Leaderboard
Denmark
Denmark
Worldwide
Rank
Level
1
Erik M. L. Johannesen
Hovedstaden
79
3
Theis Hansen
Hovedstaden
65
5
Kåre Holm Thomsen
Hovedstaden
59
8
User
Syddanmark
50
10
User
Midtjylland
41
11
User
Hovedstaden
41
12
User
Hovedstaden
39
13
User
Hovedstaden
38
14
User
Sjælland
38
15
Ixnatifual
Midtjylland
38
16
ole jensen
Hovedstaden
37
17
User
Hovedstaden
37
18
User
Hovedstaden
37
19
User
Hovedstaden
36
20
User
Midtjylland
36
21
User
Nordjylland
35
22
Dyrlund
Hovedstaden
35
23
User
Syddanmark
34
24
User
Hovedstaden
34
25
User
Hovedstaden
34
26
Teodor Andersen
Hovedstaden
33
27
User
Hovedstaden
33
28
User
Hovedstaden
32
29
User
Hovedstaden
32
30
User
Hovedstaden
30
31
User
Hovedstaden
30
32
Pedro Rodrigues
Syddanmark
30
33
Ida Maria Soelje
Hovedstaden
27
34
User
Nordjylland
27
35
Magnus Kanstrup
Sjælland
27
36
Logan
Nordjylland
27
37
User
Hovedstaden
27
38
Tommy Carlsen
Hovedstaden
26
39
User
Midtjylland
26
40
User
Sjælland
26
41
User
Hovedstaden
26
42
Clara Jensen
Hovedstaden
25
43
User
Syddanmark
25
44
User
Midtjylland
25
45
User
Hovedstaden
25
46
User
Hovedstaden
25
47
User
Hovedstaden
24
48
User
Midtjylland
24
49
Christian Alexander Yabar Mejer 2018e
Sjælland
24
50
User
Midtjylland
24
51
User
Hovedstaden
24
52
User
Hovedstaden
24
53
User
Midtjylland
24
54
User
Midtjylland
23
55
User
Hovedstaden
23
56
User
Hovedstaden
23
57
Kasper Rasmussen
Nordjylland
22
58
User
Midtjylland
22
59
User
Hovedstaden
22
60
Laura Gutiérrez
Hovedstaden
21
61
Jesper Ramsing
Syddanmark
21
62
User
Sjælland
20
63
User
Midtjylland
20
64
Achmel
Hovedstaden
20
65
User
Hovedstaden
20
66
User
Midtjylland
20
67
Iulia Prundurel
Hovedstaden
19
68
NRad
Midtjylland
19
69
Kasper
Syddanmark
19
70
Laura Turner
Hovedstaden
19
71
User
Hovedstaden
19
72
User
Hovedstaden
19
73
Alexander Mønster
Hovedstaden
19
74
User
Midtjylland
19
75
Foxylock
Nordjylland
19
76
User
Hovedstaden
19
77
marcus1x33@gmail.com
Hovedstaden
19
78
User
Sjælland
18
79
Christian Mejer
Sjælland
18
81
User
Midtjylland
18
82
User
Hovedstaden
18
83
User
Nordjylland
17
84
User
Hovedstaden
17
85
User
Midtjylland
17
86
User
Hovedstaden
17
87
User
Syddanmark
16
88
Asmus Rosendal
Midtjylland
16
89
User
Nordjylland
16
90
User
Hovedstaden
16
91
User
Syddanmark
16
92
User
Hovedstaden
16
93
User
Hovedstaden
16
94
User
Hovedstaden
16
95
User
Midtjylland
15
96
User
Hovedstaden
15
97
Ayushman Chaudhary - 8A
Nordjylland
15
98
User
Hovedstaden
15
99
User
Midtjylland
15
100
User
Hovedstaden
15
101
User
Hovedstaden
15
102
User
Hovedstaden
15
103
User
Nordjylland
14
104
User
Hovedstaden
14
105
User
Hovedstaden
14
106
User
Hovedstaden
14
107
User
Hovedstaden
14
108
Person
Nordjylland
14
109
User
Hovedstaden
14
110
User
Syddanmark
14
111
User
Midtjylland
14
