Leaderboard
Georgia
Rank
Level
1
Peter Andersson
Atlanta
87
2
Blaziken Boi
Ringgold
86
3
Jessica He
Atlanta
84
4
Srikar Satluri
East Cobb
79
5
rohaN 🐖
Marietta
79
6
Parsa Novin
East Cobb
79
7
User
Atlanta
77
8
User
Loganville
74
9
User
East Cobb
70
10
Dustin
Atlanta
69
11
User
Marietta
64
12
User
Ringgold
62
13
User
Acworth
59
14
mishmash
Atlanta
58
15
User
Loganville
57
16
User
Moultrie
57
17
JC
Atlanta
56
18
User
Lilburn
56
19
User
East Cobb
56
20
God Level
East Cobb
54
21
weird
Sandy Springs
52
22
User
Atlanta
52
23
User
Roswell
52
24
User
North Druid Hills
51
25
User
Atlanta
50
26
User
Atlanta
48
27
Pedro Rossello
Sandy Springs
48
28
User
Atlanta
48
29
User
Atlanta
48
30
Jake Hoffman
Atlanta
48
31
User
Atlanta
46
32
User
Atlanta
46
33
User
Atlanta
45
34
Terpski624
Atlanta
45
35
User
Johns Creek
44
36
Abhishek Birhade
Atlanta
43
37
User
Richmond Hill
40
38
User
Atlanta
40
39
User
Alpharetta
40
40
User
Fort Valley
38
41
User
East Cobb
37
42
User
Atlanta
37
43
GURU PRASAD IYER.V
Atlanta
37
44
User
Atlanta
36
45
Yuan Yin
Atlanta
36
46
Chase Brooks
Atlanta
36
47
User
Marietta
36
48
User
Atlanta
35
49
User
Ranburne
35
50
User
Alpharetta
35
51
User
Alpharetta
35
52
User
Sandy Springs
34
53
User
Kennesaw
34
54
User
Athens
34
55
User
Atlanta
34
56
User
Atlanta
33
57
User
Atlanta
33
58
User
Roswell
33
59
User
Norcross
33
60
User
Columbus
33
61
User
Atlanta
32
62
User
Druid Hills
32
63
User
Mcdonough
31
64
User
Acworth
31
65
User
Sandersville
31
66
User
Atlanta
31
67
User
Druid Hills
31
68
User
Atlanta
31
69
User
Atlanta
31
70
User
Mcdonough
31
71
User
Chamblee
30
72
User
Sandy Springs
30
73
User
Forsyth
30
74
User
East Point
30
75
User
Columbus
30
76
User
Atlanta
30
77
User
Atlanta
29
78
User
Atlanta
29
79
User
Albany
29
80
User
Sandy Springs
28
81
User
Atlanta
28
82
User
Warner Robins
28
83
Stephen
Atlanta
28
84
Michael Jurado
Athens
28
85
User
Atlanta
28
86
User
Atlanta
27
87
User
Atlanta
27
88
User
North Druid Hills
27
89
User
Atlanta
27
90
User
Atlanta
27
91
Shashank Umesh
Marietta
26
92
User
Sandy Springs
26
93
Garret Gallion
Atlanta
26
94
User
Holly Springs
26
95
User
Druid Hills
26
96
User
Norcross
26
97
User
Atlanta
25
98
User
Athens
25
99
User
Atlanta
25
100
User
Fort Valley
25
101
User
Macon
25
102
hollists
Atlanta
25
103
User
Alpharetta
25
104
anwesha banerjee
Druid Hills
24
105
User
Tifton
24
106
User
Doraville
24
107
User
Atlanta
24
108
User
Lawrenceville
23
109
User
Atlanta
23
110
User
Dacula
23
111
User
Atlanta
23
112
User
Macon
23
113
User
Fort Valley
23
114
User
Atlanta
23
115
User
Acworth
22
116
User
Candler-Mcafee
22
117
User
Brookhaven
22
118
User
Johns Creek
22
119
User
Athens
22
120
User
Decatur
22
121
User
Macon
22
122
User
Atlanta
22
123
User
Atlanta
22
124
User
Atlanta
22
125
User
Druid Hills
22
126
Yvo
Martinez
22
127
User
Atlanta
22
128
Ariana Morgan
Grovetown
22
129
User
Atlanta
22
130
User
Atlanta
22
131
User
Albany
21
132
User
Suwanee
21
133
User
Athens
21
134
User
Milledgeville
21
135
Kirby Cyber
Woodstock
21
136
User
Fort Valley
21
137
Noah Davis
Stockbridge
21
138
User
Atlanta
21
139
User
North Druid Hills
21
140
User
Douglasville
21
141
User
Snellville
21
142
User
Norcross
21
143
User
Doraville
21
144
User
Atlanta
21
145
User
