Leaderboard
Spain
Spain
Worldwide
Rank
Level
1
JoEp
Valenciana
84
2
Pan Mun Jom
Madrid
80
3
User
Catalunya
72
5
User
Catalunya
66
6
User
Catalunya
66
7
Eduardo Gallego
Valenciana
66
8
Gloudust HD
Andalucía
66
10
Andreas Oikonomou
Catalunya
61
11
Carles Pinyol Mas
Catalunya
59
12
User
Catalunya
58
13
GUILLEM
Catalunya
58
14
Bart ter Huurne
Valenciana
57
15
User
Catalunya
52
16
User
Catalunya
50
17
User
Madrid
48
19
User
Madrid
48
20
User
Madrid
48
21
User
Madrid
47
22
Sanjana Nandwani
Andalucía
46
23
User
Madrid
45
24
Luis Fett
Andalucía
45
25
User
Catalunya
44
26
Pol de Bas
Catalunya
44
27
Dani Díaz
Madrid
43
28
User
Madrid
43
29
Víctor Ballester Ribó
Catalunya
43
30
Xavier
Catalunya
43
32
User
Catalunya
42
33
Jaime Benabent
Andalucía
42
34
Ainhoa
Euskal Herria
42
35
User
Catalunya
42
36
User
Madrid
41
37
User
Asturias
41
38
User
Madrid
41
39
maria
Madrid
41
40
Juan Vecino
Madrid
41
41
ricardo
Madrid
40
43
User
Madrid
39
44
Charlie
Catalunya
38
45
User
Catalunya
38
46
Alex
Catalunya
37
47
User
Catalunya
34
48
Tim
Andalucía
34
49
Merce Ribera
Catalunya
34
50
Leonas Marius Iosif
Murcia
34
51
Parke Aiken
Catalunya
34
52
Rom
Catalunya
33
53
NACHO RUIZ
Andalucía
33
54
User
Catalunya
32
55
User
Catalunya
32
56
User
Madrid
32
57
Ule 17
Catalunya
31
58
Tomás Ramos Llerandi
Madrid
31
59
User
Madrid
31
60
User
Catalunya
31
61
User
Catalunya
31
62
User
Catalunya
30
63
Dinosaureo
Madrid
30
64
Bartt
Valenciana
30
65
valkyrie
Euskal Herria
30
66
Sirdeblas
Catalunya
30
67
Miguel Pérez Riera
Catalunya
29
68
Pau Herrera
Catalunya
29
69
User
Andalucía
29
70
User
Madrid
29
71
User
Catalunya
29
72
Marta Cogollor
Asturias
29
73
User
Cantabria
27
74
Japheth
Asturias
27
75
Poldeblas
Catalunya
27
76
User
Madrid
27
77
User
Catalunya
27
78
Carlos Perelló
Valenciana
26
79
User
Cantabria
26
80
User
Madrid
26
81
Gerard palau Martinez
Catalunya
26
82
User
Catalunya
26
83
User
Catalunya
26
84
GuillerLT
Madrid
25
85
User
Catalunya
25
86
User
Catalunya
25
88
Roberto
Catalunya
25
89
José Miguel Gomis
Valenciana
25
90
Benjamin Bourasseau
Andalucía
24
91
User
Catalunya
24
92
User
Catalunya
24
93
User
Murcia
24
94
Enya Roncero Olmo
Catalunya
24
95
User
Madrid
24
96
User
Madrid
23
97
User
Catalunya
23
98
User
Madrid
23
99
Marcel Sandoval
Catalunya
23
100
User
Catalunya
23
101
User
Catalunya
22
102
User
Catalunya
22
103
User
Catalunya
22
104
User
Madrid
22
105
User
Madrid
22
106
User
Madrid
22
107
Aadi Khemlani
Andalucía
22
108
User
Catalunya
22
109
Wayra uhlig
Catalunya
22
110
Monica Casanova
Catalunya
22
111
User
Madrid
22
112
Luis Fett
Andalucía
22
113
User
Catalunya
21
114
User
Catalunya
21
115
User
Catalunya
21
116
MMolero
Madrid
21
117
User
Galicia
21
118
User
Cantabria
21
119
Nicholas Brauner
Catalunya
21
120
User
Andalucía
21
121
CarlosNT
Madrid
21
122
User
Madrid
21
123
User
Madrid
21
125
Alejandro Lucas
Asturias
20
126
User
Illes Balears
20
127
User
Andalucía
20
128
Rubén Lorenzo García
Asturias
20
129
Helena Yuste
Catalunya
20
130
User
Catalunya
20
131
User
Catalunya
20
132
User
Catalunya
20
133
User
Catalunya
20
134
Eduard Flores Juncadella
Catalunya
19
135
Miquel Timoner Vaquer
Catalunya
19
136
User
Valenciana
19
137
User
Murcia
19
138
HULIO
Andalucía
19
140
User
Murcia
19
141
User
Catalunya
