Leaderboard
Germany
Germany
Worldwide
Rank
Level
1
Jan K
Hessen
95
2
User
Hessen
95
4
Daniel
Bayern
86
6
Dekaz Turuz
Hessen
84
7
Sebastian_B
Bayern
84
8
User
Hamburg
84
9
moormaster
Baden-Württemberg
83
10
User
Berlin
83
11
Seth Robles
Bayern
82
12
User
Bayern
82
13
ooer
Bremen
82
15
Timmy
Niedersachsen
80
16
User
Bayern
80
17
User
Bayern
75
18
User
Bayern
73
19
Elfy
Hamburg
69
20
User
Bayern
69
21
User
Bremen
69
22
User
Sachsen
68
23
Leifur Eriksson
Berlin
63
24
User
Bayern
63
25
User
Bremen
63
26
User
Hessen
62
27
User
Hessen
62
28
Jakob Bohnacker
Hessen
62
29
WhatWillHappen?
Bayern
62
30
User
Bremen
61
31
User
Bremen
61
32
User
Bremen
60
33
User
Hessen
59
34
User
Bayern
58
35
I am going to be top
Rheinland-Pfalz
58
36
User
Bremen
58
37
User
Bremen
58
38
User
Bremen
57
39
phipsae
Bayern
57
40
User
Bremen
57
41
User
Bremen
57
43
User
Bremen
56
44
User
Bremen
56
45
James Krano
Bayern
55
47
User
Bayern
55
48
User
Bayern
55
49
User
Bremen
55
50
Warm Tea
Berlin
54
51
User
Bremen
54
52
User
Bremen
54
53
User
Bremen
54
54
User
Hessen
53
56
User
Bayern
52
57
User
Bremen
52
58
pete pete
Niedersachsen
52
59
User
Bremen
52
60
Serhat
Hessen
52
61
User
Bayern
51
62
User
Berlin
51
67
User
Hessen
50
68
User
Hessen
50
69
dgdgdffrg
Sachsen
50
70
User
Bremen
50
72
Eric
Hessen
49
74
User
Bremen
49
75
Abhilasha Singh
Hamburg
49
76
User
Bremen
49
77
User
Hessen
49
78
Ostroco maths team two
Baden-Württemberg
48
79
User
Bremen
48
80
Joschka Ziesche
Rheinland-Pfalz
48
84
User
Bremen
47
85
Maik Schneider
Berlin
46
86
Mátyás Mérőfi
Baden-Württemberg
46
88
User
Hessen
46
89
User
Bayern
46
90
User
Bayern
46
91
User
Hessen
45
92
Jan
Bayern
45
93
User
Bayern
45
94
User
Bremen
45
95
User
Bremen
45
96
Helios Chromatic
Baden-Württemberg
44
97
Login Online
Nordrhein-Westfalen
44
100
User
Hessen
44
103
User
Bremen
44
104
Michael Pencil
Schleswig-Holstein
44
105
User
Bayern
43
107
NoC4
Sachsen
43
109
User
Hamburg
43
110
User
Hessen
43
111
Miguel Sanz
Baden-Württemberg
43
112
User
Bayern
43
113
Chris
Bayern
43
114
User
Bremen
43
115
Sören Krüger
Niedersachsen
43
116
User
Bremen
43
118
User
Bremen
42
120
User
Bayern
41
122
Felix Frenzel
Nordrhein-Westfalen
41
123
User
Hessen
41
124
User
Hessen
41
125
User
Hessen
41
126
User
Berlin
41
127
User
Niedersachsen
41
128
User
Sachsen
41
130
User
Bayern
41
132
User
Bayern
40
133
User
Hamburg
40
134
User
Bremen
40
135
User
Hessen
39
137
User
Niedersachsen
38
138
Chris
Hessen
38
139
User
Bayern
38
140
User
Bayern
38
141
User
Bremen
38
143
Platen
Niedersachsen
37
144
Victor ARMINJON
Bayern
37
146
User
Bayern
37
149
User
Niedersachsen
37
150
User
Bayern
37
151
User
Berlin
36
152
Wolle
Niedersachsen
36
154
Max
Bayern
36
155
zormit
Hessen
36
157
User
Bremen
36
158
Harsha Kumar
