Leaderboard
Netherlands
Netherlands
Worldwide
Rank
Level
1
Jessey S.
Noord-Brabant
99
5
Sakboy TB
Noord-Holland
90
6
Sergio Tam
Zuid-Holland
87
7
Maarten Markering
Gelderland
86
8
Empty Suit
Noord-Brabant
86
9
P.R. Keus
Zuid-Holland
85
11
User
Zuid-Holland
82
13
Ruben Bieze
Noord-Holland
81
14
David
Noord-Holland
81
15
Carlos Alberto Neto
Zuid-Holland
81
16
Victor Harmsen
Zuid-Holland
80
17
Pavels Stecenko
Noord-Holland
80
18
Fritsie boiker
Utrecht
79
20
User
Utrecht
78
24
User
Gelderland
77
27
User
Utrecht
76
28
Coen Sonneveld
Noord-Brabant
76
29
Bram Klein Kranenbarg
Noord-Holland
75
31
User
Groningen
74
32
User
Overijssel
73
34
User
Region
71
35
User
Zuid-Holland
70
37
Mark
Zuid-Holland
69
38
User
Utrecht
68
39
Bas Nijman
Noord-Holland
68
40
User
Utrecht
68
42
User
Utrecht
66
44
User
Utrecht
65
45
User
Zuid-Holland
65
46
User
Zuid-Holland
64
49
User
Zuid-Holland
63
50
User
Zuid-Holland
63
51
User
Zuid-Holland
63
52
User
Zuid-Holland
62
53
User
Region
62
54
User
Zuid-Holland
61
56
Wisse
Zuid-Holland
59
57
User
Zuid-Holland
58
58
Nick Timmer
Drenthe
58
59
Prashanth Neelakantan
Zuid-Holland
58
61
Hajime Sasaki
Limburg
58
62
John Doe
Zuid-Holland
57
63
Freek
Zuid-Holland
57
65
User
Utrecht
57
66
Dirk Noom
Utrecht
57
67
User
Zuid-Holland
56
68
User
Overijssel
56
69
User
Overijssel
56
70
finnley
Noord-Holland
56
71
User
Gelderland
56
72
User
Overijssel
56
73
User
Utrecht
56
74
User
Utrecht
56
75
User
Overijssel
55
77
User
Overijssel
55
78
User
Zuid-Holland
55
80
User
Utrecht
55
81
Armo J
Noord-Holland
55
82
Anano Basilaia
Utrecht
54
84
Thomas
Zuid-Holland
54
85
User
Zuid-Holland
54
86
User
Utrecht
53
87
Dominique Langeslag
Noord-Holland
53
88
gamer
Region
53
89
User
Overijssel
53
90
User
Zuid-Holland
53
91
Paul Vergouwe
Zuid-Holland
53
92
User
Zuid-Holland
53
94
Stan Schalij
Noord-Holland
52
95
Rowan Mommers
Utrecht
52
96
User
Region
51
98
Jeroen van der Gevel
Utrecht
51
99
Wouter Janssen
Overijssel
51
100
User
Utrecht
50
101
User
Zuid-Holland
50
102
User
Overijssel
50
103
asddsa
Zuid-Holland
49
104
User
Zuid-Holland
49
105
User
Overijssel
49
106
Max Crolla
Noord-Holland
49
107
User
Utrecht
49
108
User
Zuid-Holland
49
109
User
Noord-Holland
49
110
User
Noord-Holland
48
111
User
Noord-Holland
48
112
User
Noord-Holland
48
113
User
Noord-Holland
48
114
User
Utrecht
48
115
User
Noord-Holland
47
117
User
Gelderland
47
118
User
Noord-Holland
46
119
Tartagnan
Utrecht
46
120
User
Overijssel
46
121
User
Noord-Holland
45
122
User
Overijssel
45
123
User
Overijssel
45
124
User
Noord-Holland
45
125
Andrei Cojocaru
Gelderland
45
126
Bhavya Jain
Zuid-Holland
45
127
User
Zuid-Holland
44
128
User
Zuid-Holland
44
129
Bram
Gelderland
44
130
User
Noord-Holland
44
131
User
Overijssel
44
132
User
Noord-Holland
44
133
User
Overijssel
44
134
User
Overijssel
44
135
User
Overijssel
44
136
Sven Polak
Noord-Holland
44
137
User
Zuid-Holland
44
138
User
Utrecht
44
139
User
Noord-Holland
44
140
Daan Wimmers
Noord-Brabant
43
141
User
Noord-Holland
43
142
User
Noord-Holland
43
143
User
Noord-Holland
43
145
User
Noord-Holland
43
146
User
Gelderland
43
147
Igor Quint
Noord-Holland
43
148
User
Noord-Holland
43
149
User
Region
42
150
Mátyás Tóth
Noord-Holland
42
151
Tim
Overijssel
42
152
User
Utrecht
42
153
User
Noord-Holland
42
154
User
Noord-Holland
42
155