19
142
User
Madrid
19
143
User
Madrid
19
144
User
Catalunya
19
145
Damia de angulo
Catalunya
19
146
Daniel Torres Martín
Madrid
18
147
User
Galicia
18
148
User
Catalunya
18
149
User
Catalunya
18
150
User
Madrid
18
151
Adrià Gasull
Catalunya
18
152
User
Catalunya
18
153
Vinyet
Catalunya
18
154
Joana Moya Tirado
Andalucía
18
155
User
Catalunya
18
156
User
Catalunya
18
157
gerard merino
Catalunya
18
158
User
Madrid
17
159
User
Andalucía
17
160
User
Andalucía
17
161
User
Valenciana
17
162
User
Madrid
17
163
David serra
Catalunya
17
164
Yeha Yeha
Andalucía
17
165
User
Valenciana
17
166
User
Canarias
17
167
User
Catalunya
17
168
DoNotDeleteMe
Valenciana
17
169
User
Valenciana
17
170
User
Madrid
16
171
User
Catalunya
16
172
Martí Lahoz
Catalunya
16
173
Sergio Valencia Jaraba
Catalunya
16
174
User
Andalucía
16
175
peng pai
Madrid
16
176
User
Madrid
16
177
MANUEL GONZÁLEZ
Catalunya
16
178
User
Euskal Herria
16
179
Joan Gallego Vazquez
Catalunya
16
180
User
Madrid
16
181
Hector Sansano Parra
Catalunya
16
182
User
Andalucía
16
183
User
Catalunya
16
184
User
Catalunya
16
185
Gaurav Poona
Valenciana
16
186
User
Catalunya
16
187
User
Andalucía
15
188
User
Madrid
15
189
User
Valenciana
15
190
User
Catalunya
15
191
User
Madrid
15
192
User
Madrid
15
193
User
Catalunya
15
194
User
Catalunya
15
195
Juanjo
Madrid
15
197
User
Catalunya
15
198
User
Catalunya
15
199
User
Catalunya
15
200
User
Catalunya
15
201
Àngel Oribe Godàs
Catalunya
15
202
Iñigo Diez
Madrid
15
203
Ayman Krikaz
Catalunya
15
204
User
Catalunya
15
205
User
Madrid
15
206
User
Madrid
15
207
User
Catalunya
14
208
User
Asturias
14
209
User
Madrid
14
210
User
Catalunya
14
211
User
Valenciana
14
212
User
Madrid
14
213
User
Catalunya
14
214
Jorge Calvar
Madrid
14
215
Jaime
Andalucía
14
216
José Joaquín Ruzafa
Murcia
14
217
User
Catalunya
14
218
marti serra
Catalunya
14
219
User
Madrid
14
220
MaxMas
Catalunya
14
221
User
Madrid
13
222
User
Madrid
13
223
User
Valenciana
13
224
User
Catalunya
13
225
User
Euskal Herria
13
226
ego
Galicia
13
227
User
Madrid
13
228
User
Catalunya
13
229
User
Catalunya
13
230
User
Andalucía
13
231
User
Catalunya
13
232
User
Andalucía
13
233
User
Madrid
13
234
User
Andalucía
13
235
User
Catalunya
13
236
User
Madrid
13
237
User
Catalunya
13
238
User
Euskal Herria
13
239
User
Catalunya
12
240
User
Catalunya
12
241
gortrux
Catalunya
12
242
Jep Lezaun Sola
Catalunya
12
243
laia
Catalunya
12
244
William Eddleston
Nafarroako Foru Komunitatea*
12
245
User
Valenciana
12
247
Lluís Pinyol
Catalunya
12
248
User
Catalunya
12
249
User
Catalunya
12
250
User
Canarias
12
251
User
Andalucía
12
252
Javier Cuerva Heredia
Madrid
12
253
User
Galicia
12
254
User
Catalunya
12
255
User
Valenciana
12
256
roller 11
Catalunya
12
257
User
Catalunya
12
258
Arnau Estella Baeza
Catalunya
12
259
Arnau Beulas Vinaixa
Catalunya
12
260
User
Catalunya
12
261
BIZARRE MIND
Madrid
12
262
Jorge ruiz
Asturias
12
263
User
Madrid
12
264
User
Euskal Herria
12
265
User
Galicia
12
266
Wayra
Catalunya
12
267
User
Cantabria
12
268
User
Valenciana
12
269
User
Andalucía
12
270
Jon Goodey
Murcia
12
271
User
Catalunya
12
272
User
Catalunya
12
273
Illya Kostetskiy
Madrid
12
274
User
Catalunya
12
275
User
Catalunya
12
276
LUNA ELIGIO CALZADO
Catalunya
12