Nordrhein-Westfalen
36
159
undefined
Hamburg
36
160
User
Berlin
36
161
Peter Hans
Baden-Württemberg
35
162
User
Hamburg
35
163
User
Niedersachsen
35
164
CP
Hessen
35
165
User
Hessen
35
167
User
Bayern
35
168
User
Niedersachsen
35
170
User
Bayern
35
171
User
Hamburg
35
173
User
Hessen
35
174
Ghostdog
Schleswig-Holstein
35
175
User
Hessen
34
176
User
Bremen
34
177
User
Hessen
34
178
User
Hessen
34
179
User
Niedersachsen
34
180
User
Saarland
34
181
User
Berlin
34
182
User
Niedersachsen
34
183
f mustermann
Nordrhein-Westfalen
33
184
User
Hessen
33
186
User
Hessen
33
187
User
Hessen
33
190
User
Bremen
32
191
User
Hessen
32
192
User
Hessen
32
193
User
Bayern
32
195
User
Hamburg
32
196
User
Bremen
32
197
User
Berlin
32
198
User
Hamburg
31
200
Lidan nadiL
Nordrhein-Westfalen
31
201
User
Niedersachsen
31
204
User
Berlin
31
205
User
Bayern
31
206
User
Berlin
31
207
User
Hamburg
31
209
User
Bayern
31
210
User
Hessen
31
212
User
Hessen
31
214
User
Hessen
30
216
User
Sachsen
30
219
User
Bremen
30
220
User
Bayern
30
221
User
Berlin
30
222
User
Bayern
30
223
User
Berlin
30
224
User
Berlin
30
226
Gergely Horváth
Baden-Württemberg
29
227
User
Hamburg
29
228
User
Bayern
29
229
User
Hessen
29
231
Andreas W.
Hessen
29
232
Karim
Rheinland-Pfalz
29
235
User
Hessen
29
238
User
Hessen
28
239
Ildiko Benke
Bayern
28
240
Hi
Berlin
28
242
mathdumb
Berlin
28
244
User
Berlin
28
245
Marcus Bannenberg
Nordrhein-Westfalen
28
246
User
Hessen
28
248
User
Hessen
28
250
User
Hessen
28
251
User
Hamburg
28
252
User
Bayern
28
253
User
Berlin
28
255
User
Berlin
27
256
User
Bayern
27
259
User
Hamburg
27
260
User
Berlin
26
262
User
Hessen
26
263
User
Berlin
26
264
User
Bayern
26
266
User
Berlin
26
267
User
Hessen
26
268
User
Hessen
26
269
User
Berlin
25
271
User
Berlin
25
274
User
Berlin
25
275
moormaster
Baden-Württemberg
25
276
User
Bayern
25
278
User
Berlin
25
279
User
Berlin
25
280
User
Berlin
25
281
User
Bayern
25
282
Plus Plus
Nordrhein-Westfalen
25
283
Kubko
Hamburg
25
285
User
Hessen
24
287
James Whynot
Nordrhein-Westfalen
24
288
User
Sachsen
24
289
User
Hessen
24
290
User
Hamburg
24
291
Grahmbler
Baden-Württemberg
24
292
User
Niedersachsen
24
293
Irina Lebedeva
Hessen
24
295
User
Niedersachsen
24
296
Samuel Scandola
Bayern
23
297
User
Bremen
23
298
User
Bayern
23
300
User
Hamburg
23
301
User
Bayern
23
302
User
Hessen
23
303
User
Thüringen
23
304
User
Berlin
23
305
User
Berlin
23
306
User
Berlin
23
307
User
Hessen
23
308
User
Bayern
23
309
User
Berlin
22
312
User
Bayern
22
314
User
Berlin
22
316
User
Berlin
22
317
User
Bayern
22
318
User
Berlin
22
321
User
Niedersachsen
22
323
User
Berlin
22
324
Anne-Sophie Korotov
Berlin
21
327
User
Brandenburg
21
330
User
Hessen
21
331
User
Thüringen
21
333
User
Berlin
21
335
User
Berlin
21
