Leaderboard
Netherlands
Netherlands
Worldwide
Rank
Level
1
Jessey S.
Noord-Brabant
99
3
User
Groningen
96
7
Sakboy TB
Noord-Holland
90
8
Anano Basilaia
Utrecht
89
9
Sergio Tam
Zuid-Holland
87
10
sven_polak
Noord-Holland
86
12
Ernst
Region
86
13
Maarten Markering
Gelderland
86
14
Empty Suit
Noord-Brabant
86
15
Robin David
Overijssel
85
17
Alvin
Gelderland
84
18
User
Zuid-Holland
84
20
User
Groningen
84
21
User
Region
83
23
User
Zuid-Holland
82
25
User
Overijssel
81
26
Ruben Bieze
Noord-Holland
81
27
David
Noord-Holland
81
28
Carlos Alberto Neto
Zuid-Holland
81
29
Victor Harmsen
Zuid-Holland
80
30
Pavels Stecenko
Noord-Holland
80
32
User
Utrecht
79
33
Fritsie boiker
Utrecht
79
35
User
Utrecht
78
39
User
Region
77
40
User
Gelderland
77
43
Xinyu Hu
Zuid-Holland
76
44
User
Utrecht
76
45
Coen Sonneveld
Noord-Brabant
76
47
User
Overijssel
73
48
User
Overijssel
72
49
Pim Kouwenhoven
Zuid-Holland
71
50
Dominique Langeslag
Noord-Holland
71
51
Yal Gnoj
Utrecht
70
52
User
Zuid-Holland
70
53
User
Overijssel
69
55
Mark
Zuid-Holland
69
56
Babba babs
Noord-Holland
68
57
User
Utrecht
68
58
Bas Nijman
Noord-Holland
68
59
User
Utrecht
68
61
N. F.
Zuid-Holland
66
63
User
Utrecht
66
64
User
Utrecht
65
65
User
Zuid-Holland
65
66
Rishab Khandelwal
Noord-Brabant
64
69
User
Zuid-Holland
63
70
User
Zuid-Holland
63
71
User
Zuid-Holland
63
72
Pie pie
Zuid-Holland
62
73
John Doe
Noord-Holland
62
74
User
Region
62
75
User
Zuid-Holland
62
76
User
Region
62
77
Remmie
Noord-Holland
61
78
User
Zuid-Holland
61
79
Nick Timmer
Noord-Brabant
60
81
Jur Tasseron
Gelderland
59
83
User
Zuid-Holland
59
85
Wisse
Zuid-Holland
59
86
User
Zuid-Holland
58
87
Prashanth Neelakantan
Zuid-Holland
58
89
Hajime Sasaki
Limburg
58
90
User
Gelderland
57
92
User
Zuid-Holland
57
94
User
Zuid-Holland
57
95
Freek
Zuid-Holland
57
97
User
Utrecht
57
98
Dirk Noom
Utrecht
57
99
Bram Sjaak
Gelderland
56
100
User
Drenthe
56
101
User
Zuid-Holland
56
102
User
Overijssel
56
103
User
Overijssel
56
104
finnley
Noord-Holland
56
105
User
Overijssel
56
106
User
Utrecht
56
107
User
Utrecht
56
108
User
Noord-Holland
55
109
Lukas Bieliauskas
Noord-Holland
55
110
User
Utrecht
55
111
User
Region
55
112
User
Overijssel
55
113
User
Noord-Holland
55
114
User
Overijssel
55
115
User
Zuid-Holland
55
116
Wins
Noord-Holland
55
117
User
Utrecht
55
118
Armo J
Noord-Holland
55
119
User
Zuid-Holland
54
120
User
Zuid-Holland
54
121
User
Groningen
54
123
User
Noord-Brabant
54
124
Thomas
Zuid-Holland
54
125
User
Zuid-Holland
54
126
User
Utrecht
53
127
gamer
Region
53
128
User
Overijssel
53
129
User
Zuid-Holland
53
130
Paul Vergouwe
Zuid-Holland
53
131
User
Zuid-Holland
53
132
User
Zuid-Holland
52
133
User
Gelderland
52
134
User
Zuid-Holland
52
135
User
Zuid-Holland
52
136
Erik J.
