18
287
User
Madrid
18
288
User
Madrid
17
289
Oscar de Mier
Catalunya
17
290
Jorge F
Valenciana
17
291
User
Catalunya
17
292
Marc
Catalunya
17
293
User
Catalunya
17
294
User
Catalunya
17
295
User
Catalunya
17
296
User
Catalunya
17
297
User
Catalunya
17
298
Adam El Messaoudi
Catalunya
17
299
User
Catalunya
17
300
User
Extremadura
17
301
User
Andalucía
17
302
User
Andalucía
17
303
Pablo Ávila
Andalucía
17
304
User
Andalucía
17
305
User
Madrid
17
306
User
Valenciana
17
307
User
Catalunya
17
308
User
Catalunya
16
309
Arnau Martos Agustina
Catalunya
16
310
User
Catalunya
16
311
Irene Sanchez Sanchez
Catalunya
16
312
laia llansana zamora
Catalunya
16
313
User
Catalunya
16
314
User
Madrid
16
315
User
Catalunya
16
316
User
Catalunya
16
317
User
Canarias
16
318
User
Madrid
16
319
Keesjan lod
Euskal Herria
16
320
User
Madrid
16
321
User
Catalunya
16
322
User
Catalunya
16
323
User
Galicia
16
324
User
Andalucía
16
325
User
Andalucía
16
326
User
La Rioja
16
327
User
Catalunya
15
328
User
Catalunya
15
329
User
Catalunya
15
330
Ezequiel Ribas
Catalunya
15
331
User
Catalunya
15
332
User
Valenciana
15
333
Nico Varela
Madrid
15
334
User
Catalunya
15
335
User
Catalunya
15
336
Pere Molins
Catalunya
15
337
Juan
Murcia
15
338
User
Andalucía
15
339
User
Andalucía
15
340
Miquel Farreras
Catalunya
15
341
User
Andalucía
15
342
Harry Twigg
Illes Balears
15
343
User
Catalunya
15
344
User
Asturias
15
345
User
Catalunya
15
346
Gaurav Poona
Valenciana
14
347
JuanJo
Madrid
14
348
User
Catalunya
14
349
Noa Uhlig
Catalunya
14
350
User
Valenciana
14
351
Daniel Díaz López
Catalunya
14
352
User
Catalunya
14
353
User
Catalunya
14
354
Jana You
Catalunya
14
355
User
Catalunya
14
356
User
Catalunya
14
357
User
Catalunya
14
358
User
Catalunya
14
359
User
Madrid
14
360
Iván Expósito
Euskal Herria
14
361
User
Andalucía
14
362
User
Andalucía
14
363
User
Madrid
14
364
User
Catalunya
14
365
User
Madrid
14
366
User
Madrid
14
367
Javier Ruata
Catalunya
13
368
Noemí Vázquez
Catalunya
13
370
User
Madrid
13
371
Marta Bucar Velasco
Catalunya
13
372
Elia Garcia
Catalunya
13
373
User
Catalunya
13
374
User
Madrid
13
375
User
Catalunya
13
376
User
Catalunya
13
377
User
Madrid
13
378
User
Andalucía
13
379
Morel Hernandes
Castilla-La Mancha
13
380
User
Asturias
13
381
User
Catalunya
12
382
User
Madrid
12
383
User
Catalunya
12
384
User
Madrid
12
385
User
Catalunya
12
386
User
Madrid
12
387
User
Madrid
12
388
User
Andalucía
12
389
User
Madrid
12
390
User
Catalunya
12
391
User
Madrid
12
392
User
Madrid
12
393
User
Valenciana
12
394
User
Catalunya
12
396
User
Catalunya
12
397
User
Canarias
12
398
User
Catalunya
11
399
Leonard Balm
Catalunya
11
400
User
Madrid
11
401
Western Revival
Andalucía
11
402
User
Catalunya
11
403
Gabriel Torres Agustina
Catalunya
11
404
User
Catalunya
11
405
User
Catalunya
11
406
Gael gomez asuncion
Catalunya
11
407
User
Catalunya
11
408
Adria Seseras
Catalunya
11
409
User
Catalunya
11
410
Esteve Rodon Salazar
Catalunya
11
411
User
Catalunya
11
412
User
Catalunya
11
413
User
Catalunya
11
414
User
Madrid
11
415
User
Madrid
11
416
User
Andalucía
11
417
User
Andalucía
11
418
User
Madrid
11
419
Iñigo Ruiz
Andalucía
11
420
User
Catalunya
11
421
User