112
User
Hovedstaden
14
113
User
Hovedstaden
14
114
Finn Nørgaard-Pedersen
Hovedstaden
14
115
User
Hovedstaden
13
116
User
Hovedstaden
13
117
User
Sjælland
13
118
User
Hovedstaden
13
119
Rebekka Lykke
Hovedstaden
13
120
User
Midtjylland
12
121
User
Nordjylland
12
122
User
Sjælland
12
123
User
Hovedstaden
12
124
User
Midtjylland
12
125
User
Hovedstaden
12
126
User
Syddanmark
12
127
User
Syddanmark
11
128
User
Hovedstaden
11
129
User
Hovedstaden
11
130
User
Hovedstaden
11
131
User
Hovedstaden
11
132
User
Hovedstaden
11
133
User
Hovedstaden
11
134
User
Hovedstaden
11
135
User
Syddanmark
11
136
User
Midtjylland
11
137
User
Hovedstaden
11
138
User
Hovedstaden
11
139
User
Hovedstaden
10
140
Rina
Hovedstaden
10
141
User
Midtjylland
10
142
User
Hovedstaden
10
143
User
Hovedstaden
10
144
User
Midtjylland
10
145
Kreas
Sjælland
10
146
User
Hovedstaden
10
147
kenneth.bruus@gmail.com
Midtjylland
10
148
User
Hovedstaden
10
149
User
Hovedstaden
10
150
Danny Nygård Hansen
Midtjylland
9
151
Mikkel Risgaard
Syddanmark
9
152
User
Hovedstaden
9
153
Ida Marie Soelje
Hovedstaden
9
154
User
Midtjylland
9
155
User
Midtjylland
9
156
User
Hovedstaden
9
157
User
Hovedstaden
9
158
User
Syddanmark
9
159
User
Midtjylland
9
160
Lucazzz
Hovedstaden
9
161
User
Midtjylland
9
162
User
Hovedstaden
9
163
User
Syddanmark
9
164
User
Hovedstaden
9
165
User
Hovedstaden
9
166
User
Syddanmark
9
167
User
Hovedstaden
9
168
User
Nordjylland
9
169
User
Hovedstaden
9
170
User
Hovedstaden
9
171
User
Hovedstaden
9
172
User
Hovedstaden
9
173
User
Hovedstaden
9
174
User
Midtjylland
9
175
Mona
Hovedstaden
9
176
User
Hovedstaden
9
177
Caroline CC
Hovedstaden
9
178
User
Nordjylland
9
179
User
Hovedstaden
8
180
User
Nordjylland
8
181
User
Hovedstaden
8
182
User
Hovedstaden
8
183
Ozzy
Midtjylland
8
184
User
Hovedstaden
8
185
User
Hovedstaden
8
186
User
Hovedstaden
8
187
Karl Levinsen
Hovedstaden
8
188
User
Hovedstaden
8
189
User
Midtjylland
8
190
User
Hovedstaden
8
191
User
Syddanmark
8
192
User
Hovedstaden
8
193
User
Syddanmark
8
194
User
Midtjylland
8
195
User
Hovedstaden
8
196
User
Hovedstaden
8
197
User
Hovedstaden
8
198
User
Syddanmark
7
199
User
Syddanmark
7
200
Gustav Haugan
Hovedstaden
7
201
User
Hovedstaden
7
202
User
Syddanmark
7
203
User
Midtjylland
7
204
User
Hovedstaden
7
205
User
Sjælland
7
206
User
Hovedstaden
7
207
User
Hovedstaden
7
208
August Soelje
Hovedstaden
7
209
User
Hovedstaden
7
210
User
Hovedstaden
7
211
User
Midtjylland
7
212
User
Midtjylland
7
213
User
Sjælland
7
214
User
Hovedstaden
7
215
User
Hovedstaden
7
216
User
Nordjylland
7
217
User
Hovedstaden
7
218
User
Midtjylland
7
219
User
Hovedstaden
7
220
Mergus
Syddanmark
7
221
User
Hovedstaden
7
222
User
Hovedstaden
6
223
User
Sjælland
6
224
User
Hovedstaden
