Leaderboard
Portugal
Portugal
Worldwide
Rank
Level
1
MartimG
Lisboa
100
2
Miguel Juncais
Lisboa
91
3
Dinis Santos
Leiria
90
4
User
Aveiro
86
5
User
Aveiro
80
6
BichodoMato
Porto
75
7
João Alves
Lisboa
74
8
User
Lisboa
71
9
Miguel Vasconcelos
Lisboa
67
10
User
Lisboa
66
11
Fernando Bacelar de Sousa
Lisboa
65
12
User
Lisboa
62
13
User
Lisboa
62
14
User
Santarém
60
15
User
Faro
60
16
João Tiago Bairras Gorgulho
Porto
54
17
User
Porto
53
18
User
Faro
51
19
Pedro Caeiro
Lisboa
51
20
User
Santarém
50
21
User
Lisboa
49
22
Vitor Hugo
Porto
48
23
User
Viseu
48
24
User
Faro
47
25
Tiago
Setúbal
47
26
Tiago Morais
Porto
47
27
User
Aveiro
46
28
Afonso CV
Castelo Branco
44
29
User
Porto
43
30
User
Lisboa
42
31
User
Lisboa
42
32
User
Lisboa
42
33
User
Lisboa
42
34
User
Lisboa
41
35
Vilgeforte Águas-Santas
Lisboa
41
36
Solange Silva
Lisboa
41
37
Joaquim Silva
Lisboa
40
38
Inês Martins
Lisboa
39
39
Carolina Paes de Faria
Lisboa
39
40
User
Porto
39
41
User
Lisboa
39
42
User
Porto
39
43
joao.miguel
Setúbal
39
44
User
Porto
38
45
User
Lisboa
38
46
User
Porto
37
47
User
Lisboa
37
48
User
Porto
36
49
Shahrukh khan
Lisboa
36
50
User
Lisboa
35
51
User
Porto
35
52
User
Porto
35
53
User
Coimbra
35
54
User
Lisboa
35
55
User
Lisboa
35
56
User
Lisboa
34
57
User
Lisboa
34
58
User
Porto
34
59
Paulo Pesqueira
Lisboa
34
60
User
Porto
34
61
User
Porto
33
62
User
Porto
33
63
User
Porto
33
64
User
Porto
32
65
User
Porto
32
66
User
Porto
32
67
User
Porto
32
68
User
Vila Real
32
69
User
Santarém
32
70
User
Lisboa
32
71
User
Lisboa
32
72
User
Porto
31
73
User
Porto
31
74
User
Porto
31
75
User
Porto
31
76
User
Porto
31
77
User
Porto
31
78
User
Lisboa
31
79
User
Santarém
31
80
Tiago Cabo
Porto
31
81
User
Lisboa
31
82
Andreia Nogueira
Porto
31
83
User
Porto
30
84
User
Lisboa
30
85
User
Porto
30
86
User
Aveiro
30
87
User
Lisboa
30
88
User
Lisboa
30
89
edmond turquieh
Santarém
30
90
User
Coimbra
30
91
User
Porto
30
92
User
Lisboa
30
93
User
Porto
29
94
User
Porto
29
95
User
Porto
29
96
User
Lisboa
29
97
User
Porto
29
98
User
Faro
29
99
User
Porto
29
100
User
Porto
29
101
User
Porto
28
102
User
Lisboa
28
103
User
Setúbal
28
104
User
Porto
28
105
User
Porto
28
106
User
Porto
28
107
User
Lisboa
28
108
User
Lisboa
28
109
User
Faro
28
110
User
Lisboa
28
111
User
Porto
28
112
User
Porto
27
113
User
Porto
27
114
User
Lisboa
27
115
User
Coimbra
27
116
User
Porto
27
117
User
Porto
27
118
User
Lisboa
27
119
User
Porto
27
120
User
Porto
27
121
User
Porto
27
122
Iara Serbaneci
Faro
27
123
User
Porto
27
124
User
Viseu
27
125
User
Porto
26
126
User
Lisboa
26
127
User
Porto
26
128
User
Porto
26
129
User
Porto
26
130
User
Porto
26
131