Atlanta
21
146
User
Atlanta
21
147
Taylor Lawson
Atlanta
21
148
User
Atlanta
21
149
User
Atlanta
21
150
KeNorian Davis
Mcintyre
21
151
Pigmilk Man
Lilburn
21
152
User
Columbus
21
153
User
Atlanta
20
154
User
Johns Creek
20
155
User
Alpharetta
20
156
User
Atlanta
20
157
User
Macon
20
158
User
Woodstock
20
159
User
Dunwoody
20
160
User
Atlanta
20
161
User
Atlanta
20
162
User
Atlanta
20
163
User
Atlanta
20
164
User
Atlanta
20
165
User
Atlanta
20
166
cedric
Fort Valley
20
167
User
Holly Springs
20
168
User
East Cobb
20
169
User
Marietta
19
170
User
Dunwoody
19
171
User
Calhoun
19
172
User
Doraville
19
173
User
Duluth
19
174
User
Conyers
19
175
User
Atlanta
19
176
User
Conyers
19
177
User
East Cobb
19
178
User
Atlanta
19
179
User
Fort Valley
19
180
User
Mcintyre
19
181
User
Sandy Springs
18
182
User
Marietta
18
183
User
Warner Robins
18
184
User
East Cobb
18
185
User
Decatur
18
186
User
Marietta
18
187
Allen Gonzalez
Doraville
18
188
User
Atlanta
18
189
User
Atlanta
18
190
User
Atlanta
18
191
User
Atlanta
18
192
User
Atlanta
18
193
User
East Point
18
194
User
Marietta
18
195
User
Stonecrest
17
196
User
Atlanta
17
197
User
Marietta
17
198
User
Commerce
17
199
Patrick
Atlanta
17
200
User
Suwanee
17
201
User
Stockbridge
17
202
User
Bremen
17
203
User
Atlanta
17
204
User
Doraville
17
205
User
Atlanta
17
206
User
Atlanta
17
207
User
Atlanta
17
208
User
Atlanta
17
209
User
Duluth
17
210
User
Byron
17
211
User
Sandy Springs
17
212
User
Stockbridge
17
213
Bas Weijerman
Atlanta
17
214
User
Atlanta
17
215
User
Atlanta
17
216
User
Atlanta
16
217
Matthew Wise
Augusta
16
218
User
Lilburn
16
219
User
Atlanta
16
220
User
Atlanta
16
221
User
Atlanta
16
222
User
Atlanta
16
223
User
Atlanta
16
224
Cheronda Reynolds
Irwinton
16
225
User
Sandy Springs
15
226
User
Cumming
15
227
User
Atlanta
15
228
User
Doraville
15
229
User
North Decatur
15
230
User
Atlanta
15
231
User
Atlanta
15
232
User
Atlanta
15
233
User
Fort Valley
15
234
User
Atlanta
15
235
User
Buford
15
236
User
Atlanta
15
237
User
Lawrenceville
15
238
User
Marietta
15
239
User
Atlanta
15
240
User
Griffin
14
241
User
Griffin
14
242
User
East Cobb
14
243
User
Atlanta
14
244
User
Fort Valley
14
245
User
Fort Valley
14
246
Irving
Conyers
14
247
User
Conyers
14
248
User
Atlanta
14
249
User
Atlanta
14
250
User
Atlanta
14
251
User
Pooler
14
252
User
Atlanta
14
253
User
Cumming
14
254
User
Statesboro
14
255
User
Byron
14
256
User
Vinings
14
257
User
Brookhaven
14
258
User
Marietta
14
259
User
Atlanta
13
260
User
Atlanta
13
261
Mark Simmons
Woodstock
13
262
User
Athens
13
263
User
Marietta
13
264
User
Atlanta
13
265
User
Macon
13
266
User
Atlanta
13
267
User
Atlanta
13
268
User
Atlanta
13
269
User
Atlanta
13
270
User
Atlanta
13
271
User
Atlanta
13
272
User
Atlanta
13
273
User
Fort Valley
13
274
User
City
13
275
User
Druid Hills
13
276
User
Augusta
13
277
User
Conyers
13
278
User
Atlanta
13
279
User
Atlanta
13
280
User
Druid Hills
13
281
Guillermo Reynoso-Gonzalez
Irwinton
13
282
User
East Point
12
283
User
Riverdale
12
284
User
Atlanta
12
285
User
Rome
12
286
User
Fort Valley
12
287
Trent Davis
Atlanta
12
288
User
Macon
12
289
User
Atlanta
12
290
User
Atlanta
12
291
User
Atlanta
12
292
User
Atlanta
12
293
User
Fort Valley
12
294