6
225
User
Midtjylland
6
226
User
Sjælland
6
227
User
Hovedstaden
6
228
User
Hovedstaden
6
229
User
Hovedstaden
6
230
User
Nordjylland
6
231
User
Hovedstaden
6
232
User
Hovedstaden
6
233
User
Hovedstaden
6
234
User
Hovedstaden
6
235
User
Hovedstaden
6
236
User
Hovedstaden
6
237
Lasse Andersen
Syddanmark
6
238
User
Hovedstaden
6
239
User
Syddanmark
6
240
User
Syddanmark
6
241
User
Hovedstaden
6
242
User
Syddanmark
6
243
User
Hovedstaden
6
244
Johan Sørensen
Hovedstaden
6
245
User
Hovedstaden
6
246
User
Hovedstaden
6
247
Torben Musolf
Syddanmark
6
248
User
Hovedstaden
6
249
User
Hovedstaden
6
250
User
Syddanmark
6
251
User
Hovedstaden
6
252
fishcannon gaming
Hovedstaden
6
253
User
Midtjylland
6
254
User
Midtjylland
6
255
User
Hovedstaden
6
256
User
Hovedstaden
6
257
Max Hansen
Syddanmark
6
258
User
Midtjylland
6
259
User
Hovedstaden
5
260
User
Hovedstaden
5
261
User
Midtjylland
5
262
User
Hovedstaden
5
263
User
Hovedstaden
5
264
User
Midtjylland
5
265
User
Hovedstaden
5
266
User
Hovedstaden
5
267
User
Hovedstaden
5
268
User
Midtjylland
5
269
User
Hovedstaden
5
270
User
Midtjylland
5
271
User
Hovedstaden
5
272
User
Midtjylland
5
273
User
Hovedstaden
5
274
User
Hovedstaden
5
275
User
Sjælland
5
276
User
Hovedstaden
5
277
User
Midtjylland
5
278
User
Midtjylland
5
279
User
Hovedstaden
5
280
User
Midtjylland
5
281
User
Hovedstaden
5
282
User
Hovedstaden
5
283
User
Syddanmark
5
284
User
Syddanmark
5
285
User
Hovedstaden
5
286
User
Hovedstaden
5
287
User
Syddanmark
5
288
User
Syddanmark
5
289
User
Hovedstaden
5
290
User
Hovedstaden
5
291
User
Hovedstaden
5
292
User
Hovedstaden
5
293
User
Hovedstaden
5
294
User
Hovedstaden
5
295
User
Hovedstaden
5
296
Jacob Østergaard Holm
Midtjylland
5
297
User
Syddanmark
5
298
User
Hovedstaden
5
299
User
Hovedstaden
5
300
User
Syddanmark
5
301
User
Midtjylland
5
302
User
Nordjylland
5
303
User
Syddanmark
5
304
User
Hovedstaden
5
305
User
Hovedstaden
5
306
User
Hovedstaden
5
307
User
Hovedstaden
5
308
User
Hovedstaden
4
309
User
Hovedstaden
4
310
User
Midtjylland
4
311
User
Hovedstaden
4
312
User
Hovedstaden
4
313
User
Hovedstaden
4
314
User
Hovedstaden
4
315
User
Midtjylland
4
316
User
Hovedstaden
4
317
User
Nordjylland
4
318
User
Hovedstaden
4
319
User
Hovedstaden
4
320
User
Hovedstaden
4
321
User
Sjælland
4
322
User
Hovedstaden
4
323
User
Hovedstaden
4
324
User
Hovedstaden
4
325
User
Midtjylland
4
326
User
Syddanmark
4
327
User
Hovedstaden
4
328
User
Syddanmark
4
329
mikkel.vigel@hotmail.com
Hovedstaden
4
330
User
Nordjylland
4
331
User
Syddanmark
4
332
User
Hovedstaden
4
333
User
Hovedstaden
4
334
User
Hovedstaden
4
335
User
Hovedstaden
4
336
User
Midtjylland
4
337
User
Hovedstaden
4
338
Ricky
Hovedstaden
4
339
User
Hovedstaden
4
340
User
Region