User
Porto
26
132
User
Lisboa
26
133
User
Porto
26
134
User
Porto
26
135
User
Lisboa
26
136
User
Porto
25
137
User
Lisboa
25
138
User
Bragança
25
139
User
Lisboa
25
140
User
Porto
25
141
User
Porto
25
142
User
Porto
25
143
User
Porto
25
144
User
Lisboa
25
145
User
Porto
25
146
User
Porto
25
147
User
Porto
25
148
User
Porto
25
149
User
Porto
25
150
User
Porto
25
151
User
Porto
25
152
Matthew Brews
Faro
25
153
User
Lisboa
25
154
User
Porto
25
155
User
Porto
25
156
User
Santarém
25
157
User
Porto
25
158
User
Porto
25
159
User
Lisboa
25
160
User
Porto
25
161
User
Porto
24
162
User
Porto
24
163
User
Porto
24
164
User
Porto
24
165
User
Porto
24
166
User
Porto
24
167
User
Porto
24
168
User
Faro
24
169
User
Lisboa
24
170
User
Porto
24
171
User
Porto
24
172
User
Porto
24
173
User
Porto
24
174
User
Porto
24
175
User
Porto
24
176
User
Porto
24
177
User
Porto
24
178
User
Porto
24
179
User
Porto
24
180
User
Porto
24
181
User
Porto
24
182
User
Porto
24
183
User
Porto
24
184
User
Lisboa
23
185
User
Porto
23
186
User
Porto
23
187
User
Porto
23
188
User
Porto
23
189
adr
Lisboa
23
190
User
Porto
23
191
User
Porto
23
192
User
Porto
23
193
Teresa Sa
Lisboa
23
194
User
Lisboa
23
195
User
Porto
23
196
User
Porto
22
197
User
Porto
22
198
User
Porto
22
199
User
Setúbal
22
200
NM
Lisboa
22
201
User
Lisboa
22
202
User
Porto
22
203
User
Porto
22
204
User
Lisboa
22
205
User
Porto
22
206
User
Porto
22
207
User
Porto
22
208
User
Coimbra
22
209
User
Porto
22
210
User
Porto
22
211
User
Porto
22
212
User
Porto
22
213
User
Porto
22
214
User
Porto
22
215
User
Porto
22
216
User
Porto
22
217
User
Porto
22
218
User
Porto
22
219
User
Porto
22
220
User
Porto
22
221
User
Porto
22
222
User
Porto
22
223
User
Porto
21
224
User
Porto
21
225
Hes pt
Braga
21
226
User
Porto
21
227
User
Porto
21
228
User
Porto
21
229
User
Lisboa
21
230
User
Porto
21
231
User
Porto
21
232
User
Porto
21
233
User
Porto
21
234
User
Porto
21
235
User
Porto
21
236
User
Lisboa
21
237
User
Lisboa
21
238
User
Porto
21
239
null
Lisboa
21
240
User
Lisboa
20
241
User
Porto
20
242
User
Porto
20
243
User
Porto
20
244
User
Braga
20
245
User
Porto
20
246
User
Lisboa
20
247
User
Porto
20
248
User
Porto
20
249
User
Lisboa
20
250
User
Porto
20
251
User
Porto
20
252
User
Lisboa
20
253
User
Porto
20
254
User
Lisboa
20
255
User
Porto
20
256
User
Porto
20
257
User
Porto
20
258
User
Porto
20
259
User
Porto
20
260
User
Porto
20
261
User
Lisboa
20
262
User
Lisboa
20
263
User
Setúbal
20
264
User
Porto
20
265
User
Lisboa
20
266
User
Porto
20
267
User
Porto
20
268
User
Lisboa
20
269
User
Lisboa
20
270
User
Porto
20
271
Ricardo Correia
Aveiro
19
272
User
Porto
19
273
