Leaderboard
Pakistan
Pakistan
Worldwide
Rank
Level
1
User
Sindh
82
2
Malik Muhammad Kashif Saeed
Islāmābād
81
3
Zaaid Ahmed
Islāmābād
63
4
User
Sindh
52
5
User
Sindh
45
6
User
Sindh
43
7
User
Panjāb
42
8
User
Panjāb
42
11
User
Panjāb
40
12
User
Panjāb
39
13
User
Sindh
38
14
User
Sindh
38
15
User
Panjāb
37
16
Faraz
Islāmābād
37
17
User
Islāmābād
36
18
User
Sindh
36
19
mariam raza
Panjāb
35
20
User
Panjāb
35
21
Shahid Anwer
Islāmābād
34
22
User
Sindh
34
23
User
Sindh
34
24
User
Panjāb
34
25
User
Panjāb
34
26
User
Islāmābād
33
27
User
Islāmābād
33
28
User
Islāmābād
33
29
User
Sindh
32
30
Ali Hanif
Sindh
31
31
User
Sindh
30
32
User
Panjāb
30
33
User
Sindh
30
34
User
Panjāb
29
35
User
Sindh
29
36
User
Sindh
29
37
User
Sindh
28
38
User
Panjāb
28
39
User
Sindh
27
40
User
Islāmābād
27
41
User
Panjāb
26
42
User
Panjāb
26
43
User
Panjāb
25
44
malik muhammad kashif saeed
Islāmābād
25
45
User
Sindh
25
46
User
Sindh
25
47
User
Sindh
25
48
User
Panjāb
25
49
User
Islāmābād
24
50
User
Panjāb
23
51
User
Sindh
23
52
User
Panjāb
23
53
User
Panjāb
23
54
User
Panjāb
23
55
User
Sindh
23
56
User
Panjāb
22
57
User
Sindh
22
58
User
Panjāb
22
59
User
Panjāb
22
60
User
Panjāb
22
61
User
Sindh
22
62
User
Panjāb
21
63
User
Sindh
21
64
User
Sindh
21
65
User
Sindh
21
66
User
Panjāb
21
67
User
Panjāb
21
68
Tai moor
Region
21
69
User
Sindh
20
70
User
Sindh
20
71
User
Panjāb
20
72
User
Islāmābād
20
73
User
Panjāb
20
74
User
Sindh
20
75
User
Panjāb
20
76
User
Islāmābād
20
77
User
Panjāb
19
78
Palwasha Faizan
Sindh
19
79
User
Sindh
19
80
User
Panjāb
19
81
User
Islāmābād
19
82
User
Panjāb
19
83
User
Sindh
19
85
User
Sindh
18
86
User
Sindh
18
87
Aribah Noman
Sindh
18
88
User
Sindh
18
89
User
Sindh
17
90
User
Sindh
17
91
User
Panjāb
17
92
User
Panjāb
17
93
User
Sindh
17
94
User
Sindh
17
95
User
Islāmābād
17
96
User
Panjāb
16
97
User
Sindh
16
98
User
Sindh
16
99
User
Panjāb
16
100
User
Panjāb
16
101
User
Sindh
16
102
User
Islāmābād
15
103
Muhammad Razi
Panjāb
15
104
User
Sindh
15
105
User
Sindh
15
106
User
Panjāb
15
107
Fahd
Islāmābād
15
108
User
Panjāb
15
109
User
Panjāb
15
110
User
Panjāb
15
111
User
Panjāb
15
112
User
Sindh
15
113
User
Panjāb
14
114
User
Sindh
14
115
User
Sindh
14
116
User
Islāmābād
14
117
User
Sindh
14
118
User
Sindh
14
119
User
Panjāb
14
120
User
Sindh
14
121
User
Sindh
14
122
User
Sindh
13
123
User
Panjāb
13
124
User
Panjāb
13
125
User
Panjāb
13
126
User
Panjāb
13
127
User
Sindh
13
128
User
Panjāb
13
129
User
Panjāb
13
130
User
Sindh
13
131
User
Panjāb
13
132
User
Sindh
13
133
User
Panjāb
12
134
User
Islāmābād
12
135
User
Panjāb
12
136
User
Sindh
12
137
User
Sindh
12
138
User
Panjāb
12
139
User
Sindh
12
140
User
Sindh
12
141
User
Sindh
12
142
User
Panjāb
12
143
User
Panjāb
12
144
Farhaj Ahmed
Sindh
12
145
User
Islāmābād
12
146
User
Sindh
12
147
User
Panjāb
12
148
User
Sindh
11