Anniek van Eijkern
Zuid-Holland
42
156
User
Utrecht
42
157
User
Gelderland
41
158
User
Noord-Brabant
41
159
User
Zuid-Holland
41
160
User
Overijssel
41
161
Vince
Limburg
41
162
User
Noord-Holland
41
163
Jefferson Vegas
Noord-Brabant
41
164
Tom Bour
Limburg
40
165
User
Zuid-Holland
40
166
User
Noord-Holland
40
167
User
Noord-Holland
40
168
User
Overijssel
40
169
User
Noord-Holland
40
170
User
Zuid-Holland
40
171
User
Noord-Holland
40
172
User
Noord-Holland
39
173
User
Zuid-Holland
39
174
User
Region
39
175
User
Noord-Holland
39
176
Sebastiaan Vaz
Noord-Holland
39
177
User
Gelderland
39
178
User
Noord-Holland
39
179
User
Gelderland
38
180
John Doe
Zuid-Holland
38
181
User
Noord-Brabant
38
182
Dániel Erdélyi
Zuid-Holland
38
183
Ketchup
Noord-Holland
38
184
Jantine Boomsma
Zuid-Holland
38
185
vicc
Utrecht
38
186
User
Zuid-Holland
38
187
User
Zuid-Holland
38
188
User
Zuid-Holland
38
189
Robin David
Utrecht
37
190
User
Utrecht
37
191
User
Noord-Holland
37
192
User
Noord-Holland
37
193
User
Noord-Holland
37
194
Susan Harris
Utrecht
37
195
User
Noord-Brabant
37
196
User
Noord-Holland
37
197
Guus
Noord-Holland
37
198
User
Overijssel
37
199
User
Zuid-Holland
37
200
User
Noord-Holland
37
201
User
Zuid-Holland
37
202
User
Noord-Holland
37
203
Gary Mikoen
Zuid-Holland
37
204
User
Zuid-Holland
36
206
User
Noord-Holland
36
207
Rabea Semmler
Limburg
36
208
User
Zuid-Holland
36
209
User
Noord-Brabant
36
210
User
Noord-Holland
36
211
User
Zuid-Holland
36
212
User
Noord-Holland
36
213
MF
Gelderland
36
214
Matt Tee
Zuid-Holland
35
215
User
Utrecht
35
216
User
Noord-Holland
35
217
User
Utrecht
35
218
User
Overijssel
35
219
User
Overijssel
35
220
User
Noord-Holland
35
221
User
Zuid-Holland
35
222
User
Zuid-Holland
35
223
User
Noord-Brabant
35
224
User
Utrecht
34
225
User
Noord-Holland
34
226
User
Noord-Holland
34
227
User
Gelderland
34
228
User
Overijssel
34
229
User
Zuid-Holland
34
230
RJ Bodewes
Noord-Brabant
34
231
User
Noord-Brabant
33
232
Jelmer
Utrecht
33
233
User
Noord-Holland
33
234
User
Noord-Holland
33
235
User
Noord-Holland
33
236
User
Zuid-Holland
33
237
User
Noord-Brabant
33
238
User
Zuid-Holland
33
239
User
Noord-Holland
33
240
User
Noord-Holland
32
241
User
Noord-Holland
32
242
User
Noord-Brabant
32
243
User
Noord-Brabant
32
244
User
Gelderland
32
245
User
Noord-Brabant
32
246
User
Zeeland
32
247
User
Noord-Holland
32
248
User
Overijssel
32
249
User
Noord-Holland
32
250
zita lukacs
Noord-Holland
32
251
valharrry
Zuid-Holland
32
252
User
Noord-Holland
31
253
User
Noord-Holland
31
254
alain faddegon
Zuid-Holland
31
255
User
Noord-Holland
31
256
User
Groningen
31
257
robert van wijk
Overijssel
31
258
User
Noord-Holland
31
259
User
Noord-Brabant
31
260
User
Noord-Holland
31
261
User
Zuid-Holland
31
262
User
Zuid-Holland
31
263
User
Noord-Holland
31
264
User
Noord-Holland
31
265
User
Noord-Holland
31
266
User
Noord-Holland
30
267
User
Noord-Brabant
30
268
User
Noord-Holland
30
269
User
Noord-Holland
30
270
Nick Hennen
Noord-Brabant
30
271
User
Zuid-Holland
30
272
User
Noord-Holland
30
273
User
Noord-Holland
30
274
User
Zuid-Holland
30
275
Roan Paulus
Noord-Holland
30
276
User
Noord-Holland
30
277
User
Noord-Holland
29
278
User
Noord-Holland
29
279
User
Zuid-Holland
29
280
User
Noord-Holland
29
281
Gert
Gelderland
29
282
User
Noord-Brabant