User
East Cobb
12
295
User
Atlanta
12
296
User
Demorest
12
297
User
Jonesboro
12
298
User
Atlanta
11
299
User
Johns Creek
11
300
User
North Druid Hills
11
301
User
Columbus
11
302
User
Conyers
11
303
User
Woodstock
11
304
User
Druid Hills
11
305
User
Atlanta
11
306
User
Macon
11
307
User
Roswell
11
308
User
Atlanta
11
309
User
Atlanta
11
310
User
Atlanta
11
311
User
Atlanta
11
312
User
Atlanta
11
313
User
Atlanta
11
314
User
Atlanta
11
315
User
Atlanta
11
316
User
Alpharetta
11
317
User
Savannah
11
318
User
Atlanta
11
319
User
Atlanta
11
320
Adriana Garcia Ceja
Atlanta
11
321
User
Warner Robins
11
322
MJ
Dekalb County
11
323
User
Powder Springs
10
324
User
Columbus
10
325
User
Johns Creek
10
326
User
Atlanta
10
327
User
Roswell
10
328
User
Athens
10
329
User
Atlanta
10
330
User
Atlanta
10
331
User
Atlanta
10
332
User
Dunwoody
10
333
User
Atlanta
10
334
User
Atlanta
10
335
User
Atlanta
10
336
User
Atlanta
10
337
User
Atlanta
10
338
User
Watkinsville
10
339
User
Holly Springs
10
340
User
Atlanta
10
341
User
Acworth
10
342
User
Johns Creek
10
343
User
Atlanta
10
344
User
Atlanta
10
345
User
Savannah
10
346
User
Sandy Springs
9
347
User
Columbus
9
348
User
Atlanta
9
349
User
Brookhaven
9
350
User
Roswell
9
351
User
Gainesville
9
352
User
East Cobb
9
353
User
Atlanta
9
354
User
Johns Creek
9
355
User
Milton
9
356
Bobby J
Gainesville
9
357
Aaliyah McDowell
Hiram
9
358
User
Atlanta
9
359
User
Atlanta
9
360
User
Macon
9
361
User
Duluth
9
362
User
Atlanta
9
363
User
Atlanta
9
364
User
Atlanta
9
365
User
Atlanta
9
366
User
Atlanta
9
367
edward
Fort Valley
9
368
User
Fort Valley
9
369
User
Conyers
9
370
User
Statesboro
9
371
User
Statesboro
9
372
User
Roswell
9
373
User
Carrollton
9
374
User
Jefferson
9
375
User
Fayetteville
9
376
User
Roswell
9
377
User
Atlanta
8
378
User
Lawrenceville
8
379
User
Atlanta
8
380
User
Johns Creek
8
381
User
Atlanta
8
382
User
Woodstock
8
383
User
Powder Springs
8
384
User
Roswell
8
385
User
Dunwoody
8
386
User
Atlanta
8
387
User
Atlanta
8
388
User
Atlanta
8
389
User
Atlanta
8
390
User
Atlanta
8
391
User
Atlanta
8
392
User
Atlanta
8
393
User
Statesboro
8
394
User
Atlanta
8
395
User
Atlanta
8
396
User
Peachtree Corners
8
397
User
Roswell
7
398
User
East Cobb
7
399
User
Woodstock
7
400
User
Atlanta
7
401
Sandreka Brown
Powder Springs
7
402
User
East Cobb
7
403
User
Atlanta
7
404
User
Roswell
7
405
User
Atlanta
7
406
User
Atlanta
7
407
User
Athens
7
408
User
Druid Hills
7
409
User
North High Shoals
7
410
User
Atlanta
7
411
Janine Jasmin
Mcdonough
7
412
User
Dunwoody
7
413
User
Macon
7
414
User
Moultrie
7
415
User
Atlanta
7
416
User
Holly Springs
7
417
User
Statesboro
7
418
User
Alpharetta
7
419
User
Stockbridge
7
420
User
Druid Hills
7
421
User
Atlanta
7
422
User
Atlanta
7
423
User
Evans
7
424
joshua wood
Irwinton
7
425
User
Smyrna
6
426
User
Lawrenceville
6
427
User
Marietta
6
428
User
Canton
6
429
User
Valdosta
6
430
User
Atlanta
6
431
User
Demorest
6
432
User
Palmetto
6
433
User
Dunwoody
6
434
User
Atlanta
6
435
User
Woodstock
6
436
User
Duluth
6
437
User
Atlanta
6
438
User
Atlanta
6
439
User
Atlanta
6
440
User
Augusta
6
441
User
Moultrie
6
442
User
Atlanta
6
443
User
Atlanta
6
444
User
Atlanta
6
445
User
Atlanta
6
446
User
Cumming
6
