Leaderboard
India
India
Worldwide
Rank
Level
1
User
Delhi
99
2
Rajaram
West Bengal
98
3
Abhishek Singh
West Bengal
98
4
Mo
Delhi
97
7
User
Tamil Nadu
95
8
Dhruv Arora
Punjab
94
10
Sanchi Wankhade
Maharashtra
92
11
User
Maharashtra
92
12
User
West Bengal
91
13
User
Bihar
91
14
MyNameIsMe
Telangana
91
15
Narasimha
Karnataka
91
16
User
Bihar
90
17
User
Bihar
90
18
User
West Bengal
90
19
User
Maharashtra
90
20
ROBIN KUMAR
Karnataka
89
21
User
Bihar
89
22
User
Bihar
89
23
Neeraj Thakur
Delhi
89
24
RUSHI
Telangana
88
25
User
Bihar
87
26
Sachit Mehrotra
Maharashtra
87
27
User
Goa
87
28
Anya Sanghvi (CK)
Madhya Pradesh
86
29
User
Bihar
86
30
User
Bihar
85
31
Sai Jahnavi Yeruva
Telangana
85
32
User
Bihar
85
33
User
Bihar
85
36
User
Bihar
84
37
User
West Bengal
84
38
Abhilash IITP
Bihar
84
39
User
Maharashtra
83
40
Vivek Nautiyal
Karnataka
83
41
Bipin Bihari che15(IIT bhu)
Uttar Pradesh
83
42
User
Bihar
82
43
shubham bansal
Maharashtra
82
44
Sachin Dubey
Madhya Pradesh
81
45
User
Maharashtra
81
46
User
Maharashtra
81
47
User
Maharashtra
81
48
Shrivathsan V
Tamil Nadu
80
49
ARKA CHAKRABORTY
West Bengal
80
50
Satya swarup Jena
Odisha
80
51
Eeshan Zele
Maharashtra
80
52
User
Maharashtra
80
53
User
Telangana
80
55
bigblackclock
Karnataka
79
56
User
Chandigarh
79
57
Monica
Maharashtra
79
58
Jagat
Karnataka
78
59
User
Tamil Nadu
78
61
User
Bihar
78
62
User
Gujarat
78
63
Rakesh Singh
West Bengal
78
64
User
Delhi
77
65
aditya jain
Maharashtra
77
66
Ayushi Gupta
Karnataka
77
67
User
Tamil Nadu
77
68
User
Delhi
77
69
sudharsan ss
Tamil Nadu
77
70
User
Rajasthan
77
71
User
Rajasthan
77
72
Ankit Ahuja
Delhi
77
74
Rahul Dugar
Karnataka
76
75
User
Maharashtra
76
76
User
Maharashtra
76
77
Jayasheelli k
Karnataka
76
78
User
Bihar
76
79
User
Kerala
76
80
Mahesh Kanade
Madhya Pradesh
76
81
User
West Bengal
76
82
User
Gujarat
76
83
Vinay Indrala
Telangana
75
84
sawan agarwal
Rajasthan
75
85
User
Maharashtra
75
86
User
Karnataka
75
87
GitGud
Karnataka
75
88
Hari Sankar Narayanan
Tamil Nadu
75
89
MUKILA M
Tamil Nadu
75
90
Ganesh Kulakarni
Karnataka
74
91
Tejas ___
Maharashtra
74
92
Tejas
Maharashtra
74
93
User
Bihar
74
94
User
Maharashtra
74
95
Sumant Kumar
Tamil Nadu
74
96
User
Tamil Nadu
73
97
User
Tamil Nadu
73
98
User
West Bengal
73
99
User
Telangana
73
100
User
Delhi
73
101
User
Maharashtra
73
102
User
Bihar
73
103
User
Bihar
73
104
User
Telangana
72
105
User
Delhi
72
106
User
West Bengal
72
107
User
Maharashtra
72
108
User
West Bengal
72
109
Aayush Mani
Chandigarh
72
110
User
Maharashtra
72
111
Surya Narayan Sen
West Bengal
72
112
User
Kerala
71
113
User
Maharashtra
71
114
User
Delhi
71
115
User
Maharashtra
71
116
velidi murali krishna
Telangana
71
117
User
Haryana
71
118
Sachin kumar Gupta
Madhya Pradesh
71
120
User
Maharashtra
70
121
Yousef c t
Kerala
70
122
User
Tamil Nadu
70
123
User
Maharashtra
70
124
User
Tamil Nadu
70
125
Mohit Singhal
Rajasthan
70
126
User
Maharashtra
70
127
Sharan
Chandigarh
70
128
User
Maharashtra
70
129
User
Tamil Nadu
69
130
User
Maharashtra
69
131
User
Rajasthan
69
132
User
Uttarakhand
69
133
User
Rajasthan
69
134
User
Delhi
69
135
Ujjwal Bansal
Uttarakhand
69
136
User
Maharashtra
69
137
hotessy
Delhi
69
138
User
Tamil Nadu
69
139
User
Maharashtra
69
141
Nilesh Ramesh Gavali
Karnataka
68
142
Manay Divatia
Kerala
68
144
Raghunath Kaushik
Tamil Nadu