29
283
User
Noord-Brabant
29
284
User
Noord-Holland
29
285
User
Noord-Holland
29
286
User
Zuid-Holland
29
287
User
Zuid-Holland
29
288
User
Noord-Holland
29
289
User
Noord-Brabant
29
290
User
Flevoland
29
291
User
Noord-Holland
29
292
Tobias Pütz
Zuid-Holland
29
293
User
Overijssel
28
294
User
Zuid-Holland
28
295
User
Noord-Holland
28
296
Jojo
Limburg
28
297
User
Utrecht
28
298
User
Noord-Brabant
28
299
Guus Veldhuijzen
Utrecht
28
300
User
Zuid-Holland
28
301
User
Noord-Brabant
28
302
User
Flevoland
28
303
User
Utrecht
28
304
User
Zuid-Holland
28
305
Portugees
Flevoland
28
306
User
Noord-Holland
27
307
Bram
Noord-Holland
27
308
User
Noord-Holland
27
309
User
Noord-Holland
27
310
User
Limburg
27
311
User
Zuid-Holland
27
312
User
Noord-Holland
27
313
User
Zuid-Holland
27
314
User
Zuid-Holland
27
315
User
Noord-Holland
27
316
User
Gelderland
27
317
User
Zuid-Holland
27
318
User
Noord-Holland
27
319
User
Noord-Holland
27
320
User
Limburg
27
321
User
Noord-Holland
26
322
User
Zuid-Holland
26
323
User
Noord-Holland
26
324
User
Zuid-Holland
26
325
User
Flevoland
26
326
User
Noord-Holland
26
327
Alexander Schwarze
Limburg
26
328
User
Noord-Holland
26
329
User
Noord-Holland
26
330
User
Noord-Holland
26
331
User
Zuid-Holland
26
332
User
Noord-Holland
26
333
User
Noord-Holland
26
334
User
Zuid-Holland
26
335
User
Noord-Holland
26
336
User
Noord-Brabant
26
337
User
Zuid-Holland
26
338
Floris-Jan Carton
Noord-Holland
26
339
User
Noord-Holland
26
340
User
Zuid-Holland
26
341
User
Zuid-Holland
26
342
Bas bakker
Gelderland
25
343
User
Noord-Holland
25
344
Ronald Schotman
Zuid-Holland
25
345
User
Noord-Holland
25
346
User
Noord-Holland
25
347
User
Groningen
25
348
User
Noord-Holland
25
349
User
Zuid-Holland
25
350
User
Noord-Holland
25
351
User
Noord-Holland
25
352
User
Zuid-Holland
25
353
User
Noord-Brabant
25
354
User
Noord-Brabant
25
355
User
Zuid-Holland
25
356
User
Gelderland
24
357
User
Zuid-Holland
24
358
User
Noord-Holland
24
359
User
Zuid-Holland
24
360
User
Noord-Brabant
24
361
User
Limburg
24
362
User
Groningen
24
363
User
Drenthe
24
364
User
Noord-Brabant
24
365
User
Zuid-Holland
24
366
User
Noord-Brabant
24
367
User
Noord-Brabant
24
368
User
Noord-Holland
24
369
User
Noord-Holland
24
370
User
Noord-Holland
24
371
User
Zuid-Holland
24
372
User
Limburg
24
373
User
Overijssel
23
374
User
Zuid-Holland
23
375
User
Noord-Holland
23
376
User
Zuid-Holland
23
377
User
Zuid-Holland
23
378
User
Zuid-Holland
23
379
User
Zuid-Holland
23
380
User
Gelderland
23
381
User
Noord-Holland
23
383
User
Zuid-Holland
23
384
User
Utrecht
23
385
User
Limburg
23
386
User
Limburg
23
387
User
Noord-Holland
23
388
User
Zuid-Holland
23
389
User
Overijssel
23
390
User
Noord-Holland
23
391
User
Flevoland
23
392
Pieter
Gelderland
23
393
User
Gelderland
23
394
User
Limburg
23
395
Austin Menzies
Noord-Holland
22
396
User
Zuid-Holland
22
397
User
Zuid-Holland
22
398
User
Noord-Holland
22
399
User
Noord-Holland
22
400
Jaco Fourie
Noord-Holland
22
401
User
Zuid-Holland
22
402
User
Utrecht
22
403
User
Noord-Holland
22
404
User
Noord-Holland
22
405
User
Noord-Holland
22
406
User
Zuid-Holland
22
407
User
Noord-Holland
22
408
User
Utrecht
22
409
Tracy Gee
Noord-Holland
22
410
User
Noord-Holland
22
411
User
Zuid-Holland
21
412
User
Zuid-Holland
21
413
User
Zuid-Holland