336
User
Berlin
21
337
User
Bremen
21
339
overwatchKatze
Hessen
21
341
User
Berlin
20
344
User
Hamburg
20
345
János Géczi
Baden-Württemberg
20
346
User
Hamburg
20
350
User
Berlin
20
351
User
Bremen
20
352
User
Bayern
20
354
road text
Rheinland-Pfalz
20
355
Justyna Kurzela
Hessen
20
356
User
Niedersachsen
20
357
Jihan Gervasoni
Hamburg
20
358
User
Bayern
20
359
User
Bremen
20
360
User
Berlin
20
361
User
Hamburg
20
364
User
Bayern
19
365
User
Hessen
19
366
User
Hamburg
19
367
User
Berlin
19
368
User
Bayern
19
371
User
Berlin
19
373
User
Niedersachsen
19
374
User
Bayern
19
376
User
Berlin
19
379
User
Bayern
18
381
User
Hamburg
18
382
User
Bayern
18
383
User
Hessen
18
386
User
Berlin
18
388
Stella Matthiessen Krausenecker
Bayern
18
390
User
Berlin
18
391
User
Bayern
18
392
User
Sachsen
18
393
NJOI
Hamburg
17
395
User
Bayern
17
396
User
Bayern
17
399
User
Bayern
17
400
User
Niedersachsen
17
404
User
Hessen
17
407
User
Hamburg
17
408
User
Bayern
17
409
User
Sachsen
17
410
User
Bayern
16
411
User
Bayern
16
413
User
Berlin
16
414
cherrycoke
Berlin
16
415
Caio Martins
Baden-Württemberg
16
416
User
Bayern
16
417
User
Berlin
16
418
User
Bayern
16
419
Mic
Region
16
420
User
Bayern
16
425
User
Hessen
16
426
User
Bayern
16
427
FARES
Hessen
16
428
User
Hessen
16
429
User
Bremen
16
432
User
Hessen
15
433
Im Doing This For Yuna
Hessen
15
434
User
Region
15
435
User
Bayern
15
436
User
Berlin
15
437
User
Niedersachsen
15
439
User
Bayern
15
444
User
Hessen
15
447
User
Sachsen
15
448
User
Saarland
14
449
User
Region
14
453
User
Hessen
14
454
User
Bayern
14
455
User
Bayern
14
456
Nigel Armino
Nordrhein-Westfalen
14
457
User
Berlin
14
458
User
Region
14
459
User
Bayern
14
463
User
Berlin
14
464
User
Berlin
14
465
User
Bayern
14
466
User
Hessen
14
467
User
Bayern
13
468
User
Hamburg
13
469
Noah Afkir
Nordrhein-Westfalen
13
470
User
Niedersachsen
13
471
User
Sachsen
13
475
User
Berlin
13
476
User
Hessen
13
477
User
Berlin
13
479
User
Niedersachsen
13
480
User
Hessen
13
481
User
Hessen
13
483
User
Hessen
13
484
User
Bremen
13
485
User
Bayern
13
487
User
Berlin
13
488
User
Hamburg
13
489
User
Hessen
13
490
User
Niedersachsen
13
493
User
Bayern
13
494
User
Berlin
12
497
User
Bayern
12
498
User
Bayern
12
499
User
Bayern
12
500
User
Bayern
12
503
User
Bayern
12
504
User
Berlin
12
506
User
Berlin
12
507
User
Bayern
12
508
User
Bayern
12
509
User
Hessen
12
511
User
Bayern
12
512
User
Berlin
12
516
User
Berlin
12
520
Eugen Klein
Baden-Württemberg
11
521
User
Bayern
11
525
Felix Urbasik
Nordrhein-Westfalen
11
526
User
Bayern
11
527
User
Berlin
11
529
User
Bayern
11
532
User
Bayern
11
533
User
Berlin
11
535
Tim Wind
Bayern
11
538
User
Hessen
11
540
User
Berlin
11
541
User
Bayern
11
543
User
Berlin
11
545
User
Berlin