149
User
Panjāb
11
150
User
Islāmābād
11
151
KTX ANNAN
Sindh
11
152
User
Panjāb
11
154
User
Panjāb
11
155
User
Sindh
11
156
User
Panjāb
11
157
User
Panjāb
11
158
User
Sindh
11
159
User
Islāmābād
11
160
Yousuf Khawaja
Panjāb
11
161
more what
Region
11
162
Zahra
Region
11
163
User
Panjāb
10
164
User
Islāmābād
10
165
User
Sindh
10
166
User
Sindh
10
167
User
Islāmābād
10
168
User
Sindh
10
169
User
Sindh
10
170
User
Sindh
10
171
User
Panjāb
10
172
User
Panjāb
10
173
User
Sindh
10
174
User
Sindh
10
175
User
Sindh
10
176
User
Sindh
10
177
User
Sindh
10
178
User
Sindh
10
179
User
Panjāb
10
180
User
Panjāb
10
181
User
Sindh
10
182
User
Sindh
9
183
User
Panjāb
9
184
User
Panjāb
9
185
User
Sindh
9
186
User
Panjāb
9
188
User
Sindh
9
189
User
Sindh
9
190
User
Sindh
9
191
User
Islāmābād
9
192
User
Islāmābād
9
193
User
Sindh
9
194
User
Panjāb
9
195
User
Sindh
9
197
User
Sindh
8
198
User
Sindh
8
199
User
Sindh
8
200
User
Sindh
8
201
Aaseya
Panjāb
8
202
User
Panjāb
8
203
User
Sindh
8
204
User
Panjāb
8
207
User
Panjāb
8
208
User
Panjāb
8
209
User
Sindh
8
210
User
Sindh
8
211
User
Sindh
8
212
Hina Tahir
Sindh
8
213
User
Sindh
8
214
Sadia Mamji
Sindh
8
215
User
Islāmābād
8
216
User
Sindh
8
217
User
Islāmābād
8
218
User
Islāmābād
8
219
User
Islāmābād
7
220
User
Panjāb
7
221
User
Islāmābād
7
222
User
Sindh
7
223
User
Panjāb
7
224
User
Sindh
7
225
User
Sindh
7
226
User
Panjāb
7
227
User
Sindh
7
228
User
Panjāb
7
229
User
Panjāb
7
230
User
Panjāb
7
232
User
Sindh
7
233
User
Panjāb
7
234
User
Panjāb
7
235
User
Panjāb
7
236
User
Sindh
7
237
User
Panjāb
7
238
User
Sindh
7
239
User
Islāmābād
7
240
User
Sindh
7
241
User
Sindh
7
242
User
Sindh
7
243
User
Islāmābād
7
244
User
Panjāb
7
245
User
Panjāb
7
246
User
Sindh
7
247
User
Sindh
7
248
User
Panjāb
7
249
User
Panjāb
7
250
User
Islāmābād
6
251
User
Panjāb
6
252
User
Sindh
6
253
Ahmed AbuBakr
Panjāb
6
254
User
Panjāb
6
255
User
Islāmābād
6
256
User
Panjāb
6
257
User
Sindh
6
258
User
Islāmābād
6
259
aaima adil
Sindh
6
260
User
Panjāb
6
261
User
Sindh
6
262
User
Sindh
6
263
User
Sindh
6
264
User
Sindh
6
265
User
Sindh
6
266
User
Islāmābād
6
267
User
Sindh
6
268
User
Sindh
6
269
User
Panjāb
6
270
User
Panjāb
6
271
Saad Akhtar
Sindh
6
272
User
Sindh
6
273
User
Sindh
6
274
User
Panjāb
6
275
User
Sindh
6
276
User
Islāmābād
6
277
User
Sindh
6
278
User
Panjāb
6
279
User
Sindh
6
280
User
Sindh
6
281
User
Sindh
6
282
User
Panjāb
6
283
User
Panjāb
6
284
User
Sindh
6
285
User
Sindh
6
286
User
Sindh
6
287
User
Islāmābād
6
288
User
Panjāb
6
289
User
Panjāb
6
290
User
Sindh
6
291
User
Sindh
6
292
User
Islāmābād
6
293
User
Sindh
6
294
User
Sindh
6
295
User
Panjāb
5
296
User
Sindh
5
297
User
Islāmābād
5
298
User
Sindh
5
299
User
Sindh
5
300
User
Sindh
5
301
User
Sindh
5
302
User
Sindh
5
303
User
Sindh
5
304
User
Sindh
5
305
User
Sindh
5
306
User
Sindh
5