277
User
Aragón
12
278
User
Valenciana
11
279
User
Andalucía
11
281
User
Catalunya
11
282
User
Madrid
11
283
User
Catalunya
11
284
User
Catalunya
11
285
User
Catalunya
11
286
User
Canarias
11
287
Ana Núñez
Catalunya
11
288
Jon Iriondo Aragonés
Euskal Herria
11
290
User
Andalucía
11
291
User
Galicia
11
292
User
Illes Balears
11
293
Guillem Parra
Catalunya
11
294
Alejo Soriano
Catalunya
11
295
User
Catalunya
11
296
Carlos Torrecillas Vidal
Valenciana
11
297
User
Asturias
11
298
User
Andalucía
11
299
User
Andalucía
11
300
Angel Sanz Barragan
Catalunya
11
301
User
Madrid
11
302
User
Madrid
11
303
User
Madrid
11
304
User
Catalunya
11
305
User
Catalunya
11
306
User
Catalunya
11
307
User
Catalunya
11
308
Woody Barking Brown
Andalucía
11
309
User
Madrid
11
310
User
Madrid
11
311
User
Madrid
11
312
User
Asturias
11
313
User
Madrid
11
314
User
Galicia
11
315
User
Madrid
11
316
Juan Del Guayo
Madrid
11
318
User
Murcia
11
319
User
Catalunya
10
320
Vera
Madrid
10
321
Adrián Ramírez Marín
Andalucía
10
322
User
Valenciana
10
323
Javier Fernandez
Madrid
10
324
User
Catalunya
10
325
Alex
Catalunya
10
326
User
Catalunya
10
327
MARC ORIOL YBERN
Catalunya
10
328
Inma Segura
Andalucía
10
329
User
Valenciana
10
330
User
Madrid
10
331
User
Aragón
10
332
Ignasi Valls
Catalunya
10
333
Nikolas Perez
Andalucía
10
334
User
Madrid
10
335
User
Catalunya
10
337
User
Catalunya
10
338
User
Andalucía
10
339
User
Catalunya
10
340
User
Catalunya
10
341
User
Catalunya
10
342
User
Catalunya
10
343
User
Andalucía
10
345
User
Catalunya
10
346
User
Madrid
10
347
User
Madrid
10
348
User
Valenciana
10
349
Oscar de Mier
Catalunya
10
350
User
Andalucía
10
351
User
Andalucía
10
352
User
Madrid
10
353
User
Madrid
10
354
Donald Trump
Asturias
10
355
David Peralta
Catalunya
10
356
User
Catalunya
10
357
Jorge F
Valenciana
10
358
User
Madrid
10
359
User
Catalunya
10
360
User
Catalunya
10
361
User
Madrid
10
362
User
Andalucía
10
363
User
Madrid
10
364
User
Catalunya
10
365
Laia Riesgo García
Asturias
10
366
Maria Corona
Catalunya
10
367
Mònica Serra
Catalunya
10
368
User
Madrid
10
369
User
Madrid
10
370
User
Catalunya
10
371
User
Madrid
9
372
User
Catalunya
9
373
User
Extremadura
9
374
User
Catalunya
9
375
Gonzalo Calvar
Madrid
9
376
User
Catalunya
9
377
User
Catalunya
9
378
User
Andalucía
9
379
User
Madrid
9
380
Marc
Catalunya
9
381
User
Catalunya
9
382
User
Catalunya
9
384
User
Catalunya
9
385
User
Madrid
9
386
User
Catalunya
9
388
User
Valenciana
9
389
User
Madrid
9
390
User
Murcia
9
391
User
Catalunya
9
392
Arnau Martos Agustina
Catalunya
9
393
User
Valenciana
9
394
User
Catalunya
9
395
User
Catalunya
9
396
User
Valenciana
9
397
User
Catalunya
9
398
Javier Ruata
Catalunya
9
400
User
Catalunya
9
401
User
Catalunya
9
402
laia llansana zamora
Catalunya
9
403
Adam El Messaoudi
Catalunya
9
404
User
Catalunya
9
405
User
Andalucía
9
406
User
Madrid
9
407
User
Madrid
9
408
Noemí Vázquez
Catalunya
9
409
User
Andalucía
9
410
User
Catalunya
9
411
User
Catalunya
9
412
User
Madrid
9
413
User
Catalunya
9
414
User
Catalunya
9
416
peru ubillos
Castilla-La Mancha
9
417
User
Catalunya
9
418
User
Catalunya
9
419
User
Madrid
9
420
User
Catalunya
9
421
silvia
Catalunya
9
422
User
Catalunya
9
423
User
Catalunya
8
424
User
Madrid
8
425
