Leaderboard
Georgia
Rank
Level
1
Cameron Moats
Lawrenceville
76
2
Jessica He
Atlanta
63
3
Srikar Satluri
East Cobb
61
4
rohaN 🐖
Marietta
61
5
Parsa Novin
East Cobb
60
6
User
Atlanta
59
7
User
Atlanta
58
8
User
Martinez
57
9
User
Austell
56
10
Howard Clark
Atlanta
54
11
User
Atlanta
53
12
User
East Cobb
53
13
User
Sandy Springs
52
14
User
Lithia Springs
52
15
User
Loganville
50
16
A kunk
Grovetown
49
17
User
Atlanta
49
18
Matt C
Atlanta
49
19
User
Marietta
48
20
God Level
East Cobb
44
21
Kai Shen
Vinings
44
22
User
Acworth
44
23
User
Loganville
43
24
User
Suwanee
43
25
JC
Atlanta
43
26
weird
Sandy Springs
39
27
Blaziken Boi
Ringgold
39
28
User
Martinez
37
29
User
Marietta
37
30
User
Tucker
37
31
Peter Andersson
Atlanta
36
32
User
Atlanta
35
33
User
Ringgold
35
34
Pedro Rossello
Sandy Springs
34
35
User
Forest Park
34
36
User
Atlanta
33
37
User
Atlanta
33
38
Abhishek Birhade
Atlanta
32
39
User
Suwanee
30
40
User
Atlanta
30
41
User
Atlanta
29
42
User
Atlanta
29
43
User
Atlanta
28
44
User
Atlanta
28
45
User
Atlanta
28
46
User
Sandy Springs
26
47
User
Atlanta
26
48
User
Sandy Springs
25
49
User
Tucker
25
50
User
Moultrie
24
51
User
Lilburn
24
52
User
Chamblee
24
53
User
Atlanta
24
54
Soham Bose
Atlanta
24
55
User
Mcdonough
24
56
User
Lagrange
23
57
User
Richmond Hill
23
58
User
Acworth
23
59
User
Atlanta
23
60
User
Sandy Springs
22
61
User
Sandersville
22
62
User
Atlanta
22
63
mishmash
Atlanta
22
64
User
Atlanta
22
65
User
Johns Creek
22
66
User
Atlanta
22
67
User
East Cobb
21
68
User
East Cobb
21
69
User
Atlanta
21
70
User
Druid Hills
21
71
SavageBall21 ON YT
Stockbridge
20
72
User
Sandy Springs
20
73
Shashank Umesh
Marietta
20
74
User
Atlanta
20
75
User
Atlanta
20
76
User
Marietta
20
77
User
Atlanta
19
78
User
Atlanta
19
79
User
Sandy Springs
19
80
User
Atlanta
18
81
User
Atlanta
18
82
User
Sandy Springs
18
83
User
Atlanta
18
84
User
Ranburne
18
85
Jake Hoffman
Atlanta
18
86
User
Lawrenceville
18
87
User
Kennesaw
18
88
User
Roswell
18
89
User
North Druid Hills
17
90
Terpski624
Atlanta
17
91
User
Atlanta
17
92
Luis Ramirez-Herrera
Valdosta
17
93
User
Roswell
17
94
User
Norcross
16
95
GURU PRASAD IYER.V
Atlanta
16
96
User
Acworth
16
97
User
Atlanta
16
98
User
Alpharetta
16
99
User
Atlanta
16
100
User
Atlanta
16
101
User
Athens
16
102
Yuan Yin
Atlanta
16
103
User
North Decatur
16
104
User
Candler-Mcafee
16
105
User
Albany
15
106
User
Forsyth
15
107
User
Chickamauga
15
108
User
Fort Valley
15
109
User
Johns Creek
15
110
User
East Point
15
111
User
Atlanta
15
112
User
Atlanta
15
113
User
Atlanta
15
114
anwesha banerjee
Druid Hills
14
115
User
Atlanta
14
116
User
Suwanee
14
117
Garret Gallion
Atlanta
14
118
User
Warner Robins
14
119
User
Stonecrest
14
120
User
Tifton
14
121
User
Atlanta
13
122
User
Athens
13
123
User
Atlanta
13
124
User
Sandy Springs
13
125
User
Dacula
13
126
User
Atlanta
13
127
User
Marietta
13
128
User
Alpharetta
13
129
User
Commerce
13
130
User
Athens
12
131
User
Brookhaven
12
132
User
Johns Creek
12
133
User
Atlanta
12
134
User
Atlanta
12
135
User
Milledgeville
12
136
User
Statham
12
137
Kirby Cyber
Woodstock
12
138
Patrick
Atlanta
12
139
User
Suwanee
12
140
User
Druid Hills
12
141
User
Roswell
12
142
User
Thomasville
11
143
User
Fort Valley