User
Lisboa
19
274
User
Lisboa
19
275
User
Porto
19
276
User
Porto
19
277
User
Porto
19
278
User
Porto
19
279
User
Porto
19
280
User
Porto
19
281
User
Porto
19
282
User
Porto
19
283
User
Porto
19
284
User
Porto
19
285
User
Porto
19
286
Vanessa Muteia
Lisboa
19
287
User
Porto
19
288
User
Porto
19
289
User
Porto
19
290
User
Porto
19
291
User
Porto
19
292
User
Porto
19
293
User
Porto
19
294
User
Porto
19
295
User
Porto
19
296
User
Porto
19
297
User
Vila Real
18
298
User
Porto
18
299
User
Lisboa
18
300
User
Porto
18
301
Afonso Miranda
Porto
18
302
User
Lisboa
18
303
User
Porto
18
304
User
Lisboa
18
305
User
Porto
18
306
User
Porto
18
307
User
Porto
18
308
User
Porto
18
309
Afonso Elvas
Porto
18
310
User
Porto
18
311
User
Porto
18
312
User
Porto
18
313
User
Porto
18
314
User
Porto
18
315
User
Porto
18
316
User
Porto
18
317
User
Porto
17
318
User
Porto
17
319
User
Porto
17
320
User
Porto
17
321
User
Porto
17
322
User
Lisboa
17
323
User
Porto
17
324
User
Porto
17
325
User
Faro
17
326
User
Porto
17
327
User
Porto
17
328
User
Porto
17
329
User
Coimbra
17
330
User
Porto
17
331
User
Lisboa
17
332
User
Porto
16
333
User
Lisboa
16
334
User
Porto
16
335
User
Porto
16
336
User
Lisboa
16
337
User
Porto
16
338
User
Porto
16
339
User
Porto
16
340
User
Porto
16
341
User
Porto
16
342
User
Porto
16
343
User
Porto
16
344
User
Porto
16
345
User
Porto
16
346
User
Porto
16
347
User
Porto
16
348
User
Porto
16
349
User
Lisboa
16
350
User
Porto
16
351
User
Lisboa
16
352
User
Lisboa
16
353
User
Porto
16
354
User
Porto
16
355
User
Porto
16
356
Adam Grey
Porto
16
357
User
Porto
16
358
User
Porto
16
359
User
Porto
16
360
User
Lisboa
15
361
User
Lisboa
15
362
User
Lisboa
15
363
User
Porto
15
364
User
Porto
15
365
User
Porto
15
366
henrique proenca
Lisboa
15
367
User
Lisboa
15
368
User
Porto
15
369
User
Porto
15
370
User
Porto
15
371
User
Porto
15
372
User
Coimbra
15
373
User
Porto
15
374
Francis M
Bragança
15
375
User
Porto
15
376
User
Porto
15
377
User
Coimbra
15
378
User
Porto
15
379
User
Porto
15
380
User
Lisboa
15
381
User
Lisboa
15
382
User
Porto
15
383
User
Porto
15
384
User
Porto
15
385
User
Porto
15
386
User
Aveiro
15
387
User
Porto
15
388
User
Porto
15
389
User
Porto
15
390
User
Lisboa
15
391
User
Porto
15
392
User
Lisboa
14
393
User
Porto
14
394
Catarina Guerreiro
Lisboa
14
395
User
Porto
14
396
User
Porto
14
397
User
Lisboa
14
398
User
Porto
14
399
User
Porto
14
400
User
Beja
14
401
User
Porto
14
402
User
Porto
14
403
User
Porto
14
404
User
Porto
14
405
User
Porto
14
406
User
Lisboa
14
407
User
Porto
14
408
User
Porto
14
409
User
Porto
14
410
User
Lisboa
14
411
User
Porto
14
412
User
Porto
14
413
