Leaderboard
Virginia
Rank
Level
1
User
Danville
99
2
User
Danville
98
3
User
Danville
97
4
User
Danville
94
5
User
Danville
91
6
User
Norfolk
91
7
Ethan Zh
Wolf Trap
87
8
User
Danville
81
9
Timothy Gates
Warrenton
81
10
Mohamed Tchambaz
Alexandria
79
11
User
Fairfax
79
12
James Doe
Chantilly
75
13
Pickle
Aldie
74
14
Leafy Onion
Reston
73
15
User
Arlington
71
16
User
Arlington
68
17
tony huang
Herndon
68
18
User
Danville
65
19
Chrissy Day
Tuckahoe
65
20
User
Fairfax
64
21
User
Dale City
62
22
Takanori Tojikubo
Washington
61
23
User
Charlottesville
61
24
User
Fairfax
58
25
User
Danville
58
26
Christopher Lanthier
West Springfield
57
27
User
Charlottesville
56
28
Kobe
Roanoke
55
29
User
Ashburn
55
30
Clifton Conrad
Alexandria
54
31
Sharon Liu
South Riding
52
32
alphadalpha
Alexandria
51
33
User
Salem
49
34
User
Great Falls
49
35
Patrick Sliwinski
Ashburn
48
36
User
Culpeper
48
37
User
Kilmarnock
47
38
User
Charlottesville
47
39
User
Leesburg
45
40
User
Centreville
45
41
User
Chantilly
45
42
User
Blacksburg
45
43
Roy Jad
Charlottesville
44
44
Arleth Salinas
Charlottesville
44
45
Yo Learn
Charlottesville
44
46
Nick Novicov
Virginia Beach
44
47
Caroline
Blacksburg
43
48
Mi Back
Reston
42
49
User
Culpeper
41
50
seeberger
Reston
39
51
User
Portsmouth
38
52
Kenyah Calhoun
West Falls Church
38
53
User
Burke
38
54
Jason Brown
Arlington
37
55
User
Alexandria
37
56
User
Culpeper
37
57
User
Tuckahoe
36
58
User
Charlottesville
36
59
User
Charlottesville
36
60
User
Tuckahoe
36
61
Arabelle Sarreal
West Falls Church
36
62
User
Chatham
36
63
User
Richmond
35
64
User
Wolf Trap
35
65
User
Blacksburg
35
66
User
Arlington
35
67
PA
Arlington
35
68
User
Reston
35
69
User
Culpeper
35
70
Nick N
Portsmouth
35
71
User
Portsmouth
34
72
User
Fredericksburg
34
73
User
Ashburn
34
74
User
Tysons
34
75
User
Reston
33
76
User
Mclean
33
77
Guillermo Verdasco
Tuckahoe
33
78
Chung-chen Fang
Charlottesville
33
79
User
Ashburn
33
80
vinhnguyen
Mclean
33
81
User
Chantilly
32
82
User
Woodbridge
31
83
User
Mclean
31
84
User
Charlottesville
31
85
User
Tuckahoe
31
86
User
Culpeper
31
87
User
Fort Hunt
31
88
User
Ashburn
31
89
User
Danville
30
90
User
Arlington
30
91
User
Culpeper
30
92
User
Charlottesville
30
93
Skapple
Wyndham
29
94
Paul Luongo
Tysons
29
95
User
Norfolk
29
96
User
Cave Spring
29
97
User
Arlington
29
98
smh tbh
Midlothian
29
99
User
Roanoke
28
100
Jason Lee
Ashburn
28
101
User
Woodlake
28
102
User
Burke
28
103
User
Burke
28
104
User
Annandale
28
105
Michael Lippman
Herndon
28
106
User
Reston
28
107
kamran
Arlington
28
108
User
Arlington
27
109
User
Leesburg
27
110
User
Tysons
27
111
User
Herndon
27
112
User
Charlottesville
27
113
User
Ashburn
27
114
User
Reston
27
115
User
Culpeper
27
116
User
Charlottesville
27
117
Jason Odum
Alexandria
27
118
User
Herndon
27
119
User
Herndon
27
120
User
Virginia Beach
27
121
User
Williamsburg
27
122
User
Lynchburg
27
123
User
Wyndham
26
124
User
Tysons
26
125
User
Blacksburg
26
126
User
Virginia Beach
26
127
Romeo Delacroix
Mclean
26
128
User
Tuckahoe
26
129
User
Franconia
26
130
Unknown User
Culpeper
26
131
User
Arlington
26
132
User
Culpeper
26
133
User
Culpeper
26
134
User
Reston
26
135
User
Blacksburg
25
136
User
Burke
25
137
Jai Chintagunta
Blacksburg
25
138
User
Herndon
25
139
User
Potomac Falls
25
140
User
Arlington
25
141
User
Ashburn