307
User
Sindh
5
308
User
Panjāb
5
309
User
Sindh
5
311
User
Islāmābād
5
312
User
Sindh
5
313
User
Sindh
5
314
User
Panjāb
5
315
User
Islāmābād
5
316
User
Panjāb
5
317
User
Sindh
5
318
User
Panjāb
5
319
User
Sindh
5
320
User
Sindh
5
321
User
Panjāb
5
322
User
Panjāb
5
323
User
Sindh
5
324
User
Islāmābād
5
325
User
Sindh
5
326
User
Sindh
5
327
User
Sindh
5
328
User
Panjāb
5
329
User
Panjāb
5
330
User
Sindh
5
331
User
Panjāb
4
332
User
Sindh
4
333
User
Panjāb
4
334
User
Panjāb
4
335
User
Islāmābād
4
336
User
Sindh
4
337
User
Panjāb
4
338
User
Sindh
4
339
User
Sindh
4
340
User
Panjāb
4
341
User
Sindh
4
342
User
Sindh
4
343
User
Sindh
4
344
User
Sindh
4
345
User
Sindh
4
346
User
Panjāb
4
348
User
Sindh
4
349
User
Sindh
4
350
User
Sindh
4
351
User
Panjāb
4
352
hi
Sindh
4
353
User
Sindh
4
354
User
Panjāb
4
355
User
Sindh
4
356
User
Sindh
4
357
User
Sindh
4
358
User
Sindh
4
359
User
Islāmābād
4
360
User
Balochistan
4
361
User
Sindh
4
362
Muhammad Jafri
Sindh
4
363
User
Sindh
4
364
User
Sindh
4
365
User
Sindh
4
366
User
Sindh
4
367
User
Islāmābād
4
368
User
Sindh
4
369
User
Sindh
4
370
User
Sindh
4
371
User
Sindh
4
372
User
Islāmābād
3
373
Baryal Sher
Khaībar Pakhtūnkhwā
3
374
User
Panjāb
3
375
User
Sindh
3
376
User
Sindh
3
377
User
Sindh
3
378
User
Sindh
3
379
User
Panjāb
3
380
User
Sindh
3
381
User
Sindh
3
382
User
Sindh
3
383
User
Islāmābād
3
384
User
Sindh
3
385
User
Sindh
3
386
User
Sindh
3
387
User
Sindh
3
388
User
Islāmābād
3
390
User
Sindh
3
391
User
Panjāb
3
392
User
Sindh
3
393
User
Sindh
3
394
User
Sindh
3
395
User
Sindh
3
396
User
Islāmābād
3
397
User
Sindh
3
398
User
Panjāb
3
399
User
Panjāb
3
401
User
Panjāb
3
402
User
Panjāb
3
403
User
Sindh
3
404
User
Panjāb
3
405
User
Sindh
3
408
User
Sindh
3
409
User
Sindh
3
410
User
Sindh
3
411
User
Islāmābād
3
412
User
Panjāb
3
413
Lovely
Sindh
3
414
User
Sindh
3
415
User
Sindh
3
416
User
Sindh
3
417
User
Sindh
3
418
User
Sindh
3
419
User
Sindh
3
420
User
Sindh
3
421
User
Sindh
3
422
User
Panjāb
3
423
User
Panjāb
3
424
User
Panjāb
3
425
User
Panjāb
3
426
User
Sindh
3
427
User
Sindh
3
428
User
Sindh
3
429
User
Sindh
3
430
User
Panjāb
3
431
User
Islāmābād
3
432
User
Sindh
3
433
User
Panjāb
3
434
User
Panjāb
3
435
User
Panjāb
3
436
User
Panjāb
3
437
User
Islāmābād
3
438
User
Sindh
3
439
User
Sindh
3
440
User
Sindh
3
441
User
Panjāb
3
442
User
Islāmābād
3
443
Ayesha Anwar
Region
3
444
User
Panjāb
2
445
User
Panjāb
2
446
User
Islāmābād
2
447
User
Sindh
2
448
User
Panjāb
2
449
User
Sindh
2
450
User
Panjāb
2
451
User
Sindh
2
452
User
Panjāb
2
453
User
Panjāb
2
454
User
Sindh
2
455
User
Sindh
2
456
User
Panjāb
2
457
User
Sindh
2
458
User
Islāmābād
2
459
User
Panjāb
2
460
User
Sindh
2
461
User
Panjāb
2
462
Anwer Hussain
Sindh
2
463
User
Sindh
2
464
User
Sindh
2
465
User
Sindh
2
466
User
Sindh
2
467
User
Sindh
2
468
User