Zuid-Holland
52
137
User
Noord-Brabant
52
138
Stan Schalij
Noord-Holland
52
139
Rowan Mommers
Utrecht
52
140
Piet
Noord-Holland
51
141
Radmir Leushuis
Utrecht
51
142
User
Noord-Brabant
51
143
Jeroen van der Gevel
Utrecht
51
144
Wouter Janssen
Overijssel
51
145
Olivier van der Vet
Zuid-Holland
50
146
Bas bakker
Gelderland
50
147
User
Utrecht
50
148
User
Zuid-Holland
50
149
User
Overijssel
50
150
User
Zuid-Holland
49
151
Max Dijkstra
Noord-Holland
49
152
User
Noord-Holland
49
153
Sci Sil
Region
49
154
User
Noord-Holland
49
155
User
Zuid-Holland
49
156
DJ Bread
Zuid-Holland
49
157
User
Zuid-Holland
49
158
asddsa
Zuid-Holland
49
159
User
Zuid-Holland
49
160
User
Overijssel
49
161
Max Crolla
Noord-Holland
49
162
User
Utrecht
49
163
User
Zuid-Holland
49
164
User
Noord-Holland
49
165
User
Zuid-Holland
48
166
User
Zuid-Holland
48
167
Michael Wijtzes
Zuid-Holland
48
168
User
Noord-Holland
48
169
User
Noord-Holland
48
170
User
Noord-Holland
48
171
User
Noord-Holland
48
172
User
Utrecht
48
173
User
Zuid-Holland
47
174
User
Zuid-Holland
47
175
Sven Witteveen
Zuid-Holland
47
176
User
Zuid-Holland
47
177
User
Noord-Holland
47
178
Casper Steenbergen
Noord-Holland
47
179
Olivier Kuijpers
Zuid-Holland
47
180
User
Noord-Holland
47
182
User
Gelderland
47
183
User
Zuid-Holland
46
184
User
Zuid-Holland
46
185
Jan Hoekman
Zuid-Holland
46
186
User
Noord-Holland
46
187
Tartagnan
Utrecht
46
188
User
Overijssel
46
189
User
Zuid-Holland
45
190
Mats Heijmans
Noord-Holland
45
191
Doodle
Noord-Brabant
45
192
User
Noord-Brabant
45
193
ingmar Heikens
Groningen
45
194
User
Noord-Brabant
45
195
Austin Menzies
Noord-Holland
45
196
sky sky
Zuid-Holland
45
197
User
Overijssel
45
198
User
Overijssel
45
199
User
Noord-Holland
45
200
Andrei Cojocaru
Gelderland
45
201
Bhavya Jain
Zuid-Holland
45
203
Dwarrel
Noord-Holland
44
204
User
Zuid-Holland
44
205
User
Zuid-Holland
44
206
User
Zuid-Holland
44
207
User
Region
44
208
User
Utrecht
44
209
Giovanni Scheepers
Flevoland
44
210
User
Zuid-Holland
44
211
User
Zuid-Holland
44
212
User
Noord-Holland
44
213
User
Overijssel
44
214
User
Noord-Holland
44
215
User
Overijssel
44
216
User
Overijssel
44
217
User
Overijssel
44
218
User
Zuid-Holland
44
219
User
Utrecht
44
220
User
Noord-Holland
44
221
User
Zuid-Holland
43
222
User
Zuid-Holland
43
223
Stevan Jevremov
Zuid-Holland
43
224
Daan Wimmers
Noord-Brabant
43
225
User
Noord-Holland
43
226
User
Noord-Holland
43
227
User
Noord-Holland
43
229
User
Noord-Holland
43
230
User
Gelderland
43
231
Igor Quint
Noord-Holland
43
232
User
Noord-Holland
43
233
Johan Loots
Groningen
42
234
Jeroen van der Does de Willebois
Noord-Holland
42
235
User
Zuid-Holland
42
236
User
Zuid-Holland
42
237
User
Zuid-Holland
42
238
User
Gelderland
42
239
Mátyás Tóth
Noord-Holland
42
240
Tim
Overijssel
42
241
User
Utrecht
42
242
User
Noord-Holland
42
243
User
Noord-Holland
42
244
Anniek van Eijkern
Zuid-Holland
42
245
User
Utrecht
42
246
User
Noord-Holland
41
247
User
Zuid-Holland
41
248
User
Zuid-Holland
41
249
User
Zuid-Holland
41
250
User
Noord-Brabant
41
251
User
Gelderland
41
252
User
Noord-Brabant
41
253
User
Zuid-Holland
41
254
User
Overijssel
41
255
Vince
Limburg
41
256
User
Noord-Holland
41
257
Jefferson Vegas
Noord-Brabant
41
259
User
Zuid-Holland
40
260
Daan Klaver
Zuid-Holland
40
261
Tom Bour
Limburg
40
262
User
Zuid-Holland
40
263
User
Noord-Holland
40
264
User
Noord-Holland
40
265
User
Overijssel
40
266
User
Noord-Holland
40
267
User
Zuid-Holland
40
268
User
Noord-Holland
40