11
144
User
Atlanta
11
145
User
Johns Creek
11
146
User
Atlanta
11
147
User
Johns Creek
11
148
User
Atlanta
11
149
User
Atlanta
11
150
User
Athens
11
151
Stephen
Atlanta
11
152
User
Sandy Springs
11
153
Chase Brooks
Atlanta
11
154
User
Marietta
11
155
User
Atlanta
11
156
User
Atlanta
11
157
User
Decatur
10
158
Noah Davis
Stockbridge
10
159
User
Atlanta
10
160
User
Atlanta
10
161
User
North Druid Hills
10
162
User
Atlanta
10
163
User
Dunwoody
10
164
User
Atlanta
10
165
User
Douglasville
10
166
User
Marietta
10
167
User
Macon
10
168
User
Atlanta
10
169
User
Griffin
10
170
User
Atlanta
10
171
User
Holly Springs
10
172
User
Calhoun
10
173
User
Duluth
10
174
User
Doraville
10
175
User
East Point
10
176
User
Snellville
10
177
User
Doraville
9
178
User
Fort Valley
9
179
User
Macon
9
180
Michael Jurado
Athens
9
181
User
Lawrenceville
9
182
User
North Druid Hills
9
183
User
Columbus
9
184
User
Griffin
9
185
User
Atlanta
9
186
User
Atlanta
9
187
User
East Cobb
9
188
User
Macon
9
189
User
Columbus
9
190
User
Warner Robins
9
191
Mark Simmons
Woodstock
9
192
User
Duluth
9
193
User
Brookhaven
9
194
User
Alpharetta
9
195
User
Macon
9
196
User
East Cobb
9
197
User
Riverdale
9
198
User
Martinez
9
199
User
Atlanta
9
200
User
Johns Creek
9
201
User
Douglasville
9
202
User
Woodstock
9
203
User
Atlanta
9
204
User
Atlanta
9
205
User
North Druid Hills
9
206
User
Fort Valley
9
207
hollists
Atlanta
9
208
User
Roswell
9
209
User
Decatur
9
210
User
Powder Springs
8
211
User
Statesboro
8
212
User
Columbus
8
213
User
Norcross
8
214
User
Stockbridge
8
215
User
Doraville
8
216
User
Atlanta
8
217
User
Atlanta
8
218
User
Atlanta
8
219
Taylor Lawson
Atlanta
8
220
User
Atlanta
8
221
User
Johns Creek
8
222
User
Atlanta
8
223
User
Dunwoody
8
224
User
Atlanta
8
225
User
Atlanta
8
226
User
Atlanta
8
227
User
Atlanta
8
228
User
Atlanta
8
229
User
Atlanta
8
230
User
Atlanta
8
231
User
Atlanta
8
232
cedric
Fort Valley
8
233
User
Atlanta
8
234
User
Columbus
8
235
User
Atlanta
8
236
User
Bremen
8
237
User
Conyers
8
238
User
Cumming
8
239
User
Conyers
8
240
User
Rome
8
241
User
Atlanta
8
242
User
Atlanta
8
243
User
Druid Hills
8
244
User
Roswell
8
245
User
Atlanta
8
246
User
Atlanta
8
247
Irving
Conyers
8
248
User
Fort Valley
8
249
User
Marietta
8
250
User
Atlanta
7
251
User
Sandy Springs
7
252
Matthew Wise
Augusta
7
253
User
Holly Springs
7
254
User
Druid Hills
7
255
User
Lilburn
7
256
Allen Gonzalez
Doraville
7
257
User
Columbus
7
258
User
Atlanta
7
259
User
Atlanta
7
260
User
Atlanta
7
261
User
Atlanta
7
262
User
Atlanta
7
263
User
Atlanta
7
264
User
Athens
7
265
User
Brookhaven
7
266
User
Conyers
7
267
User
Woodstock
7
268
User
Atlanta
7
269
Yvo
Martinez
7
270
User
Atlanta
7
271
User
Marietta
7
272
User
East Cobb
7
273
User
Atlanta
7
274
Trent Davis
Atlanta
7
275
User
Athens
7
276
User
Georgetown
7
277
User
Fort Valley
7
278
User
Roswell
7
279
User
Gainesville
7
280
User
East Point
7
281
User
Doraville
7
282
User
Atlanta
7
283
User
Atlanta
7
284
User
Atlanta
7
285
User
Atlanta
7
286
User
East Cobb
7
287
User
Atlanta
7
288
User
East Cobb
7
289
User
Atlanta
7
290
User
Atlanta
6
291
User
Johns Creek
6
292
User
Atlanta
6
293
User
Atlanta
6
294
User
Atlanta
6
295
User
Atlanta
6
296
User
Atlanta
6