68
145
User
Tamil Nadu
68
146
User
Tamil Nadu
68
147
RKT
Maharashtra
68
148
User
Bihar
68
149
ajinkya awachar
Madhya Pradesh
67
150
User
Telangana
67
151
User
Delhi
67
152
User
Karnataka
67
153
Divya Goel
Delhi
67
154
User
Goa
67
155
User
Maharashtra
67
156
Anupam Dhar
Bihar
66
157
D Abhi
Telangana
66
158
User
Delhi
66
159
Rahul Godara
Delhi
66
160
User
Tamil Nadu
66
161
KSingh
Karnataka
66
162
User
Haryana
66
163
User
Kerala
66
164
User
Bihar
66
165
User
Maharashtra
66
168
User
West Bengal
66
169
Arjit Kapoor
Delhi
66
170
hardik vashishtha
Haryana
65
171
User
Tamil Nadu
65
172
User
Gujarat
65
173
SidMan
Maharashtra
65
174
User
Chandigarh
65
175
User
Delhi
65
176
User
Rajasthan
65
177
User
West Bengal
65
178
User
Delhi
65
179
Soumya
Uttar Pradesh
65
180
User
Maharashtra
65
181
User
West Bengal
65
182
User
Maharashtra
65
183
Raghav Dhawan
Andhra Pradesh
65
184
User
Maharashtra
65
185
Yash Shukla
Uttar Pradesh
64
186
User
Rajasthan
64
187
User
Goa
64
189
Ronit Chidara
Telangana
64
190
Krishna M.V.
Tamil Nadu
64
191
User
Chandigarh
64
192
Arham Kothari
Telangana
64
193
User
Maharashtra
64
194
User
Bihar
64
195
User
Bihar
64
196
User
Tamil Nadu
64
198
User
Maharashtra
64
199
User
Rajasthan
63
200
User
Karnataka
63
201
User
Tamil Nadu
63
203
White Beard
Karnataka
63
204
User
West Bengal
63
205
User
Rajasthan
63
206
User
Tamil Nadu
63
207
User
Uttar Pradesh
63
208
CHITRANGAD
West Bengal
63
209
Dishank Agrawal
West Bengal
63
210
User
Maharashtra
63
211
User
Chandigarh
63
213
Bharadwaj Gandepalli
Telangana
62
214
Bhagyashri Patil
Maharashtra
62
215
Ashraful Islam
Assam
62
216
User
Delhi
62
217
User
Karnataka
62
218
Abhishek Manwani
Madhya Pradesh
62
219
User
West Bengal
62
220
John Louis
Tamil Nadu
62
221
Shaurya Vijay
Bihar
62
222
Aarya Singh
Uttar Pradesh
62
223
User
Delhi
62
224
User
Delhi
62
225
User
Telangana
62
226
User
Chandigarh
62
227
Sai Yashwanth
Telangana
62
228
User
Odisha
62
229
User
Kerala
61
230
User
Rajasthan
61
231
User
Tamil Nadu
61
232
User
Rajasthan
61
233
Soumya
Uttar Pradesh
61
234
Vedavyas
Haryana
61
235
User
Delhi
61
236
Priyanshu Gupta
Karnataka
61
237
User
Maharashtra
61
238
User
Tamil Nadu
61
239
babnish19
Bihar
61
240
User
Maharashtra
61
241
User
Tamil Nadu
61
242
User
West Bengal
61
243
Pratik Kumar
Delhi
61
244
User
Karnataka
60
245
User
Rajasthan
60
246
User
Delhi
60
247
suresh
Telangana
60
248
User
Delhi
60
249
Harjot singh Bhatti
Maharashtra
60
250
User
Telangana
60
251
User
West Bengal
60
252
User
Tamil Nadu
60
253
Sreenidhi V
Tamil Nadu
60
254
User
Tamil Nadu
60
255
User
Delhi
60
256
User
Karnataka
60
257
krishna bhati
Rajasthan
60
258
User
West Bengal
60
259
User
Karnataka
60
261
User
West Bengal
60
262
User
Maharashtra
60
263
User
Bihar
60
264
Raman Bhansali
Maharashtra
59
265
User
West Bengal
59
266
User
Maharashtra
59
267
User
West Bengal
59
268
Madhu Rawat
Maharashtra
59
269
User
Uttar Pradesh
59
271
User
Tamil Nadu
59
272
User
Delhi
59
273
User
West Bengal
59
274
User
West Bengal
59
275
Kolisettiramalingam Ramalingam
Tamil Nadu
59
276
Adarsh Verma
Uttar Pradesh
59
278
User
West Bengal
58
279
User
Delhi
58
280
Milind Shukla
Uttar Pradesh
58
281
Sushant
Delhi
58
282
User
Karnataka
58
283
User
West Bengal
58
284
User
Bihar
58
285
User
Telangana
58
286
User
Maharashtra
58
287
User
Gujarat
58
288
User
Telangana
58