25
142
User
South Riding
25
143
Daniel Roman
Fort Lee
25
144
User
Centreville
25
145
User
Charlottesville
25
146
User
Ashburn
25
147
User
Chantilly
24
148
User
Stephens City
24
149
User
Leesburg
24
150
Jaden Lau
Ashburn
24
151
Papa Big Balls
Alexandria
24
152
User
Tysons
24
153
User
Reston
24
154
User
Short Pump
24
155
User
Annandale
24
156
User
Richmond
24
157
User
Charlottesville
24
158
User
Reston
24
159
User
Culpeper
24
160
User
Charlottesville
24
161
User
Charlottesville
24
162
User
Prince William County
24
163
User
Charlottesville
23
164
User
Tysons
23
165
User
Fort Lee
23
166
User
Charlottesville
23
167
User
Roanoke
23
168
User
Woodbridge
23
169
User
South Riding
23
170
User
Culpeper
23
171
User
Herndon
23
172
Molly Hoffman
Kilmarnock
23
173
User
Arlington
22
174
User
Dale City
22
175
User
Arlington
22
176
McKrayJ
Reston
22
177
User
Chantilly
22
178
User
Charlottesville
22
179
Abhay Sri
Tuckahoe
22
180
User
Arlington
22
181
User
Tysons
22
182
User
Alexandria
22
183
User
Virginia Beach
22
184
User
Newport News
22
185
User
Alexandria
22
186
User
Chantilly
22
187
User
Richmond
22
188
User
Fairfax Station
21
189
User
Chantilly
21
190
User
Fairfax
21
191
User
Virginia Beach
21
192
User
Fort Lee
21
193
User
Fort Lee
21
194
User
Fort Lee
21
195
User
Blacksburg
21
196
User
Arlington
21
197
User
Charlottesville
21
198
Tokohama
City
21
199
User
Charlottesville
21
200
User
Winchester
21
201
Sanjib Sahoo
Ashburn
21
202
User
Richmond
21
203
User
Arlington
20
204
User
Arlington
20
205
User
Richmond
20
206
User
Fort Lee
20
207
User
Fort Lee
20
208
User
Fort Lee
20
209
User
Fort Lee
20
210
User
Tuckahoe
20
211
Hazel Vineet
Chantilly
20
212
User
West Falls Church
20
213
User
Culpeper
20
214
User
Culpeper
20
215
User
Culpeper
20
216
User
Richmond
20
217
User
Charlottesville
20
218
User
Arlington
20
219
User
Virginia Beach
20
220
User
Wyndham
19
221
User
Charlottesville
19
222
User
Fairfax
19
223
User
Fort Lee
19
224
User
Fort Lee
19
225
User
Alexandria
19
226
User
Gloucester
19
227
User
Culpeper
19
228
J
Virginia Beach
18
229
Christian Lessard
Charlottesville
18
230
User
Aquia Harbour
18
231
dig
Potomac Falls
18
232
K Kennedy
Manassas
18
233
User
Virginia Beach
18
234
User
Fort Lee
18
235
User
Fort Lee
18
236
User
Lorton
18
237
User
Merrifield
18
238
User
New Kent
18
239
User
Culpeper
18
240
User
Culpeper
18
241
User
Culpeper
18
242
User
Newport News
18
243
User
Virginia Beach
18
244
User
Richmond
18
245
User
Ashburn
18
246
User
Arlington
18
247
User
Herndon
17
248
User
Ashburn
17
249
sodo haranud
Centreville
17
250
User
Fort Lee
17
251
Hamza Elmissouab
Alexandria
17
252
User
Culpeper
17
253
User
Culpeper
17
254
User
Culpeper
17
255
User
Culpeper
17
256
User
Kinsale
17
257
User
Springfield
17
258
User
Haymarket
17
259
User
Chantilly
16
260
User
Virginia Beach
16
261
Victoria Chung
Dale City
16
262
User
Blacksburg
16
263
User
Richmond
16
264
User
Fort Lee
16
265
User
Centreville
16
266
User
Tuckahoe
16
267
User
Glen Allen
16
268
User
Culpeper
16
269
User
Brambleton
16
270
User
Charlottesville
16
271
User
Arlington
16
272
User
Virginia Beach
16
273
User
Herndon
16
274
User
Ashburn
16
275
User
Fairfax Station
16
276
User
Alexandria
16
277
User
Chesapeake
15
278
User
Reston
15
279
User
Tysons
15
280
User
Charlottesville
15
281
User
Chantilly
15
282
User
Fort Lee