269
Titus Pellegrom
Noord-Holland
39
270
User
Noord-Holland
39
271
User
Zuid-Holland
39
272
User
Noord-Holland
39
273
Sebastiaan Vaz
Noord-Holland
39
274
User
Gelderland
39
275
User
Noord-Holland
39
276
User
Zuid-Holland
38
277
User
Zuid-Holland
38
278
User
Gelderland
38
279
John Doe
Zuid-Holland
38
280
User
Noord-Brabant
38
281
Dániel Erdélyi
Zuid-Holland
38
282
Ketchup
Noord-Holland
38
283
Jantine Boomsma
Zuid-Holland
38
284
vicc
Utrecht
38
285
User
Zuid-Holland
38
286
User
Zuid-Holland
38
287
User
Zuid-Holland
38
288
User
Zuid-Holland
37
289
User
Utrecht
37
290
User
Noord-Holland
37
291
User
Noord-Holland
37
292
User
Noord-Holland
37
293
Susan Harris
Utrecht
37
294
User
Noord-Brabant
37
295
User
Noord-Holland
37
296
Guus
Noord-Holland
37
297
User
Overijssel
37
298
User
Zuid-Holland
37
299
User
Noord-Holland
37
300
User
Zuid-Holland
37
301
User
Noord-Holland
37
302
Gary Mikoen
Zuid-Holland
37
303
User
Overijssel
36
304
User
Zuid-Holland
36
305
User
Noord-Holland
36
307
User
Noord-Holland
36
308
Rabea Semmler
Limburg
36
309
User
Zuid-Holland
36
310
User
Noord-Brabant
36
311
User
Noord-Holland
36
312
User
Zuid-Holland
36
313
User
Noord-Holland
36
314
MF
Gelderland
36
315
Matt Tee
Zuid-Holland
35
316
User
Utrecht
35
317
User
Noord-Holland
35
318
User
Utrecht
35
319
User
Overijssel
35
320
User
Overijssel
35
321
User
Noord-Holland
35
322
User
Zuid-Holland
35
323
User
Zuid-Holland
35
324
User
Utrecht
34
325
User
Noord-Holland
34
326
User
Noord-Holland
34
327
User
Gelderland
34
328
User
Overijssel
34
329
User
Zuid-Holland
34
330
RJ Bodewes
Noord-Brabant
34
331
User
Zuid-Holland
33
332
User
Noord-Brabant
33
333
Jelmer
Utrecht
33
334
User
Noord-Holland
33
335
User
Noord-Holland
33
336
User
Noord-Holland
33
337
User
Zuid-Holland
33
338
User
Noord-Brabant
33
339
User
Zuid-Holland
33
340
User
Noord-Holland
33
341
User
Noord-Holland
32
342
User
Noord-Holland
32
343
User
Noord-Holland
32
344
User
Zuid-Holland
32
345
User
Zuid-Holland
32
346
User
Noord-Holland
32
347
User
Noord-Holland
32
348
User
Noord-Brabant
32
349
User
Noord-Brabant
32
350
User
Gelderland
32
351
User
Noord-Brabant
32
352
User
Zeeland
32
353
User
Noord-Holland
32
354
User
Overijssel
32
355
User
Noord-Holland
32
356
zita lukacs
Noord-Holland
32
357
valharrry
Zuid-Holland
32
358
User
Noord-Holland
31
359
User
Zuid-Holland
31
360
User
Noord-Holland
31
361
User
Noord-Holland
31
362
User
Noord-Holland
31
363
alain faddegon
Zuid-Holland
31
364
User
Noord-Holland
31
365
User
Groningen
31
366
User
Noord-Holland
31
367
User
Noord-Brabant
31
368
User
Noord-Holland
31
369
User
Zuid-Holland
31
370
User
Zuid-Holland
31
371
User
Noord-Holland
31
372
User
Noord-Holland
31
373
User
Noord-Holland
31
374
User
Noord-Holland
30
375
Nick Hennen
Gelderland
30
376
Anner
Zuid-Holland
30
377
User
Groningen
30
378
User
Gelderland
30
379
P.R. Keus
Zuid-Holland
30
380
User
Noord-Holland
30
381
User
Noord-Brabant
30
382
User
Noord-Holland
30
383
User
Noord-Holland
30
384
Nick Hennen
Noord-Brabant
30
385
User
Zuid-Holland
30
386
User
Noord-Holland
30
387
User
Noord-Holland
30
388
User
Zuid-Holland
30
389
User
Noord-Holland
30
390
User
Zuid-Holland
29
391
User
Zuid-Holland
29
392
User
Zuid-Holland
29
393
User
Noord-Holland
29
394
User
Noord-Holland
29
395
User
Zuid-Holland
29
396
User
Noord-Holland
29
397
Gert
Gelderland
29
398
User
Noord-Brabant
29
399
User
Noord-Brabant
29
400
User
Noord-Holland
29
401
User
Noord-Holland
29
402
User
Zuid-Holland
29
403
User
Zuid-Holland
29