289
User
West Bengal
58
290
Abhyuday Sharma
Rajasthan
58
291
cbzsports1098
Maharashtra
58
292
User
Maharashtra
58
293
Vikrant Shah
Maharashtra
58
294
souparna
West Bengal
57
295
User
Delhi
57
296
User
Uttar Pradesh
57
297
User
Goa
57
298
User
Delhi
57
299
User
Delhi
57
300
User
Rajasthan
57
301
User
Telangana
57
302
User
West Bengal
57
304
Krittika Saha
Delhi
57
305
User
Karnataka
57
306
User
Delhi
57
307
User
Delhi
57
308
User
Maharashtra
57
309
User
Maharashtra
57
310
Sankalp Krishnan
Maharashtra
57
311
a a
Maharashtra
57
312
Sandhya Naik
Karnataka
57
313
Sandhya
Karnataka
57
314
User
Haryana
56
315
User
Telangana
56
316
Sachin bhavre
Delhi
56
317
User
Delhi
56
318
Prateek Mohapatra
Uttar Pradesh
56
319
User
Karnataka
56
321
User
Delhi
56
322
User
Rajasthan
56
323
User
Telangana
56
324
User
West Bengal
56
325
User
West Bengal
56
326
User
Kerala
56
327
User
Maharashtra
56
328
User
Tamil Nadu
56
329
aman rathore
Uttar Pradesh
56
330
Gauransh
Uttar Pradesh
56
331
User
West Bengal
56
332
User
Kerala
55
333
Sabir Molla
West Bengal
55
334
Aurko Haldi
Maharashtra
55
335
Lavish Bhutani
Karnataka
55
337
User
Maharashtra
55
338
User
Chandigarh
55
339
Shailesh Aagarwal
Karnataka
55
340
ANKIT SANKHALA
Rajasthan
55
341
User
Maharashtra
55
342
Akhila Vijayan
Kerala
55
343
User
Maharashtra
55
344
User
West Bengal
55
345
User
Haryana
54
346
User
Maharashtra
54
347
User
Karnataka
54
348
suniti murmu
Bihar
54
349
User
Karnataka
54
351
Tejaswi B
Telangana
54
352
User
Bihar
54
353
User
Uttar Pradesh
54
354
User
Rajasthan
54
356
User
Karnataka
54
357
User
Delhi
54
358
User
Rajasthan
53
359
User
Rajasthan
53
360
Balmukund
Madhya Pradesh
53
361
Yohan Ninan
Karnataka
53
362
User
Tamil Nadu
53
363
User
Maharashtra
53
364
User
Delhi
53
365
User
Rajasthan
53
366
User
Karnataka
53
367
Tanushi Pandey
Delhi
53
368
Happy
Andhra Pradesh
53
369
User
Maharashtra
53
370
User
Rajasthan
53
371
User
Chhattisgarh
53
372
User
West Bengal
53
373
User
Maharashtra
53
374
User
West Bengal
53
375
User
West Bengal
53
376
User
Maharashtra
53
377
User
Bihar
53
378
User
West Bengal
53
379
Parismita Das
Assam
53
380
Aasif Anis
Delhi
53
381
User
Delhi
53
382
NARENDRA AGRAWAL
Andhra Pradesh
53
383
User
Rajasthan
53
384
SHUBHAM KUMAR
Bihar
53
385
User
Maharashtra
53
386
User
Maharashtra
52
387
Rajesh Yadav
Uttar Pradesh
52
388
User
Maharashtra
52
389
User
Tamil Nadu
52
390
Anubhav Bora
Uttarakhand
52
391
User
Rajasthan
52
392
Swarna Kanti Hazra
West Bengal
52
393
User
Uttar Pradesh
52
394
User
Rajasthan
52
395
Eshan
Madhya Pradesh
52
396
User
Tamil Nadu
52
397
Duke Das
Karnataka
52
398
User
Rajasthan
52
399
User
Maharashtra
52
400
PRADEEP KUMAR
Telangana
52
401
User
Tamil Nadu
52
402
User
Telangana
52
403
User
Odisha
52
404
User
Maharashtra
52
405
User
West Bengal
52
406
User
Odisha
52
407
User
Delhi
52
408
User
Maharashtra
52
409
User
Haryana
51
410
User
Kerala
51
411
User
West Bengal
51
412
User
West Bengal
51
413
User
Bihar
51
414
User
West Bengal
51
415
User
Chhattisgarh
51
416
User
West Bengal
51
417
srihari
Tamil Nadu
51
418
User
Delhi
51
420
Soham Gupta
Gujarat
51
421
Adarshdeep Singh
Karnataka
51
422
Keshav
Haryana
51
423
Prerna Pahwa
Delhi
51
424
User
Telangana
51
426
User
West Bengal
51
427
User
Gujarat
51
428
Prem Agrawal
Maharashtra
51
429
User
Maharashtra
51
430
Yedukrishnan P
Maharashtra