Islāmābād
2
469
User
Sindh
2
470
User
Islāmābād
2
471
User
Sindh
2
472
User
Sindh
2
473
User
Sindh
2
474
User
Sindh
2
475
User
Islāmābād
2
476
User
Panjāb
2
477
User
Panjāb
2
478
User
Panjāb
2
479
User
Sindh
2
481
User
Panjāb
2
482
User
Panjāb
2
483
User
Islāmābād
2
484
User
Sindh
2
485
User
Sindh
2
486
User
Sindh
2
487
User
Sindh
2
488
User
Sindh
2
489
User
Sindh
2
490
User
Sindh
2
491
User
Sindh
2
492
User
Sindh
2
493
User
Sindh
2
494
User
Panjāb
2
495
User
Sindh
2
496
User
Sindh
2
497
User
Panjāb
2
498
User
Sindh
2
499
User
Sindh
2
501
User
Panjāb
2
502
User
Panjāb
2
503
User
Panjāb
2
504
User
Panjāb
2
505
User
Panjāb
2
506
User
Islāmābād
2
507
User
Panjāb
2
508
User
Sindh
2
509
User
Sindh
2
510
User
Sindh
2
511
User
Sindh
2
513
User
Panjāb
2
514
User
Sindh
2
515
User
Panjāb
2
516
User
Panjāb
2
517
User
Sindh
2
518
User
Sindh
2
519
User
Sindh
2
520
User
Panjāb
2
521
User
Sindh
2
522
User
Sindh
2
523
User
Islāmābād
2
524
User
Sindh
2
525
User
Balochistan
2
526
User
Sindh
2
527
User
Sindh
2
528
User
Sindh
2
529
User
Islāmābād
2
530
User
Sindh
2
531
User
Sindh
2
532
User
Sindh
2
533
User
Sindh
2
534
User
Sindh
2
535
User
Panjāb
1
536
User
Panjāb
1
537
User
Panjāb
1
538
User
Panjāb
1
539
User
Panjāb
1
540
User
Panjāb
1
541
User
Sindh
1
542
User
Islāmābād
1
543
User
Islāmābād
1
544
User
Panjāb
1
545
User
Panjāb
1
546
User
Panjāb
1
547
User
Panjāb
1
548
User
Islāmābād
1
549
User
Panjāb
1
550
User
Panjāb
1
551
User
Panjāb
1
552
User
Panjāb
1
553
User
Panjāb
1
554
User
Panjāb
1
555
User
Sindh
1
556
User
Sindh
1
557
User
Sindh
1
558
User
Sindh
1
559
User
Panjāb
1
560
User
Panjāb
1
561
User
Sindh
1
562
User
Sindh
1
563
User
Panjāb
1
564
User
Sindh
1
565
User
Islāmābād
1
566
User
Panjāb
1
567
User
Islāmābād
1
568
User
Panjāb
1
569
User
Sindh
1
570
User
Panjāb
1
571
User
Islāmābād
1
572
User
Islāmābād
1
573
User
Sindh
1
574
zeest raza
Sindh
1
575
User
Sindh
1
576
User
Sindh
1
577
User
Panjāb
1
578
User
Panjāb
1
579
User
Sindh
1
580
User
Sindh
1
581
User
Sindh
1
583
User
Sindh
1
584
User
Sindh
1
585
User
Sindh
1
586
AbAn AshrAf
Sindh
1
587
User
Sindh
1
588
User
Islāmābād
1
589
User
Islāmābād
1
590
User
Islāmābād
1
591
User
Sindh
1
592
User
Sindh
1
593
User
Panjāb
1
594
User
Sindh
1
595
User
Panjāb
1
596
User
Sindh
1
597
User
Sindh
1
598
User
Sindh
1
599
User
Sindh
1
600
User
Sindh
1
601
User
Islāmābād
1
602
User
Panjāb
1
603
User
Sindh
1
604
User
Sindh
1
605
User
Sindh
1
607
User
Sindh
1
608
User
Panjāb
1
609
User
Sindh
1
610
User
Sindh
1
611
User
Panjāb
1
612
User
Sindh
1
613
User
Sindh
1
614
User
Sindh
1
615
User
Sindh
1
616
User
Sindh
1
617
User
Islāmābād
1
618
User
Sindh
1
619
User
Panjāb
1
620
User
Panjāb
1
621
User
Sindh
1
622
User
Sindh
1
623
User
Sindh
1
624
Ali Shan
Sindh
1
625
User
Sindh
1
626
ajea zahid
Region
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