User
Catalunya
8
426
User
Valenciana
8
427
User
Catalunya
8
428
User
Andalucía
8
429
User
Catalunya
8
430
User
Catalunya
8
431
User
Madrid
8
432
User
Andalucía
8
433
User
Andalucía
8
434
User
Valenciana
8
435
JuanJo
Madrid
8
436
User
Catalunya
8
437
User
Madrid
8
438
Noa Uhlig
Catalunya
8
439
User
Andalucía
8
440
User
Catalunya
8
441
Ezequiel Ribas
Catalunya
8
442
User
Catalunya
8
443
Marta Bucar Velasco
Catalunya
8
444
User
Valenciana
8
445
Elia Garcia
Catalunya
8
446
User
Aragón
8
448
Nico Varela
Madrid
8
449
User
Catalunya
8
450
User
Catalunya
8
451
User
Andalucía
8
452
User
Catalunya
8
453
User
Canarias
8
454
User
Catalunya
8
455
User
Catalunya
8
456
Daniel Díaz López
Catalunya
8
457
User
Catalunya
8
458
User
Aragón
8
459
User
Valenciana
8
460
User
Catalunya
8
461
User
Canarias
8
463
User
Madrid
8
464
User
Andalucía
8
465
User
Andalucía
8
466
User
Aragón
8
467
User
Madrid
8
468
User
Catalunya
8
469
User
Catalunya
8
470
User
Extremadura
8
471
User
Andalucía
8
472
Leonard Balm
Catalunya
8
473
Jana You
Catalunya
8
474
User
Madrid
8
475
User
Madrid
8
476
User
Catalunya
8
477
Keesjan lod
Euskal Herria
8
478
User
Andalucía
8
479
User
Catalunya
8
480
User
Canarias
8
481
User
Catalunya
8
482
User
Catalunya
8
483
User
Catalunya
8
484
Western Revival
Andalucía
8
485
User
Catalunya
7
486
User
Madrid
7
488
User
Madrid
7
489
User
Valenciana
7
490
User
Murcia
7
491
User
Madrid
7
492
User
Valenciana
7
493
User
Catalunya
7
494
User
Catalunya
7
495
User
Catalunya
7
496
User
Andalucía
7
497
User
Andalucía
7
498
User
Catalunya
7
499
User
Madrid
7
500
User
Madrid
7
501
User
Andalucía
7
502
User
Catalunya
7
503
User
Madrid
7
504
Gabriel Torres Agustina
Catalunya
7
505
User
Catalunya
7
506
User
Andalucía
7
507
User
Madrid
7
508
User
Catalunya
7
509
User
Catalunya
7
510
User
Canarias
7
511
User
Galicia
7
512
User
Madrid
7
513
User
Catalunya
7
514
User
Catalunya
7
515
User
Catalunya
7
516
User
Madrid
7
517
User
Catalunya
7
518
Gael gomez asuncion
Catalunya
7
519
Adria Seseras
Catalunya
7
520
User
Catalunya
7
521
Esteve Rodon Salazar
Catalunya
7
523
User
Valenciana
7
524
User
Catalunya
7
525
User
Catalunya
7
528
Quim Talamas
Madrid
7
529
User
Madrid
7
530
User
Catalunya
7
531
User
Catalunya
7
532
User
Catalunya
7
533
User
Madrid
7
534
User
Andalucía
7
535
User
Andalucía
7
536
Iván Expósito
Euskal Herria
7
537
Ana
Madrid
7
538
User
Madrid
7
539
Carlos
Catalunya
7
540
User
Andalucía
7
541
Bubu Boy
Catalunya
7
542
User
Andalucía
7
543
User
Madrid
7
544
Pere Molins
Catalunya
7
545
Andrea
Aragón
6
546
User
Andalucía
6
547
User
Madrid
6
548
User
Madrid
6
549
User
Valenciana
6
550
User
Catalunya
6
551
User
Madrid
6
552
User
Catalunya
6
553
User
Catalunya
6
554
User
Catalunya
6
555
User
Andalucía
6
556
User
Catalunya
6
557
User
Madrid
6
558
User
Madrid
6
559
User
Catalunya
6
560
Pablo Ávila
Andalucía
6
561
User
Catalunya
6
562
User
Madrid
6
563
Pol Rivero
Catalunya
6
564
User
Catalunya
6
565
jofre illa
Catalunya
6
566
User
Catalunya
6
567
User
Catalunya
6
568
User
Catalunya
6
569
User
Galicia
6
570
User
Andalucía
6
571
User
Catalunya
6
572
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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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