Leaderboard
Virginia
Rank
Level
1
User
Danville
94
2
Timothy Gates
Warrenton
81
3
User
Fairfax
79
4
Mohamed Tchambaz
Alexandria
76
5
Pickle
Aldie
74
6
Leafy Onion
Reston
73
7
HONGKONGNEES
Ashburn
69
8
Chrissy Day
Tuckahoe
65
9
Takanori Tojikubo
Washington
61
10
User
Charlottesville
61
11
PatrickS
Ashburn
60
12
User
Charlottesville
56
13
User
Ashburn
55
14
User
Arlington
54
15
Sharon Liu
South Riding
52
16
alphadalpha
Alexandria
51
17
User
Culpeper
48
18
User
Charlottesville
47
19
User
Centreville
45
20
User
Chantilly
45
21
User
Blacksburg
45
22
Yo Learn
Charlottesville
44
23
Nick Novicov
Virginia Beach
44
24
Caroline
Blacksburg
43
25
Clifton Conrad
Alexandria
43
26
User
Culpeper
41
27
User
Burke
38
28
Jason Brown
Arlington
37
29
User
Alexandria
37
30
User
Culpeper
37
31
User
Charlottesville
36
32
User
Charlottesville
36
33
User
Tuckahoe
36
34
Arabelle Sarreal
West Falls Church
36
35
User
Chatham
36
36
tony huang
Chantilly
35
37
User
Arlington
35
38
PA
Arlington
35
39
User
Reston
35
40
User
Culpeper
35
41
Nick N
Portsmouth
35
42
User
Fredericksburg
34
43
User
Ashburn
34
44
User
Tysons
34
45
User
Reston
33
46
User
Mclean
33
47
Guillermo Verdasco
Tuckahoe
33
48
Chung-chen Fang
Charlottesville
33
49
User
Ashburn
33
50
vinhnguyen
Mclean
33
51
Christopher Lanthier
West Springfield
32
52
User
Mclean
31
53
User
Charlottesville
31
54
User
Tuckahoe
31
55
User
Culpeper
31
56
User
Fort Hunt
31
57
User
Ashburn
31
58
User
Virginia Beach
31
59
User
Danville
30
60
User
Arlington
30
61
User
Culpeper
30
62
User
Charlottesville
30
63
Skapple
Wyndham
29
64
Paul Luongo
Tysons
29
65
User
Norfolk
29
66
User
Cave Spring
29
67
User
Arlington
29
68
smh tbh
Midlothian
29
69
User
Woodlake
28
70
User
Burke
28
71
User
Burke
28
72
User
Annandale
28
73
Michael Lippman
Herndon
28
74
User
Reston
28
75
kamran
Arlington
28
76
User
Chantilly
27
77
User
Herndon
27
78
User
Charlottesville
27
79
User
Ashburn
27
80
User
Reston
27
81
User
Culpeper
27
82
User
Charlottesville
27
83
Jason Odum
Alexandria
27
84
User
Herndon
27
85
User
Herndon
27
86
User
Virginia Beach
27
87
User
Williamsburg
27
88
User
Lynchburg
27
89
User
Virginia Beach
26
90
Romeo Delacroix
Mclean
26
91
User
Tuckahoe
26
92
User
Franconia
26
93
Unknown User
Culpeper
26
94
User
Arlington
26
95
User
Culpeper
26
96
User
Culpeper
26
97
User
Reston
26
98
User
Potomac Falls
25
99
User
Arlington
25
100
User
Ashburn
25
101
User
South Riding
25
102
Daniel Roman
Fort Lee
25
103
User
Centreville
25
104
User
Charlottesville
25
105
User
Ashburn
25
106
User
Stephens City
24
107
User
Leesburg
24
108
Jaden Lau
Ashburn
24
109
User
Tuckahoe
24
110
Papa Big Balls
Alexandria
24
111
User
Tysons
24
112
User
Reston
24
113
User
Short Pump
24
114
User
Annandale
24
115
User
Richmond
24
116
User
Charlottesville
24
117
User
Reston
24
118
User
Culpeper
24
119
User
Charlottesville
24
120
User
Charlottesville
24
121
User
Prince William County