User
Porto
14
414
User
Lisboa
14
415
User
Setúbal
14
416
User
Porto
14
417
User
Porto
13
418
Simao Sarmento
Lisboa
13
419
User
Porto
13
420
User
Porto
13
421
User
Lisboa
13
422
User
Porto
13
423
User
Porto
13
424
User
Lisboa
13
425
User
Porto
13
426
User
Porto
13
427
User
Porto
13
428
User
Porto
13
429
User
Bragança
13
430
User
Porto
13
431
User
Porto
13
432
User
Porto
13
433
User
Lisboa
13
434
User
Porto
13
435
User
Lisboa
13
436
User
Porto
13
437
User
Porto
13
438
User
Leiria
12
439
User
Lisboa
12
440
User
Porto
12
441
User
Lisboa
12
442
User
Lisboa
12
443
User
Lisboa
12
444
User
Porto
12
445
User
Braga
12
446
User
Porto
12
447
User
Porto
12
448
User
Porto
12
449
User
Porto
12
450
User
Porto
12
451
User
Porto
12
452
User
Porto
12
453
User
Porto
12
454
User
Porto
12
455
User
Lisboa
12
456
User
Porto
12
457
User
Porto
12
458
User
Lisboa
12
459
User
Porto
12
460
User
Porto
12
461
User
Porto
12
462
User
Lisboa
11
463
User
Lisboa
11
464
Mariana Pitôrra
Lisboa
11
465
User
Lisboa
11
466
User
Porto
11
467
User
Porto
11
468
User
Lisboa
11
469
User
Porto
11
470
XGariex
Lisboa
11
471
User
Porto
11
472
User
Porto
11
473
User
Porto
11
474
User
Porto
11
475
EzYRa
Faro
11
476
User
Porto
11
477
User
Porto
11
478
User
Porto
11
479
User
Porto
11
480
User
Porto
11
481
User
Porto
11
482
User
Porto
11
483
User
Lisboa
11
484
User
Porto
11
485
User
Porto
10
486
P. M.
Lisboa
10
487
User
Lisboa
10
488
User
Porto
10
489
User
Lisboa
10
490
User
Lisboa
10
491
User
Porto
10
492
User
Vila Real
10
493
User
Lisboa
10
494
User
Lisboa
10
496
User
Porto
10
497
User
Porto
10
498
User
Porto
10
499
User
Porto
10
500
User
Porto
10
501
User
Porto
10
502
Alexandre Romao
Lisboa
10
503
User
Coimbra
10
504
User
Lisboa
10
505
User
Lisboa
10
506
User
Lisboa
9
507
User
Porto
9
508
User
Porto
9
509
Rodrigo Maia da Silva
Região Autónoma dos Açores
9
510
User
Lisboa
9
511
User
Braga
9
512
User
Setúbal
9
513
User
Porto
9
514
User
Porto
9
515
User
Porto
9
516
User
Lisboa
9
517
User
Braga
9
518
User
Porto
9
519
User
Porto
9
520
User
Porto
9
521
User
Porto
9
522
User
Porto
9
523
User
Porto
9
524
User
Porto
9
525
User
Porto
9
526
User
Porto
9
527
User
Porto
9
528
User
Setúbal
8
529
User
Setúbal
8
530
User
Porto
8
531
User
Porto
8
532
User
Porto
8
533
User
Porto
8
534
User
Lisboa
8
535
User
Lisboa
8
536
User
Lisboa
8
537
User
Lisboa
8
538
User
Lisboa
8
539
User
Setúbal
8
540
User
Porto
8
541
User
Porto
8
542
User
Porto
8
543
User
Porto
8
544
User
Porto
8
545
User
Porto
8
546
User
Porto
8
547
User
Lisboa
8
548
User
Porto
8
549
User
Porto
8
550
User
Porto
8
551
User
Porto
8
552
User
Porto
7
553
Matilde Pitôrra
Lisboa
7
554
User