51
431
User
West Bengal
51
432
Harpreet Bagga
Uttar Pradesh
51
433
User
Rajasthan
51
435
tushar w
Maharashtra
50
436
User
Tamil Nadu
50
437
User
Delhi
50
438
User
West Bengal
50
439
Kishan
Gujarat
50
440
User
Delhi
50
441
Preet Khalsa
Rajasthan
50
442
User
Gujarat
50
443
User
Maharashtra
50
446
User
Jharkhand
50
447
User
West Bengal
50
448
User
Tamil Nadu
50
449
User
Maharashtra
50
450
User
Tamil Nadu
50
451
User
Uttar Pradesh
50
452
User
Haryana
50
453
User
Karnataka
50
454
User
West Bengal
50
455
Soumitra Chattopadhyay
Maharashtra
50
456
bheesetti rajkumar
Andhra Pradesh
50
457
Abhiraj Aditya
West Bengal
50
458
User
Maharashtra
49
459
Sri
Kerala
49
460
User
Gujarat
49
461
User
Tamil Nadu
49
462
Anjana C H
Maharashtra
49
463
User
Karnataka
49
464
User
Telangana
49
465
User
Delhi
49
466
YUVARAJ V
Telangana
49
467
User
Haryana
49
468
User
Maharashtra
49
469
User
Karnataka
49
470
User
Rajasthan
49
471
User
Gujarat
49
472
User
Uttar Pradesh
49
473
User
Delhi
49
474
User
Bihar
49
475
User
Tamil Nadu
49
476
User
Punjab
49
477
User
Maharashtra
49
478
User
Uttar Pradesh
49
479
Ankur Kothari
Rajasthan
49
480
User
Maharashtra
49
481
User
Chhattisgarh
49
482
Anuraag Kunchala
Uttar Pradesh
49
483
Harsh Sarda
Haryana
49
484
anshul sahu
Madhya Pradesh
49
485
Apurva Bomanwar
Kerala
49
486
Arka Provo Mukhopadhyay
Rajasthan
49
487
User
Gujarat
48
488
Andromeda
Haryana
48
489
User
Tamil Nadu
48
490
User
Tamil Nadu
48
491
sourabh yadav
Uttar Pradesh
48
492
User
Telangana
48
493
User
Telangana
48
494
User
Haryana
48
495
User
West Bengal
48
496
User
Puducherry
48
497
User
Telangana
48
498
RP Gaur
Uttar Pradesh
48
500
Sairam Sablo
Tamil Nadu
48
501
User
Gujarat
48
502
Meghana
Maharashtra
48
503
User
Maharashtra
48
504
User
Tamil Nadu
48
505
Cannibal Hamster
Telangana
48
506
User
Delhi
48
507
User
Delhi
48
508
User
Maharashtra
48
509
User
Maharashtra
48
510
User
Uttar Pradesh
48
511
User
Tamil Nadu
48
512
madhu mitha
Tamil Nadu
48
513
Banlam Nongkhlaw
Meghalaya
47
515
Avengers
Delhi
47
516
User
Karnataka
47
517
axolotl-man
Karnataka
47
518
User
Tamil Nadu
47
519
User
Karnataka
47
520
User
Uttar Pradesh
47
521
User
Bihar
47
522
kamal
Rajasthan
47
523
User
Maharashtra
47
524
User
Maharashtra
47
525
User
Maharashtra
47
526
Aakarsh Chauhan
Assam
47
527
RICK
Uttar Pradesh
47
528
User
Uttar Pradesh
47
529
User
Delhi
47
530
User
West Bengal
47
531
User
Bihar
47
532
User
Tamil Nadu
47
533
User
Tamil Nadu
46
534
User
Maharashtra
46
536
User
Kerala
46
537
User
Tamil Nadu
46
538
Amogh Halageri
Karnataka
46
539
User
Telangana
46
540
Jaideep Bankoti
Uttar Pradesh
46
541
Awesomefire
Andhra Pradesh
46
542
Jagjit Singh
West Bengal
46
543
Akriti Verma
Jammu and Kashmir
46
544
User
Chhattisgarh
46
545
User
West Bengal
46
546
User
Uttar Pradesh
46
548
User
Maharashtra
46
549
User
Uttar Pradesh
46
550
User
Delhi
46
551
User
Karnataka
46
552
User
West Bengal
46
553
User
Odisha
46
554
User
Bihar
46
555
User
Delhi
46
556
User
Maharashtra
46
557
User
Telangana
46
558
User
West Bengal
46
559
User
Tamil Nadu
46
560
User
Rajasthan
46
561
User
Karnataka
46
562
User
Tamil Nadu
46
563
User
Bihar
45
564
Abhishek Sanghvi
Madhya Pradesh
45
565
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Trust My Tips
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A Web App for Mathematics Training