447
User
Fort Valley
6
448
User
Atlanta
6
449
d j
Fort Valley
6
450
User
Conyers
6
451
User
Atlanta
6
452
User
Atlanta
6
453
User
Fort Valley
6
454
User
Dacula
6
455
User
Statesboro
6
456
User
Lilburn
6
457
User
Johns Creek
6
458
User
Conyers
6
459
User
Atlanta
6
460
User
Atlanta
5
461
User
Duluth
5
462
User
East Cobb
5
463
User
Atlanta
5
464
User
Johns Creek
5
465
User
Griffin
5
466
User
Griffin
5
467
User
Atlanta
5
468
Michelle Li
Athens
5
469
User
Johns Creek
5
470
User
Atlanta
5
471
User
Acworth
5
472
User
Woodstock
5
473
User
Union City
5
474
User
Riverdale
5
475
User
Atlanta
5
476
User
Fort Valley
5
477
User
Brookhaven
5
478
User
Canton
5
479
User
Mableton
5
480
User
Riverdale
5
481
User
Atlanta
5
482
User
Dunwoody
5
483
User
Porterdale
5
484
User
Savannah
5
485
User
Atlanta
5
486
User
Atlanta
5
487
User
Atlanta
5
488
User
Atlanta
5
489
User
Atlanta
5
490
User
Atlanta
5
491
User
Conyers
5
492
User
Atlanta
5
493
User
Dock Junction
5
494
User
Conyers
5
495
User
Statesboro
5
496
hunter
Statesboro
5
497
User
Statesboro
5
498
Anne Jean
Atlanta
5
499
User
Augusta
5
500
User
Johns Creek
5
501
User
Columbus
5
502
User
Atlanta
5
503
User
Tyrone
4
504
User
Atlanta
4
505
User
Mcdonough
4
506
User
Milton
4
507
User
Cumming
4
508
User
Mcintyre
4
509
User
Atlanta
4
510
User
Ashburn
4
511
User
Athens
4
512
User
Atlanta
4
513
User
Bolingbroke
4
514
User
Atlanta
4
515
User
Suwanee
4
516
User
Atlanta
4
517
User
Conyers
4
518
User
Statesboro
4
519
User
Atlanta
4
520
User
Conyers
4
521
User
Conyers
4
522
User
Macon
4
523
User
Lawrenceville
4
524
User
Atlanta
4
525
DongJun Kim
Atlanta
4
526
User
Atlanta
4
527
User
Atlanta
4
528
User
Dunwoody
4
529
User
Buford
4
530
User
Moultrie
4
531
User
Dalton
4
532
User
Suwanee
4
533
User
Atlanta
4
534
User
Atlanta
4
535
User
Atlanta
4
536
User
Atlanta
4
537
User
Atlanta
4
538
User
Atlanta
4
539
User
Atlanta
4
540
User
Lawrenceville
4
541
User
Atlanta
4
542
User
Fort Valley
4
543
User
Atlanta
4
544
User
Conyers
4
545
User
Grayson
4
546
User
Statesboro
4
547
User
Statesboro
4
548
User
Suwanee
4
549
User
Atlanta
4
550
User
Statesboro
4
551
User
Suwanee
4
552
User
East Cobb
4
553
User
Duluth
4
554
User
Tucker
4
555
User
Atlanta
4
556
User
Cumming
4
557
User
Atlanta
4
558
User
Atlanta
4
559
User
Fort Valley
4
560
User
Irwinton
4
561
User
Irwinton
4
562
User
Union City
4
563
User
Johns Creek
3
564
User
Atlanta
3
565
User
Alpharetta
3
566
User
Atlanta
3
567
J B
Atlanta
3
568
User
Atlanta
3
569
User
East Cobb
3
570
User
Woodstock
3
571
User
Sandy Springs
3
572
User
Atlanta
3
573
User
Conyers
3
574
User
Atlanta
3
575
User
Rome
3
576
User
Druid Hills
3
577
User
Conyers
3
578
User
Dahlonega
3
579
User
Atlanta
3
580
Lothania Roberts
Statesboro
3
581
User
Athens
3
582
User
Atlanta
3
583
Dax
Smyrna
3
584
User
Woodstock
3
585
User
Atlanta
3
586
User
Atlanta
3
587
User
Duluth
3
588
User
Atlanta
3
589
User
Augusta
3
590
User
Atlanta
3
591
User
Stockbridge
3
592
User
Atlanta
3
593
User
Atlanta
3
594
User
Conyers
3
595
User
Atlanta
3
596
User
Atlanta
3
597
User
Atlanta
3
598
User
Johns Creek
3
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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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