15
283
User
Lexington
15
284
User
Blacksburg
15
285
User
Culpeper
15
286
User
New Kent
15
287
User
Culpeper
15
288
User
Bull Run
15
289
User
Vienna
15
290
User
Arlington
15
291
User
Alexandria
15
292
Izzy Zhang
Mclean
15
293
User
Herndon
15
294
User
Dale City
14
295
User
Tysons
14
296
Abid Rizvi
Ashburn
14
297
Matthew Nolan-Cintron
Virginia Beach
14
298
User
Reston
14
299
User
Alexandria
14
300
User
Richmond
14
301
Sumo Mitro
Tysons
14
302
User
Warsaw
14
303
User
Arlington
14
304
User
Culpeper
14
305
User
Culpeper
14
306
User
Chesapeake
14
307
User
Vienna
14
308
User
Arlington
14
309
User
Harrisonburg
13
310
User
Chester
13
311
User
Midlothian
13
312
User
Vienna
13
313
User
BaileyS Crossroads
13
314
User
Alexandria
13
315
Romeo Delacroix
Annandale
13
316
User
Springfield
13
317
Bobby Altizer
Claypool Hill
13
318
User
Hampton
13
319
User
Fort Lee
13
320
User
Gainesville
13
321
User
Ashburn
13
322
User
Culpeper
13
323
User
Culpeper
13
324
User
Bon Air
13
325
User
Ashburn
13
326
User
Richmond
12
327
User
Tysons
12
328
User
Chantilly
12
329
User
Oakton
12
330
Carl
Reston
12
331
User
Arlington
12
332
User
Ashburn
12
333
Bob Smith
BaileyS Crossroads
12
334
Landon McNew
Bristol
12
335
User
Woodlake
12
336
User
Arlington
12
337
User
Arcola
12
338
User
Culpeper
12
339
User
Culpeper
12
340
User
Charlottesville
12
341
User
Chantilly
12
342
User
Aldie
12
343
User
Fairfax
11
344
User
Ashburn
11
345
User
Springfield
11
346
User
Kilmarnock
11
347
User
Tysons
11
348
User
Arlington
11
349
User
Virginia Beach
11
350
User
Ashburn
11
351
User
Franconia
11
352
User
Fort Lee
11
353
User
Arlington
11
354
User
Manassas
11
355
User
Fort Belvoir
11
356
User
Arlington
11
357
User
Culpeper
11
358
User
Blacksburg
11
359
User
Culpeper
11
360
User
Arlington
11
361
Pauline Jones
Richmond
11
362
User
Virginia Beach
11
363
User
Great Falls
11
364
User
Richmond
11
365
User
Herndon
11
366
User
Williamsburg
10
367
User
Centreville
10
368
Chris Lanthier
West Springfield
10
369
User
Tysons
10
370
User
Leesburg
10
371
User
Mount Vernon
10
372
Brian Carver
Ashburn
10
373
User
Arlington
10
374
User
Hampton
10
375
User
Culpeper
10
376
User
Williamsburg
10
377
User
Norfolk
10
378
User
Hampton
10
379
User
Charlottesville
10
380
User
Norfolk
10
381
User
King George
10
382
Natashya Singh
Tuckahoe
9
383
User
Richmond
9
384
allen
Ashburn
9
385
Justin Bangerter
Roanoke
9
386
User
Ashburn
9
387
User
Tysons
9
388
User
Virginia Beach
9
389
User
Cave Spring
9
390
User
Ashburn
9
391
User
Lake Barcroft
9
392
User
Culpeper
9
393
User
Fairfax
9
394
User
Charlottesville
9
395
User
Chester
9
396
User
Manassas Park
8
397
User
Roanoke
8
398
User
Lake Monticello
8
399
User
Manassas
8
400
Sydney Smith
Virginia Beach
8
401
User
Richmond
8
402
User
Virginia Beach
8
403
User
Reston
8
404
User
Newport News
8
405
User
Virginia Beach
8
406
User
Virginia Beach
8
407
User
Brambleton
8
408
User
Lynchburg
8
409
User
Dulles
8
410
User
Herndon
8
411
User
Culpeper
8
412
User
Culpeper
8
413
User
Arlington
8
414
User
Ashburn
8
415
User
Woodlake
8
416
User
Charlottesville
8
417
User
Charlottesville
8
418
User
Alexandria
8
419
User
Winchester
7
420
User
Great Falls
7
421
User
Ashburn
7
422
User
West Springfield
7
423
User
Hampton
7
424
User
Blacksburg
7
425
User
Alexandria
7
426
User
Falmouth
7
427
BangladeshiBoy 101
Herndon
7