404
User
Noord-Holland
29
405
User
Noord-Brabant
29
406
User
Flevoland
29
407
User
Noord-Holland
29
408
Tobias Pütz
Zuid-Holland
29
409
User
Zuid-Holland
28
410
User
Noord-Holland
28
411
User
Limburg
28
412
Bram Klein Kranenbarg
Noord-Holland
28
413
User
Zuid-Holland
28
414
User
Overijssel
28
415
User
Zuid-Holland
28
416
User
Noord-Holland
28
417
Jojo
Limburg
28
418
User
Utrecht
28
419
User
Noord-Brabant
28
420
User
Zuid-Holland
28
421
User
Noord-Brabant
28
422
User
Flevoland
28
423
User
Utrecht
28
424
User
Zuid-Holland
28
425
Portugees
Flevoland
28
426
User
Noord-Holland
27
427
User
Noord-Holland
27
428
User
Zuid-Holland
27
429
User
Zuid-Holland
27
430
User
Utrecht
27
431
User
Noord-Holland
27
432
User
Noord-Holland
27
433
Bram
Noord-Holland
27
434
User
Noord-Holland
27
435
User
Noord-Holland
27
436
User
Limburg
27
437
User
Zuid-Holland
27
438
User
Noord-Holland
27
439
User
Zuid-Holland
27
440
User
Zuid-Holland
27
441
User
Noord-Holland
27
442
User
Gelderland
27
443
User
Zuid-Holland
27
444
User
Noord-Holland
27
445
User
Noord-Holland
27
446
User
Limburg
27
447
User
Zuid-Holland
26
448
User
Noord-Holland
26
449
User
Zuid-Holland
26
450
User
Noord-Holland
26
451
User
Zuid-Holland
26
452
User
Flevoland
26
453
User
Noord-Holland
26
454
Alexander Schwarze
Limburg
26
455
User
Noord-Holland
26
456
User
Noord-Holland
26
457
User
Noord-Holland
26
458
User
Zuid-Holland
26
459
User
Noord-Holland
26
460
User
Noord-Holland
26
461
User
Zuid-Holland
26
462
User
Noord-Holland
26
463
User
Noord-Brabant
26
464
User
Zuid-Holland
26
465
Floris-Jan Carton
Noord-Holland
26
466
User
Noord-Holland
26
467
User
Zuid-Holland
26
468
User
Zuid-Holland
26
469
User
Utrecht
25
470
Nicolette Bosch
Zuid-Holland
25
471
User
Zuid-Holland
25
472
User
Noord-Holland
25
473
Ronald Schotman
Zuid-Holland
25
474
User
Noord-Holland
25
475
User
Noord-Holland
25
476
User
Groningen
25
477
User
Noord-Holland
25
478
User
Zuid-Holland
25
479
User
Noord-Holland
25
480
User
Noord-Holland
25
481
User
Zuid-Holland
25
482
User
Noord-Brabant
25
483
User
Noord-Brabant
25
484
User
Zuid-Holland
25
485
Andrej N
Zuid-Holland
24
486
User
Gelderland
24
487
User
Zuid-Holland
24
488
User
Noord-Holland
24
489
User
Zuid-Holland
24
490
User
Noord-Brabant
24
491
User
Limburg
24
492
User
Groningen
24
493
User
Drenthe
24
494
User
Noord-Brabant
24
495
User
Zuid-Holland
24
496
User
Noord-Brabant
24
497
User
Noord-Brabant
24
498
User
Noord-Holland
24
499
User
Noord-Holland
24
500
User
Noord-Holland
24
501
User
Zuid-Holland
24
502
User
Limburg
24
503
User
Noord-Holland
23
504
User
Zuid-Holland
23
505
User
Zuid-Holland
23
506
User
Utrecht
23
507
User
Zuid-Holland
23
508
User
Noord-Holland
23
509
User
Overijssel
23
510
User
Zuid-Holland
23
511
User
Noord-Holland
23
512
User
Zuid-Holland
23
513
User
Zuid-Holland
23
514
User
Zuid-Holland
23
515
User
Zuid-Holland
23
516
User
Gelderland
23
517
User
Noord-Holland
23
519
User
Zuid-Holland
23
520
User
Utrecht
23
521
User
Limburg
23
522
User
Limburg
23
523
User
Noord-Holland
23
524
User
Zuid-Holland
23
525
User
Overijssel
23
526
User
Noord-Holland
23
527
User
Flevoland
23
528
Pieter
Gelderland
23
529
User
Gelderland
23
530
User
Limburg
23
531
User
Zuid-Holland
22
532
User
Zuid-Holland
22
533
User
Zuid-Holland
22
534
User
Noord-Holland
22
535
User
Noord-Holland
22
536
Jaco Fourie
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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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