4
341
User
Hovedstaden
4
342
Torben Musolf
Syddanmark
4
343
User
Hovedstaden
4
344
User
Hovedstaden
4
345
User
Hovedstaden
4
346
User
Sjælland
4
347
User
Hovedstaden
4
348
User
Hovedstaden
4
349
User
Midtjylland
4
350
User
Midtjylland
4
351
User
Hovedstaden
4
352
User
Hovedstaden
4
353
User
Nordjylland
4
354
User
Hovedstaden
3
355
User
Hovedstaden
3
356
User
Midtjylland
3
357
User
Midtjylland
3
358
User
Hovedstaden
3
359
User
Midtjylland
3
360
User
Hovedstaden
3
361
User
Midtjylland
3
362
User
Hovedstaden
3
363
User
Hovedstaden
3
364
User
Hovedstaden
3
365
User
Syddanmark
3
366
User
Hovedstaden
3
367
User
Hovedstaden
3
368
User
Sjælland
3
369
Philip Olsson
Hovedstaden
3
370
User
Syddanmark
3
371
User
Hovedstaden
3
372
User
Midtjylland
3
373
User
Hovedstaden
3
374
User
Hovedstaden
3
375
User
Nordjylland
3
376
User
Midtjylland
3
377
User
Sjælland
3
378
User
Hovedstaden
3
379
User
Hovedstaden
3
380
User
Midtjylland
3
381
User
Hovedstaden
3
382
User
Midtjylland
3
383
User
Syddanmark
3
384
User
Syddanmark
3
385
User
Hovedstaden
3
386
User
Midtjylland
3
387
User
Hovedstaden
3
388
User
Midtjylland
3
389
User
Hovedstaden
3
390
User
Syddanmark
3
391
User
Hovedstaden
3
392
User
Hovedstaden
3
393
User
Hovedstaden
3
394
User
Syddanmark
3
395
User
Midtjylland
3
396
User
Syddanmark
3
397
User
Syddanmark
3
398
User
Hovedstaden
3
399
User
Nordjylland
3
400
User
Hovedstaden
3
401
User
Hovedstaden
3
402
User
Hovedstaden
3
403
User
Hovedstaden
2
404
User
Nordjylland
2
405
User
Hovedstaden
2
406
User
Midtjylland
2
407
User
Hovedstaden
2
408
User
Syddanmark
2
409
User
Hovedstaden
2
410
User
Midtjylland
2
411
User
Hovedstaden
2
412
User
Hovedstaden
2
413
User
Sjælland
2
414
User
Hovedstaden
2
415
User
Midtjylland
2
416
User
Hovedstaden
2
417
User
Hovedstaden
2
418
jrk
Syddanmark
2
419
User
Hovedstaden
2
420
User
Midtjylland
2
421
User
Hovedstaden
2
422
User
Syddanmark
2
423
User
Hovedstaden
2
424
User
Hovedstaden
2
425
User
Hovedstaden
2
426
User
Hovedstaden
2
427
User
Sjælland
2
428
User
Hovedstaden
2
429
User
Midtjylland
2
430
User
Syddanmark
2
431
User
Midtjylland
2
432
Marco Simmini
Hovedstaden
2
433
User
Hovedstaden
2
434
User
Region
2
435
User
Hovedstaden
2
436
User
Hovedstaden
2
437
User
Hovedstaden
2
438
bakugou
Syddanmark
2
439
Lars Glaesel
Hovedstaden
2
440
User
Midtjylland
2
441
User
Hovedstaden
2
442
User
Hovedstaden
2
443
User
Hovedstaden
2
444
User
Hovedstaden
2
445
User
Midtjylland
2
446
User
Hovedstaden
2
447
User
Hovedstaden
2
448
User
Hovedstaden
2
449
User
Syddanmark
2
450
User
Hovedstaden
2
451
Emil Brandt
Syddanmark
2
452
User
Hovedstaden
2
453
User
Midtjylland
2
454
User
Nordjylland
2
455
User
Midtjylland
2
456
User
Hovedstaden
2
457
User
Nordjylland
2
458
User
Syddanmark