428
User
Sterling
7
429
User
West Springfield
7
430
User
Burke
7
431
User
Chantilly
7
432
User
Fredericksburg
7
433
User
Hampton
7
434
User
Fort Lee
7
435
User
Arlington
7
436
User
Ashburn
7
437
User
Fort Lee
7
438
User
Ashburn
7
439
Daniel Knecht
Fairlawn
7
440
User
Arcola
7
441
User
Charlottesville
7
442
User
New Kent
7
443
User
Culpeper
7
444
User
Culpeper
7
445
User
Culpeper
7
446
User
Culpeper
7
447
User
Chesapeake
7
448
User
Vienna
7
449
User
Richmond
7
450
Nicole Bettis
Tysons
7
451
User
Arlington
7
452
User
Fairfax
6
453
User
Tysons
6
454
User
Tuckahoe
6
455
User
Bowling Green
6
456
User
Tysons
6
457
User
Arlington
6
458
User
Ashburn
6
459
User
Richmond
6
460
User
Gainesville
6
461
User
Fort Lee
6
462
User
Springfield
6
463
User
Tuckahoe
6
464
User
Arlington
6
465
User
Reston
6
466
User
Newport News
6
467
User
Virginia Beach
6
468
User
Harrisonburg
6
469
User
Annandale
6
470
User
Arlington
6
471
User
Culpeper
6
472
User
Culpeper
6
473
User
New Kent
6
474
User
New Kent
6
475
User
Arlington
6
476
User
Culpeper
6
477
User
Culpeper
6
478
User
Fort Belvoir
6
479
User
Woodlake
6
480
User
Vienna
6
481
User
Vienna
6
482
User
Vienna
6
483
User
Lynchburg
6
484
User
Dulles
6
485
User
Tuckahoe
6
486
User
Arlington
6
487
User
Reston
6
488
User
Ashburn
6
489
User
Virginia Beach
5
490
User
Blacksburg
5
491
MATHTRAINERGIVESMEEBOLA
Ashburn
5
492
User
Arlington
5
493
User
Virginia Beach
5
494
User
Alexandria
5
495
User
Virginia Beach
5
496
User
Fredericksburg
5
497
User
Brambleton
5
498
User
Richmond
5
499
User
Chester
5
500
User
Chester
5
501
User
Richmond
5
502
User
Burke
5
503
User
Merrifield
5
504
User
Glen Allen
5
505
User
Tuckahoe
5
506
User
Kilmarnock
5
507
User
Arlington
5
508
User
New Kent
5
509
User
New Kent
5
510
Arthur Bowles
Kilmarnock
5
511
User
Blacksburg
5
512
User
Blacksburg
5
513
User
Woodlake
5
514
User
Chesapeake
5
515
User
Abingdon
5
516
User
Tuckahoe
5
517
User
Richmond
5
518
User
Dale City
4
519
User
Virginia Beach
4
520
User
Richmond
4
521
User
Franconia
4
522
User
Short Pump
4
523
User
Ashburn
4
524
User
Mount Vernon
4
525
User
Chantilly
4
526
User
Lorton
4
527
User
Burke
4
528
User
Sterling
4
529
User
Chesapeake
4
530
User
Chester
4
531
Hunter Griffin
Midlothian
4
532
User
Tysons
4
533
User
Kilmarnock
4
534
Hunter Trylch
Tysons
4
535
User
Tysons
4
536
User
Arlington
4
537
User
Lake Ridge
4
538
User
Fort Lee
4
539
User
Fort Lee
4
540
User
South Riding
4
541
User
Suffolk
4
542
User
Vienna
4
543
User
Arlington
4
544
User
Arlington
4
545
User
Tuckahoe
4
546
User
New Kent
4
547
User
New Kent
4
548
Jacklyn Chen
Charlottesville
4
549
User
Culpeper
4
550
User
Culpeper
4
551
User
Culpeper
4
552
User
Broadway
4
553
User
Vienna
4
554
User
Manassas
4
555
User
Annandale
4
556
User
Chantilly
4
557
User
Arlington
4
558
hutman
Virginia Beach
4
559
User
Bristow
3
560
User
Ashburn
3
561
User
Richmond
3
562
User
Fairfax
3
563
User
Washington
3
564
User
Mclean
3
565
User
Reston
3
566
User
Sandston
3
567
User
Virginia Beach
3
568
User
Arlington
3
569
User
Charlottesville
3
570
User
Richmond
3
571
User
Midlothian
3
572
User
Annandale
3
573
Clare
Charlottesville
3
574
User
Virginia Beach
3
575
User
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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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