Do you want to be fast at mental math? Many people do, but the options for doing the necessary exercises are simply too cumbersome for all but the most dedicated of trainees. In physical fitness, many people are interested in training their bodies but allocating the time, energy, and money for it is a significant obstacle. Likewise, lugging around books and whatnot for math practice is a threshold that just doesn't meet the standards of modern life.

Training yourself to be skilled at mental math needs to be quick and convenient. mathtrainer.org is a web app that works in your browser rather than a program you have to download and install on your computer or phone. This allows users to try and use the app without having to install new software. As a web app, updates are also much simpler. There is no need to download endless updates—the website will always be the most current version.

You can access a web app from any device connected to the internet and equipped with a web browser, including smartphones, tablets, and desktop computers. Moreover, you are free to use whichever browser you prefer, including Google Chrome, Safari, Firefox, and others. Google Chrome is the recommended browser for the best maths training since it tends to lead the pack in supporting the latest web technologies.

Math Trainer is designed to offer a similar experience regardless of what you’re using to access it, whether it be Android, iOS, Windows, or another operating system. Though an on-screen touch keyboard will appear on mobile devices, you may prefer to use the app on a desktop with a keyboard. Hopefully the advantages of a web app for convenient mathematics training are apparent.

Another part of making the app easy to use is eliminating the need for signing up and logging in. Users can get started with their math training as soon as they click the start button on this page. After progressing to higher levels in the app, your progress is automatically saved so long as your return to the site through the same browser.


What's a Mental Math Tip?