24
122
User
Tysons
23
123
User
Fort Lee
23
124
User
Charlottesville
23
125
User
Roanoke
23
126
User
Woodbridge
23
127
User
South Riding
23
128
User
Culpeper
23
129
User
Herndon
23
130
Molly Hoffman
Kilmarnock
23
131
McKrayJ
Reston
22
132
User
Chantilly
22
133
User
Charlottesville
22
134
Abhay Sri
Tuckahoe
22
135
User
Arlington
22
136
User
Tysons
22
137
User
Alexandria
22
138
User
Virginia Beach
22
139
User
Newport News
22
140
User
Alexandria
22
141
User
Chantilly
22
142
User
Richmond
22
143
User
Virginia Beach
21
144
User
Fort Lee
21
145
User
Fort Lee
21
146
User
Fort Lee
21
147
User
Blacksburg
21
148
User
Arlington
21
149
User
Charlottesville
21
150
Tokohama
City
21
151
User
Charlottesville
21
152
User
Winchester
21
153
Sanjib Sahoo
Ashburn
21
154
User
Richmond
21
155
User
Arlington
21
156
User
Arlington
20
157
User
Arlington
20
158
User
Richmond
20
159
User
Fort Lee
20
160
User
Fort Lee
20
161
User
Fort Lee
20
162
User
Fort Lee
20
163
User
Tuckahoe
20
164
Hazel Vineet
Chantilly
20
165
User
West Falls Church
20
166
User
Culpeper
20
167
User
Culpeper
20
168
User
Culpeper
20
169
User
Richmond
20
170
User
Charlottesville
20
171
User
Arlington
20
172
User
Virginia Beach
20
173
User
Fairfax
19
174
User
Fort Lee
19
175
User
Fort Lee
19
176
User
Alexandria
19
177
User
Gloucester
19
178
User
Culpeper
19
179
dig
Potomac Falls
18
180
K Kennedy
Manassas
18
181
User
Virginia Beach
18
182
User
Fort Lee
18
183
User
Fort Lee
18
184
User
Lorton
18
185
User
Merrifield
18
186
User
New Kent
18
187
User
Culpeper
18
188
User
Culpeper
18
189
User
Culpeper
18
190
User
Newport News
18
191
User
Virginia Beach
18
192
User
Richmond
18
193
User
Ashburn
18
194
User
Arlington
18
195
sodo haranud
Centreville
17
196
User
Winchester
17
197
User
Fort Lee
17
198
Hamza Elmissouab
Alexandria
17
199
User
Culpeper
17
200
User
Culpeper
17
201
User
Culpeper
17
202
User
Culpeper
17
203
User
Kinsale
17
204
User
Springfield
17
205
User
Haymarket
17
206
Jai Chintagunta
Blacksburg
16
207
User
Blacksburg
16
208
User
Richmond
16
209
User
Fort Lee
16
210
User
Centreville
16
211
User
Tuckahoe
16
212
User
Glen Allen
16
213
User
Culpeper
16
214
User
Brambleton
16
215
User
Charlottesville
16
216
User
Arlington
16
217
User
Virginia Beach
16
218
User
Herndon
16
219
User
Ashburn
16
220
User
Fairfax Station
16
221
User
Alexandria
16
222
User
Charlottesville
15
223
User
Chantilly
15
224
User
Fort Lee
15
225
User
Lexington
15
226
User
Blacksburg
15
227
User
Culpeper
15
228
User
New Kent
15
229
User
Culpeper
15
230
User
Bull Run
15
231
User
Vienna
15
232
User
Arlington
15
233
User
Alexandria
15
234
Izzy Zhang
Mclean
15
235
User
Herndon
15
236
User
Ashburn
14
237
Chris Lanthier
West Springfield
14
238
User
Tysons
14
239
Abid Rizvi
Ashburn
14
240
Matthew Nolan-Cintron
Virginia Beach
14
241
User
Reston
14
242
User
Alexandria
14
243
User
Richmond
14
244
Sumo Mitro
Tysons
14
245
User
Warsaw
14
246