297
KeNorian Davis
Mcintyre
6
298
User
Lawrenceville
6
299
User
Macon
6
300
User
Druid Hills
6
301
User
Atlanta
6
302
User
Atlanta
6
303
User
Atlanta
6
304
User
Duluth
6
305
User
Atlanta
6
306
User
Macon
6
307
User
Atlanta
6
308
User
Doraville
6
309
User
North Decatur
6
310
User
Macon
6
311
User
Atlanta
6
312
User
Atlanta
6
313
User
Atlanta
6
314
User
Atlanta
6
315
User
Roswell
6
316
User
Atlanta
6
317
User
Atlanta
6
318
User
Atlanta
6
319
User
Atlanta
6
320
User
Atlanta
6
321
User
Milton
6
322
User
Mcintyre
6
323
User
East Cobb
6
324
User
Fort Valley
6
325
Bobby J
Gainesville
6
326
User
Byron
6
327
User
Fort Valley
6
328
Aaliyah McDowell
Hiram
6
329
User
Sandy Springs
6
330
User
Johns Creek
6
331
User
Woodstock
6
332
User
Atlanta
6
333
User
Atlanta
6
334
User
Atlanta
5
335
User
Roswell
5
336
User
Atlanta
5
337
User
Dunwoody
5
338
Sandreka Brown
Powder Springs
5
339
User
Atlanta
5
340
User
Atlanta
5
341
User
Atlanta
5
342
User
Atlanta
5
343
User
Atlanta
5
344
User
Atlanta
5
345
User
Atlanta
5
346
User
Atlanta
5
347
User
Atlanta
5
348
User
Atlanta
5
349
User
Atlanta
5
350
User
Atlanta
5
351
User
Atlanta
5
352
User
Atlanta
5
353
User
Atlanta
5
354
User
Atlanta
5
355
User
Atlanta
5
356
User
Fort Valley
5
357
User
Smyrna
5
358
User
East Cobb
5
359
User
Stockbridge
5
360
User
Atlanta
5
361
User
Atlanta
5
362
User
Atlanta
5
363
User
Alpharetta
5
364
User
Savannah
5
365
User
Atlanta
5
366
User
Buford
5
367
User
Woodstock
5
368
User
Lawrenceville
5
369
User
Atlanta
5
370
User
Powder Springs
5
371
Bas Weijerman
Atlanta
5
372
User
City
5
373
User
Macon
5
374
User
Druid Hills
5
375
User
Conyers
5
376
User
Atlanta
5
377
User
Atlanta
5
378
User
Atlanta
5
379
User
Atlanta
5
380
User
Atlanta
5
381
User
Marietta
5
382
User
Roswell
5
383
User
East Cobb
5
384
User
Johns Creek
5
385
User
Augusta
5
386
User
Canton
5
387
User
Watkinsville
5
388
User
Holly Springs
5
389
User
Conyers
5
390
User
Atlanta
5
391
User
Demorest
5
392
User
Atlanta
5
393
User
Martinez
5
394
User
Valdosta
5
395
User
Atlanta
5
396
User
Roswell
5
397
User
Atlanta
5
398
User
Duluth
5
399
User
Dunwoody
4
400
User
Atlanta
4
401
User
Atlanta
4
402
User
Atlanta
4
403
User
Atlanta
4
404
User
Atlanta
4
405
User
Atlanta
4
406
User
Atlanta
4
407
User
Atlanta
4
408
edward
Fort Valley
4
409
User
Acworth
4
410
User
Atlanta
4
411
User
Johns Creek
4
412
User
Athens
4
413
User
Demorest
4
414
User
Druid Hills
4
415
User
Fairburn
4
416
User
Snellville
4
417
User
Fort Valley
4
418
Adriana Garcia Ceja
Atlanta
4
419
User
North High Shoals
4
420
User
Atlanta
4
421
User
Atlanta
4
422
User
Conyers
4
423
User
Warner Robins
4
424
User
Atlanta
4
425
User
Atlanta
4
426
User
Atlanta
4
427
User
Statesboro
4
428
Janine Jasmin
Mcdonough
4
429
User
Duluth
4
430
User
Gainesville
4
431
User
Statesboro
4
432
User
Lawrenceville
4
433
User
Druid Hills
4
434
User
Palmetto
4
435
User
Atlanta
4
436
User
Atlanta
4
437
User
Atlanta
4
438
User
Atlanta
4
439
User
Atlanta
4
440
User
Atlanta
4
441
User
Dunwoody
4
442
User
Macon
4
443
User
East Cobb
4
444
User
Atlanta
4
445
User
Dunwoody
4
446
User
Riverdale
4
447
User
Atlanta
4
448
User
Atlanta
4
449
User
Woodstock
4
450
User
Duluth
4
451
User
Atlanta
4
452
User
Atlanta
4
453
User
Griffin
4
454
User