2
459
User
Midtjylland
2
460
User
Hovedstaden
2
461
User
Hovedstaden
1
462
User
Hovedstaden
1
463
User
Nordjylland
1
464
User
Hovedstaden
1
465
User
Syddanmark
1
466
User
Hovedstaden
1
467
User
Syddanmark
1
468
User
Hovedstaden
1
469
User
Nordjylland
1
470
User
Midtjylland
1
471
User
Hovedstaden
1
472
maryama hashi
Sjælland
1
473
User
Hovedstaden
1
474
User
Midtjylland
1
475
User
Hovedstaden
1
476
User
Hovedstaden
1
477
User
Midtjylland
1
478
User
Hovedstaden
1
479
User
Midtjylland
1
480
User
Hovedstaden
1
481
User
Hovedstaden
1
482
User
Hovedstaden
1
483
User
Hovedstaden
1
484
User
Nordjylland
1
485
User
Hovedstaden
1
486
User
Hovedstaden
1
487
User
Nordjylland
1
488
User
Syddanmark
1
489
User
Hovedstaden
1
490
User
Hovedstaden
1
491
User
Hovedstaden
1
492
User
Syddanmark
1
493
User
Hovedstaden
1
494
User
Hovedstaden
1
495
User
Hovedstaden
1
496
User
Sjælland
1
497
User
Hovedstaden
1
499
User
Midtjylland
1
500
User
Sjælland
1
501
User
Hovedstaden
1
502
User
Hovedstaden
1
503
User
Hovedstaden
1
504
User
Hovedstaden
1
505
User
Hovedstaden
1
506
User
Hovedstaden
1
507
User
Hovedstaden
1
508
User
Hovedstaden
1
509
Hannesbo
Hovedstaden
1
510
User
Hovedstaden
1
511
User
Hovedstaden
1
512
User
Hovedstaden
1
513
User
Hovedstaden
1
514
User
Hovedstaden
1
515
User
Sjælland
1
516
User
Midtjylland
1
517
User
Hovedstaden
1
518
User
Syddanmark
1
519
User
Syddanmark
1
520
User
Hovedstaden
1
521
User
Midtjylland
1
522
User
Syddanmark
1
523
User
Region
1
524
User
Midtjylland
1
525
User
Hovedstaden
1
526
User
Midtjylland
1
527
User
Hovedstaden
1
528
User
Hovedstaden
1
529
User
Hovedstaden
1
530
User
Hovedstaden
1
531
User
Syddanmark
1
532
User
Hovedstaden
1
533
User
Syddanmark
1
534
User
Midtjylland
1
535
User
Hovedstaden
1
536
User
Sjælland
1
537
User
Syddanmark
1
538
User
Midtjylland
1
539
User
Hovedstaden
1
540
User
Syddanmark
1
541
Na el
Midtjylland
1
542
User
Hovedstaden
1
543
User
Midtjylland
1
544
User
Hovedstaden
1
545
User
Hovedstaden
1
546
User
Hovedstaden
1
547
User
Hovedstaden
1
548
User
Hovedstaden
1
549
User
Midtjylland
1
550
User
Syddanmark
1
551
User
Midtjylland
1
552
User
Hovedstaden
1
553
User
Midtjylland
1
554
User
Sjælland
1
555
User
Hovedstaden
1
556
User
Midtjylland
1
557
User
Nordjylland
1
558
User
Midtjylland
1
559
User
Syddanmark
1
560
User
Hovedstaden
1
561
User
Syddanmark
1
562
User
Hovedstaden
1
563
Emil Lykke Grann
Midtjylland
1
564
User
Region
1

Free

Free basic access

Login Optional

Progress saved automatically

Students

Works great in classrooms

25,000,000
Questions answered
100,000
Users
100
Difficulty levels

Cross-Platform

Use any operating system with Chrome, Safari, or Firefox

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.


Privacy Policy