Catalunya
11
422
Carlos
Catalunya
10
423
Bubu Boy
Catalunya
10
424
User
Madrid
10
425
Andrea
Aragón
10
426
User
Catalunya
10
427
User
Catalunya
10
428
Pol Rivero
Catalunya
10
429
jofre illa
Catalunya
10
430
User
Catalunya
10
431
User
Galicia
10
432
MarcMunta03
Catalunya
10
433
User
Catalunya
10
434
Borja Varela
Catalunya
10
435
User
Catalunya
10
436
User
Valenciana
10
437
User
Catalunya
10
438
User
Madrid
10
439
User
Illes Balears
10
440
User
Catalunya
10
441
User
Andalucía
10
442
User
Andalucía
10
444
User
Madrid
10
445
User
Madrid
10
446
User
Andalucía
10
447
User
Andalucía
10
448
User
Catalunya
10
449
User
Catalunya
10
450
User
Madrid
10
451
User
Catalunya
9
452
User
Andalucía
9
453
User
Galicia
9
454
User
Catalunya
9
455
User
Catalunya
9
456
User
Catalunya
9
457
wahel
Catalunya
9
458
Judith Catalán Pacheco
Catalunya
9
459
User
Catalunya
9
460
User
Catalunya
9
461
Carlos Guadarrama Gallegos
Catalunya
9
462
User
Madrid
9
463
User
Catalunya
9
464
User
Catalunya
9
465
User
Catalunya
9
466
clara fernández
Catalunya
9
467
User
Catalunya
9
468
User
Andalucía
9
469
User
Madrid
9
470
User
Catalunya
9
471
User
Catalunya
9
473
User
Valenciana
9
474
User
Andalucía
9
475
User
Madrid
9
476
User
Valenciana
8
477
User
Madrid
8
478
User
Canarias
8
479
One Admin
Catalunya
8
480
User
Catalunya
8
481
Alda Diaz Cuevas
Valenciana
8
482
User
Catalunya
8
483
Aitana💜
Catalunya
8
484
User
Catalunya
8
485
User
Catalunya
8
486
User
Catalunya
8
487
User
Catalunya
8
488
User
Madrid
8
489
User
Valenciana
8
490
User
Madrid
8
491
User
Madrid
8
492
User
Madrid
8
493
User
Madrid
8
494
Adriana Medina Vega
Madrid
8
495
User
Valenciana
8
496
User
Valenciana
8
497
User
Madrid
7
498
User
Madrid
7
499
david
Catalunya
7
500
User
Madrid
7
501
Anna Crespo
Catalunya
7
502
Joel Vera Abella
Madrid
7
503
User
Catalunya
7
504
User
Catalunya
7
505
User
Catalunya
7
506
User
Catalunya
7
507
User
Catalunya
7
508
User
Madrid
7
509
Paco Javier
Catalunya
7
510
User
Andalucía
7
511
Rauane Ballebna Zahed
Catalunya
7
512
User
Catalunya
7
513
User
Catalunya
7
514
User
Catalunya
7
515
User
Catalunya
7
516
User
Catalunya
7
517
User
Catalunya
7
518
User
Catalunya
7
519
User
Madrid
7
520
User
Catalunya
7
521
User
Catalunya
7
522
User
Catalunya
7
523
User
Catalunya
7
524
User
Valenciana
7
525
User
Madrid
7
526
User
Andalucía
7
527
User
Andalucía
7
528
User
Valenciana
7
530
User
Catalunya
7
531
User
Madrid
7
532
User
Andalucía
7
534
Pablo
Andalucía
7
535
User
Andalucía
7
536
User
Madrid
7
538
User
Valenciana
6
539
User
Madrid
6
540
Ainhoa
Aragón
6
541
User
Madrid
6
542
User
Catalunya
6
543
apo
Madrid
6
544
User
Catalunya
6
545
User
Madrid
6
546
User
Madrid
6
547
Larre
Euskal Herria
6
548
User
Madrid
6
549
User
Aragón
6
550
Guillem
Andalucía
6
551
User
Catalunya
6
552
User
Catalunya
6
553
User
Madrid
6
554
User
Madrid
6
555
User
Catalunya
6
556
Laura Pineda
Catalunya
6
557
User
Catalunya
6
558
User
Catalunya
6
559
User
Catalunya
6
560
User
Catalunya
6
561
User
Catalunya
6
562
User
Catalunya
6
563
User
Catalunya
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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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