21
414
ingmar Heikens
Groningen
21
415
User
Zuid-Holland
21
416
User
Region
21
417
User
Noord-Holland
21
418
Linda Flcck
Noord-Holland
21
419
User
Zuid-Holland
21
420
User
Zeeland
21
421
User
Utrecht
21
422
User
Noord-Holland
21
423
User
Noord-Holland
21
424
User
Zuid-Holland
21
425
User
Gelderland
21
426
User
Noord-Brabant
21
427
User
Zuid-Holland
21
428
User
Noord-Holland
21
429
User
Noord-Holland
21
430
User
Noord-Holland
21
431
User
Noord-Holland
21
432
User
Limburg
21
433
User
Region
20
434
User
Zuid-Holland
20
435
Franklin S
Noord-Brabant
20
436
User
Groningen
20
437
User
Zuid-Holland
20
438
User
Overijssel
20
439
User
Zuid-Holland
20
440
User
Overijssel
20
441
User
Zuid-Holland
20
442
User
Gelderland
20
443
User
Noord-Holland
20
444
User
Zuid-Holland
20
445
User
Noord-Holland
20
446
User
Noord-Holland
20
447
User
Groningen
20
448
User
Noord-Holland
20
449
User
Noord-Brabant
20
450
User
Noord-Brabant
20
451
User
Zuid-Holland
20
452
User
Zuid-Holland
20
453
User
Zuid-Holland
20
454
Natasja de Kok
Zuid-Holland
20
455
User
Noord-Holland
20
456
User
Zuid-Holland
19
457
Han Ruder
Zuid-Holland
19
458
User
Noord-Holland
19
459
User
Noord-Holland
19
460
User
Noord-Holland
19
461
User
Overijssel
19
462
User
Gelderland
19
463
User
Noord-Holland
19
464
User
Groningen
19
465
User
Noord-Holland
19
466
User
Gelderland
19
467
User
Noord-Holland
19
468
User
Limburg
19
469
User
Noord-Holland
19
470
User
Zuid-Holland
18
471
User
Zuid-Holland
18
472
User
Groningen
18
473
User
Noord-Holland
18
474
User
Utrecht
18
475
User
Noord-Holland
18
476
User
Zuid-Holland
18
477
User
Noord-Brabant
18
478
User
Zuid-Holland
18
479
User
Zuid-Holland
18
480
User
Noord-Brabant
18
481
Practice Account
Zuid-Holland
18
482
Stoyal Znachit
Noord-Holland
18
483
User
Noord-Holland
18
484
User
Noord-Holland
18
485
User
Noord-Brabant
18
486
User
Noord-Holland
18
487
User
Zuid-Holland
18
488
Anavi Basu
Noord-Holland
18
489
User
Noord-Holland
18
490
User
Zuid-Holland
17
491
Joosje Goedhart
Noord-Holland
17
492
User
Utrecht
17
493
User
Noord-Holland
17
494
User
Noord-Holland
17
495
User
Zuid-Holland
17
496
User
Overijssel
17
497
User
Noord-Holland
17
498
User
Noord-Holland
17
499
User
Zuid-Holland
17
500
Jur Tasseron
Gelderland
17
501
User
Gelderland
16
502
User
Zuid-Holland
16
503
User
Zuid-Holland
16
504
User
Zuid-Holland
16
505
User
Utrecht
16
506
User
Zuid-Holland
16
507
User
Utrecht
16
508
User
Zuid-Holland
16
509
User
Noord-Brabant
16
510
User
Noord-Holland
16
511
User
Noord-Holland
16
512
Falk
Noord-Holland
15
513
User
Noord-Holland
15
514
Mike
Noord-Holland
15
515
User
Noord-Holland
15
516
User
Noord-Holland
15
517
User
Utrecht
15
518
User
Zuid-Holland
15
519
User
Zuid-Holland
15
520
User
Noord-Holland
15
521
User
Noord-Holland
15
522
User
Noord-Holland
15
523
User
Noord-Holland
15
524
User
Noord-Holland
15
525
User
Noord-Holland
15
526
User
Noord-Holland
15
527
User
Overijssel
15
528
User
Noord-Holland
15
529
User
Noord-Holland
15
530
User
Noord-Holland
15
531
User
Noord-Brabant
15
532
User
Noord-Holland
15
533
User
Gelderland
15
534
User
Gelderland
15
535
User
Zuid-Holland
15
536
User
Noord-Holland
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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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