11
547
Flemca
Hamburg
10
549
User
Hessen
10
553
User
Bayern
10
557
User
Bayern
10
559
User
Hessen
10
560
User
Hessen
10
562
User
Hessen
10
563
User
Niedersachsen
10
564
User
Berlin
10
566
User
Hamburg
10
568
User
Berlin
10
569
Hallucigenia
Hessen
10
571
User
Niedersachsen
10
573
User
Hamburg
10
575
User
Berlin
10
576
User
Bayern
10
579
Gnedla Abozriba
Berlin
9
580
Omar Afkir
Nordrhein-Westfalen
9
582
User
Berlin
9
585
User
Hessen
9
589
User
Bayern
9
591
User
Hessen
9
593
User
Bayern
9
594
User
Bayern
9
595
whyme
Hessen
9
596
User
Bayern
9
598
User
Berlin
9
599
User
Berlin
9
603
User
Thüringen
9
605
User
Bayern
9
607
User
Hessen
8
610
User
Hessen
8
611
User
Berlin
8
614
User
Hessen
8
615
User
Hessen
8
617
User
Berlin
8
619
User
Bayern
8
622
User
Hessen
8
626
Jamal
Berlin
7
628
User
Hessen
7
629
User
Sachsen
7
630
User
Hessen
7
632
User
Hamburg
7
633
chris-stud@gmx.de
Schleswig-Holstein
7
634
User
Berlin
7
639
User
Sachsen
7
641
User
Hessen
7
642
User
Berlin
7
645
SaraS
Berlin
7
646
User
Hessen
7
647
User
Berlin
7
649
User
Berlin
7
650
User
Bayern
7
651
User
Bremen
7
652
User
Hessen
7
653
User
Bremen
7
655
User
Hessen
7
658
User
Hessen
6
659
User
Bayern
6
660
User
Berlin
6
661
User
Bayern
6
663
User
Bayern
6
664
User
Sachsen
6
666
User
Bayern
6
667
User
Hamburg
6
669
User
Bayern
6
670
User
Bayern
6
671
User
Bayern
6
672
User
Bayern
6
674
User
Hamburg
6
676
User
Hessen
6
677
User
Bayern
6
678
User
Saarland
6
679
Solange
Bayern
6
680
User
Bayern
6
683
User
Bayern
6
684
User
Bayern
6
685
User
Bayern
6
686
User
Bayern
6
687
User
Hessen
6
690
User
Hessen
6
691
User
Bremen
6
692
User
Bayern
6
693
User
Hessen
6
695
User
Bremen
6
703
User
Bayern
5
704
User
Berlin
5
705
User
Hamburg
5
706
User
Hessen
5
707
Larrylaffer
Berlin
5
709
User
Bayern
5
710
User
Bayern
5
712
User
Sachsen
5
714
User
Hamburg
5
715
User
Hessen
5
716
User
Hessen
5
718
User
Hessen
5
719
User
Hessen
5
720
User
Berlin
5
721
User
Berlin
5
722
User
Bayern
5
725
User
Hessen
5
726
User
Berlin
5
727
Pooyan Salehi
Region
5
728
User
Hamburg
5
730
User
Berlin
5
732
User
Berlin
5
734
User
Hessen
5
735
User
Bayern
5
736
User
Hamburg
5
738
User
Bayern
5
739
User
Bremen
5
741
User
Berlin
5
743
User
Sachsen
5
745
User
Brandenburg
4
746
User
Bayern
4
747
User
Hessen
4
748
User
Berlin
4
749
User
Bayern
4
750
User
Berlin
4
751
User
Bayern
4
752
User
Bayern
4
754
User
Sachsen
4
755
User
Bayern
4
756
User
Berlin
4
757
User
Hessen
4
759
User
Brandenburg
4
760
User
Berlin
4
762
User
Bayern
4
763
User
Hamburg
4
765
User
Bayern
4
766
User
Berlin
4
768
User
Bayern
4
771
User
Hessen
4
772
User
Bayern
4
775
User
Bayern
4
776
User
Berlin
4
777
User
Berlin
4
782
User
Bremen
4
783
User
Berlin
4
784
User
Berlin
4
785
User
Hessen
4
786
User
Saarland
3
787