A mental math tip is a sequence of steps that can be taken to solve a math problem in your head. Click the arrow below to see an example for the following problem:

÷
984
3
328

A tip like this one is available for every problem in Math Trainer, so there's always help if you get stuck. With enough practice, you'll be able to predict what the tips will say—you'll have learned mental math!


Get Better at Mental Math

The ability to quickly perform mental calculations offers advantages in certain circumstances. But even without applications, getting better at mental math is a great way to stimulate one’s mind. It develops better number sense and intuition for quantifying the world around us. Practicing mental calculation will strengthen your foundation for learning more advanced maths topics.

Nonetheless, the tangible benefits of improving at mental math are many. It is certainly expected that educated people are able to do simple arithmetic without having to pull out a calculator. An inability to do so may reflect poorly on you, while being well-practiced in mental calculation will leave your contemporaries impressed. In many scientific and technical circles, mental math ability is even more highly regarded.

For students, mental calculation speed will often have a direct impact on math and science test scores. At all grade levels, it is not sufficient to know how to solve math problems when tests have a time limit on them. The highest-scoring test takers are able to answer questions both correctly and efficiently. Improving mental math skills will only benefit a student’s academic career.

Calculating the solution to an arithmetic problem in your head is often faster than pulling out a device to tell you the answer. For example, figuring out how much to tip a server at a restaurant is a straightforward arithmetic problem that many people are unable to perform without a calculator. By training your brain to solve basic math problems, you can save time in situations like these.

Mental math can also be relied upon when calculation devices are not available. Even with the conveniences of modern life, we occasionally find ourselves without access to our cell phones or other capable devices. A mind skilled in mental math is always available to you.

Finally, getting better at mental math enables a quick estimate and sanity check on results obtained from calculators. While computers are extremely reliable at solving math problems, there is always the risk of incorrectly inputting the problem to the computer. By getting better at mental mathematics, you will develop an intuition for whether the results of calculators make sense.

In fact, the ability to estimate is often sufficient to avoid using calculators altogether. While the use of computers is widespread, estimation is an increasingly valued skill in many industries. There are many situations where complex math will eventually be required, but a preliminary estimate is needed quickly. A major boost to productivity!


Use a Math Trainer

Mental math ability is a lot like physical fitness training. You may be out of shape in the beginning, but with diligent training you can and will improve. Initially you might not enjoy the exercise, but you will reap significant rewards for your effort. As you become more fit, you’ll begin to enjoy the activity much more. If you are serious about it, your mental calculation fitness could become a source of energy, galvanizing you to face the challenges of life with enthusiasm.

In physical training, you break down the fibers in your muscles during a workout session. Your muscles actually sustain tiny tears during resistance training exercises. While you rest afterwards, your body repairs the damage, rebuilding the fibers thicker and stronger.

A similar process is believed to occur for cognitive tasks. A 2016 study found "extensive evidence that brain-training interventions improve performance on the trained tasks".1 Therefore you can expect training your brain to answer mental math questions will lead to improved performance over time.

In the context of physical fitness, a "trainer" often refers to a trained professional who guides the workout and recovery process. Personal trainers are tasked with assessing a trainee's level of ability, prescribing an exercise regimen, and offering feedback as the training goes along. The word "trainer" could also refer to a system that automates the role of a personal trainer. Many aerobic exercise machines today offer interactive training programs with feedback and analysis of a user's performance.

A math trainer is needed for optimal math fitness. Like in physical fitness, the trainer should be compatible with users at a variety of skill levels and should guide them to the next level. It should give an accurate assessment of a user's strengths and weakness, as well as offer helpful feedback on where to focus one's efforts. Learning the ropes of mental maths with a math trainer should be a seamless, rewarding journey to ever-greater abilities.


Mental Calculation

Mental calculation, or mental math, is performing arithmetical calculations without the aid of tools or supplies. As opposed to using a calculator or pencil and paper, mental math is performed entirely in one’s head.

People use mental calculation when computation aides are not available, when it is faster to do so, or when they wish to practice, show off, or participate in mental math competitions. Most people perform basic mental calculation using elementary arithmetic on a daily basis. An inability to calculate mentally is a serious obstacle to many common tasks.