Griffin
4
455
User
Johns Creek
4
456
User
Statesboro
4
457
User
Vinings
4
458
User
Atlanta
4
459
User
Atlanta
4
460
User
Roswell
3
461
User
Atlanta
3
462
User
Moultrie
3
463
User
Atlanta
3
464
User
Atlanta
3
465
User
Atlanta
3
466
User
Duluth
3
467
User
Lawrenceville
3
468
User
Johns Creek
3
469
User
Atlanta
3
470
User
Atlanta
3
471
User
Perry
3
472
User
Atlanta
3
473
User
Hiram
3
474
User
Atlanta
3
475
User
Acworth
3
476
Michelle Li
Athens
3
477
User
Tyrone
3
478
User
Atlanta
3
479
User
Holly Springs
3
480
User
Peachtree Corners
3
481
User
Cumming
3
482
User
Atlanta
3
483
User
Woodstock
3
484
User
Statesboro
3
485
User
Union City
3
486
User
Riverdale
3
487
User
Atlanta
3
488
User
Atlanta
3
489
User
Dunwoody
3
490
User
Mcdonough
3
491
User
Milton
3
492
User
Fort Valley
3
493
User
Augusta
3
494
User
Atlanta
3
495
User
Dawsonville
3
496
User
Alpharetta
3
497
User
Brookhaven
3
498
User
Cumming
3
499
User
Mcintyre
3
500
User
Atlanta
3
501
User
Ashburn
3
502
User
Moultrie
3
503
User
Atlanta
3
504
User
Atlanta
3
505
User
Atlanta
3
506
User
Atlanta
3
507
User
Atlanta
3
508
User
Cumming
3
509
User
Fort Valley
3
510
User
Canton
3
511
User
Atlanta
3
512
User
Athens
3
513
User
Atlanta
3
514
User
Mableton
3
515
User
Stockbridge
3
516
User
Riverdale
3
517
User
Atlanta
3
518
d j
Fort Valley
3
519
User
Atlanta
3
520
User
Powder Springs
3
521
User
Conyers
3
522
User
Druid Hills
3
523
User
Atlanta
3
524
User
Dunwoody
3
525
User
Bolingbroke
3
526
User
Atlanta
3
527
User
Atlanta
3
528
User
Atlanta
3
529
User
Atlanta
3
530
User
Johns Creek
3
531
User
Suwanee
3
532
User
Porterdale
3
533
User
Atlanta
3
534
User
Fort Valley
3
535
User
Conyers
3
536
User
Statesboro
3
537
User
Atlanta
3
538
User
Mableton
3
539
User
Conyers
3
540
User
Conyers
3
541
User
Savannah
3
542
User
Dacula
3
543
User
Atlanta
2
544
User
Macon
2
545
User
Stockbridge
2
546
User
Statesboro
2
547
User
Valdosta
2
548
User
Atlanta
2
549
User
Johns Creek
2
550
User
Atlanta
2
551
User
Lawrenceville
2
552
User
Atlanta
2
553
User
Atlanta
2
554
User
Atlanta
2
555
User
Atlanta
2
556
User
Atlanta
2
557
User
Atlanta
2
558
User
Atlanta
2
559
User
Lilburn
2
560
User
Alpharetta
2
561
User
Sandy Springs
2
562
User
Atlanta
2
563
User
Conyers
2
564
User
Johns Creek
2
565
User
Atlanta
2
566
User
Glennville
2
567
User
Conyers
2
568
User
Atlanta
2
569
User
Dock Junction
2
570
DongJun Kim
Atlanta
2
571
User
Conyers
2
572
J B
Atlanta
2
573
User
Atlanta
2
574
User
Conyers
2
575
User
Statesboro
2
576
User
Statesboro
2
577
User
Fort Valley
2
578
User
Fort Valley
2
579
Makayla Jones
Martinez
2
580
User
East Cobb
2
581
User
Atlanta
2
582
User
Augusta
2
583
User
South Fulton
2
584
User
Woodstock
2
585
User
Sandy Springs
2
586
User
Macon
2
587
User
Atlanta
2
588
User
Johns Creek
2
589
User
Rome
2
590
User
Atlanta
2
591
User
Conyers
2
592
User
Atlanta
2
593
User
Rome
2
594
User
Druid Hills
2
595
User
Columbus
2
596
User
Atlanta
2
597
User
Macon
2
598
User
Atlanta
2
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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 math 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 math, 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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