Lisboa
7
555
Calvin Duarte
Lisboa
7
556
User
Porto
7
557
User
Porto
7
558
User
Porto
7
559
User
Porto
7
560
Mary Hopkins
Lisboa
7
561
User
Lisboa
7
562
Filipa Sebastião
Lisboa
7
563
User
Lisboa
7
564
User
Faro
7
565
User
Coimbra
7
566
User
Porto
7
567
User
Porto
7
568
User
Porto
7
569
User
Faro
7
570
User
Lisboa
6
571
User
Lisboa
6
572
User
Braga
6
573
User
Porto
6
574
User
Aveiro
6
575
User
Porto
6
576
User
Lisboa
6
577
User
Lisboa
6
578
User
Lisboa
6
579
User
Coimbra
6
580
User
Lisboa
6
581
User
Lisboa
6
582
User
Lisboa
6
583
User
Lisboa
6
584
User
Lisboa
6
585
User
Lisboa
6
586
User
Porto
6
587
User
Porto
6
588
User
Lisboa
6
589
User
Lisboa
6
590
User
Porto
6
591
User
Viseu
6
592
User
Porto
6
593
User
Lisboa
6
594
User
Lisboa
5
595
User
Lisboa
5
596
User
Lisboa
5
597
User
Coimbra
5
598
User
Vila Real
5
599
User
Lisboa
5
600
David Marques
Viseu
5
601
User
Setúbal
5
602
User
Porto
5
603
User
Beja
5
604
User
Lisboa
5
605
User
Porto
5
606
User
Porto
5
607
User
Viseu
5
608
User
Porto
5
609
Marta Rolo
Lisboa
5
610
Maria Fonseca
Lisboa
5
611
User
Lisboa
4
612
User
Lisboa
4
613
User
Faro
4
614
User
Santarém
4
615
diogo
Porto
4
616
User
Lisboa
4
617
User
Porto
4
618
User
Lisboa
4
619
User
Porto
4
620
User
Lisboa
4
621
User
Lisboa
4
622
User
Porto
4
623
User
Lisboa
4
624
User
Lisboa
4
625
User
Lisboa
4
626
User
Évora
3
627
User
Lisboa
3
628
User
Porto
3
629
sofs lobo
Porto
3
630
User
Porto
3
631
User
Viseu
3
632
User
Porto
3
633
User
Beja
3
634
User
Lisboa
3
635
User
Setúbal
3
636
User
Lisboa
3
637
User
Braga
3
638
User
Lisboa
3
639
User
Santarém
3
640
User
Porto
3
641
User
Setúbal
3
642
User
Lisboa
3
643
User
Lisboa
2
644
User
Lisboa
2
646
User
Lisboa
2
647
User
Lisboa
2
648
User
Lisboa
2
649
User
Braga
2
650
User
Lisboa
2
651
User
Lisboa
2
652
User
Lisboa
2
653
User
Setúbal
2
654
User
Lisboa
2
655
User
Braga
2
656
User
Porto
2
657
User
Lisboa
2
658
User
Lisboa
2
659
Pedro Caeiro
Lisboa
2
660
User
Santarém
2
661
User
Coimbra
2
662
User
Lisboa
2
663
User
Lisboa
2
664
User
Porto
2
665
User
Porto
2
666
User
Porto
2
667
User
Coimbra
2
668
User
Lisboa
2
669
User
Porto
2
670
User
Setúbal
2
671
User
Braga
2
672
User
Porto
2
673
User
Braga
2
675
Martim Gamboa
Lisboa
1
676
User
Porto
1
677
User
Aveiro
1
678
User
Lisboa
1
679
User
Porto
1
680
Oscar Cavalao
Lisboa
1
681
User
Lisboa
1
682
User
Lisboa
1
683
User
Lisboa
1
685
User
Braga
1
686
User
Lisboa
1
687
User
Setúbal
1
688
User
Viseu
1
689
User
Aveiro
1
690
User
Porto
1
691
User
Porto
1
692
User
Setúbal
1
693
User
Lisboa
1
694
User
Lisboa
1
695
User
Porto
1
696
edmond turquieh
Santarém
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