In U.S. schools, mental calculation is taught only for the most elementary arithmetic, such as single-digit addition and multiplication of two numbers between 0 and 12. To solve addition problems involving multiple digits, you are taught to add columns of digits from right to left, carrying the tens digit if the column sum exceeds 9. For example, how would you approach this addition problem?

Example of two-digit
addition problem

If you were trained like many of us were, you’d add the right column to obtain 12. Since that’s two digits, you’d write the 2 under the right column and carry the 10 by writing a 1 above the left column. Finally, you’d add the two tens digits and the carried 1 to obtain the answer, 52.

To solve an addition problem mentally, it’s best to add the columns from left to right. In our example, you could add the tens digit of the second number, 30, to the first number, 14, to obtain 44. This is easier than the full problem because you’re just doing one mental calculation and tacking on the 4 from the 14 as the singles digit. Then you’d add the remaining ones digit of the second number, 8, to 44 to arrive at the answer, 52.

Which approach seems simpler to you? Can you do the first approach without pulling out a pencil and paper? It turns out the same advantages of left-to-right addition apply to much larger numbers as well. It’s unlikely that difficult addition problems can be solved right to left without needing to write it all out, which of course is more time consuming.

Mental math should be distinguished from the memorization of math facts such as multiplication tables. A foundation of memorized answers to simple math problems will make mental math easier, but performing maths in your head requires both memorized facts and the manipulation of numbers and operations to solve problems. This combination of skill and memory allows us to solve far more complex math questions than can be answered with readily-memorized math facts.

Many mental math tricks are specific to particular numbers or types of problems, usually dependent on the base of the number system used. In the decimal numeral system, for example, it is trivially easy to multiply by 10—just add a 0 to the end of the number. This mental math trick wouldn’t work in the hexadecimal numeral system, though, because the base is 16 instead of 10.

Therefore mental calculation is the ability to manipulate complex arithmetic problems in such a way that they can be resolved using simple memorized math facts.


Arithmetic

Arithmetic is the branch of mathematics concerning basic number operations: addition, subtraction, multiplication, and division. As kids, we are taught to do arithmetic because real-world math problems depend on a mastery of elementary arithmetic. Higher-level study of arithmetic and the integers, or whole numbers, is known as number theory.

Though the math kids initially study is arithmetic, the word is rarely used in this context anymore. Originally it comes from the Greek arithmos, meaning “number”. It has however been included in the “three Rs” of elementary Western education: reading, writing, and arithmetic.

There is evidence prehistoric humans were using arithmetic as hunter-gatherers. Archaeologists have uncovered a tally stick, believed to be over 20,000 years old, which may exhibit the earliest known sequences of prime numbers. An understanding of prime numbers, which are only divisible by themselves and the number 1, requires knowledge of the operation in arithmetic known as division.

From tally marks came base-10 numerals such as those used in Egypt over 5,000 years ago. Number systems based on 10 probably arose because humans have ten “digits” as fingers on their hands (or toes on their feet). A later advance in arithmetic was positional notation, which allowed the same symbols to represent different magnitudes depending on their position in the written number. These numeral systems allowed complex arithmetic to be communicated, recorded, and applied to the challenges faced by our ancestors.

The basic operation of arithmetic is addition. It combines two or more numbers into one, the sum of the terms. The terms can be added in any order, which is known as the commutative property of arithmetic. On a number line, the sum of two numbers is the total distance from zero covered by both numbers.

The inverse arithmetical operation of addition is subtraction. It finds the difference between two numbers. Subtraction is not commutative because the order of the numbers determines whether the answer is positive or negative. On a number line, the difference between two numbers is the distance between their positions.

A second basic operation of arithmetic is multiplication, which scales a number by another number. This second number is called a factor. Like addition, multiplication is commutative—you can change the order of the factors and still get the same answer. Multiplication on a number line can be viewed as adding the first number a total number of times equal to the second factor.

Finally, division is an arithmetical operation that is essentially the inverse of multiplication. Rather than scaling a number, it is divided into a number of pieces equal to the second number. Dividing by the number 0 is not defined in arithmetic because dividing something into zero pieces is impossible.

Basic arithmetic allows us to evaluate the answers to an unlimited number of mathematical expressions. Arithmetical expressions can be purely mathematical, as in 2 + 2, or they can represent quantities in the physical world, such as two items plus two more. Understanding the laws of arithmetic is tremendously powerful.


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