User
Bayern
3
788
User
Berlin
3
789
User
Hessen
3
793
User
Hessen
3
794
Bernat
Hessen
3
795
Alexander Boyd
Bayern
3
799
User
Hessen
3
800
User
Brandenburg
3
801
User
Hamburg
3
803
User
Hessen
3
804
User
Berlin
3
807
User
Hessen
3
808
User
Bayern
3
810
User
Hessen
3
812
User
Berlin
3
813
User
Brandenburg
3
814
Umair Janjua
Hamburg
3
815
User
Bayern
3
816
User
Hamburg
3
817
User
Bayern
3
818
User
Hessen
3
819
User
Brandenburg
3
820
User
Bayern
3
824
User
Berlin
3
825
User
Berlin
3
827
User
Berlin
3
828
User
Sachsen
3
830
User
Bremen
3
831
User
Hamburg
3
832
User
Bremen
3
833
User
Bayern
3
835
User
Berlin
3
836
User
Hessen
3
837
User
Berlin
3
839
User
Hamburg
3
842
User
Hessen
3
843
karim ibrahim
Hamburg
3
844
User
Bayern
3
845
User
Hessen
3
846
User
Bremen
3
851
User
Berlin
2
852
User
Berlin
2
853
User
Bayern
2
854
User
Bayern
2
855
User
Berlin
2
857
User
Berlin
2
858
User
Hessen
2
860
User
Hessen
2
862
User
Brandenburg
2
863
User
Hessen
2
864
User
Bayern
2
865
User
Bayern
2
868
User
Hamburg
2
870
User
Berlin
2
871
User
Hessen
2
873
User
Hessen
2
877
User
Bayern
2
878
User
Thüringen
2
879
User
Bayern
2
880
User
Berlin
2
884
User
Berlin
2
885
User
Hessen
2
888
User
Bayern
2
889
User
Berlin
2
892
User
Berlin
2
893
User
Bayern
2
894
User
Hamburg
2
896
User
Brandenburg
2
902
User
Hessen
2
903
User
Hamburg
2
904
User
Hamburg
2
905
User
Hessen
2
906
User
Bayern
2
908
User
Bayern
2
909
User
Bayern
2
911
User
Berlin
2
913
User
Hessen
2
914
User
Hessen
2
916
User
Berlin
2
917
User
Berlin
2
930
User
Bayern
2
931
User
Berlin
2
932
User
Hamburg
2
933
User
Bayern
2
934
User
Berlin
2
936
User
Saarland
2
938
User
Hamburg
2
940
User
Bremen
2
942
User
Hamburg
2
943
User
Bremen
2
944
User
Bayern
2
945
User
Berlin
2
949
User
Hessen
2
950
User
Bayern
2
952
User
Thüringen
2
954
User
Bayern
2
956
User
Bayern
2
957
User
Bayern
2
960
User
Berlin
2
961
User
Hessen
1
962
User
Bayern
1
964
User
Berlin
1
966
Nicole Nagdyman
Bayern
1
967
User
Berlin
1
968
User
Hessen
1
970
User
Hessen
1
971
User
Berlin
1
972
User
Bayern
1
973
User
Berlin
1
974
User
Bayern
1
975
User
Bayern
1
976
User
Berlin
1
977
User
Bayern
1
979
User
Hamburg
1
980
User
Sachsen
1
981
User
Bayern
1
984
User
Bayern
1
985
User
Hessen
1
987
User
Hessen
1
989
Johannes Fesenmeyer
Baden-Württemberg
1
990
User
Bayern
1
992
Bruno Jahn
Berlin
1
993
GRIND GANG
Baden-Württemberg
1
994
Ashwin Kor
Berlin
1
995
Alexej Stolipin
Baden-Württemberg
1
996
B AB
Bayern
1
998
43trh
Berlin
1

Free

Free basic access

Login Optional

Progress saved automatically

Students

Works great in classrooms

20,000,000
Response times
650,000
Unique questions
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 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.


Privacy Policy