User
Arlington
14
247
User
Culpeper
14
248
User
Culpeper
14
249
User
Chesapeake
14
250
User
Vienna
14
251
User
Arlington
14
252
User
BaileyS Crossroads
13
253
User
Alexandria
13
254
Romeo Delacroix
Annandale
13
255
User
Springfield
13
256
Bobby Altizer
Claypool Hill
13
257
User
Hampton
13
258
User
Fort Lee
13
259
User
Gainesville
13
260
User
Ashburn
13
261
User
Culpeper
13
262
User
Culpeper
13
263
User
Bon Air
13
264
User
Ashburn
13
265
User
Oakton
12
266
Carl
Reston
12
267
User
Arlington
12
268
User
Ashburn
12
269
Bob Smith
BaileyS Crossroads
12
270
Landon McNew
Bristol
12
271
User
Woodlake
12
272
User
Arlington
12
273
User
Arcola
12
274
User
Culpeper
12
275
User
Culpeper
12
276
User
Charlottesville
12
277
User
Chantilly
12
278
User
Aldie
12
279
User
Chantilly
11
280
User
Tysons
11
281
User
Arlington
11
282
User
Virginia Beach
11
283
User
Ashburn
11
284
User
Franconia
11
285
User
Fort Lee
11
286
User
Arlington
11
287
User
Manassas
11
288
User
Fort Belvoir
11
289
User
Arlington
11
290
User
Culpeper
11
291
User
Blacksburg
11
292
User
Culpeper
11
293
User
Arlington
11
294
Pauline Jones
Richmond
11
295
User
Virginia Beach
11
296
User
Great Falls
11
297
User
Richmond
11
298
User
Herndon
11
299
User
Danville
10
300
User
Mount Vernon
10
301
Brian Carver
Ashburn
10
302
User
Arlington
10
303
User
Hampton
10
304
User
Culpeper
10
305
User
Williamsburg
10
306
User
Norfolk
10
307
User
Hampton
10
308
User
Charlottesville
10
309
User
Norfolk
10
310
User
King George
10
311
allen
Ashburn
9
312
Justin Bangerter
Roanoke
9
313
User
Ashburn
9
314
User
Annandale
9
315
User
Tysons
9
316
User
Virginia Beach
9
317
User
Cave Spring
9
318
User
Ashburn
9
319
User
Lake Barcroft
9
320
User
Culpeper
9
321
User
Fairfax
9
322
User
Charlottesville
9
323
User
Chester
9
324
User
Manassas
8
325
Sydney Smith
Virginia Beach
8
326
User
Richmond
8
327
User
Virginia Beach
8
328
User
Reston
8
329
User
Newport News
8
330
User
Virginia Beach
8
331
User
Virginia Beach
8
332
User
Brambleton
8
333
User
Lynchburg
8
334
User
Dulles
8
335
User
Herndon
8
336
User
Culpeper
8
337
User
Culpeper
8
338
User
Arlington
8
339
User
Ashburn
8
340
User
Woodlake
8
341
User
Charlottesville
8
342
User
Charlottesville
8
343
User
Alexandria
8
344
User
Vienna
8
345
User
Ashburn
8
346
User
Lynchburg
8
347
User
Alexandria
8
348
User
Falmouth
7
349
BangladeshiBoy 101
Herndon
7
350
User
Sterling
7
351
User
West Springfield
7
352
User
Burke
7
353
User
Chantilly
7
354
User
Fredericksburg
7
355
User
Hampton
7
356
User
Fort Lee
7
357
User
Arlington
7
358
User
Ashburn
7
359
User
Fort Lee
7
360
User
Ashburn
7
361
Daniel Knecht
Fairlawn
7
362
User
Arcola
7
363
User
Charlottesville
7
364
User
New Kent
7
365
User
Culpeper
7
366
User
Culpeper
7
367
User
Culpeper
7
368
User
Culpeper
7
369
User
Chesapeake
7
370
User
Vienna
7
371
User
Richmond
7
372
Nicole Bettis
Tysons
7
373
User
Arlington
7
374
User
Altavista
7
375
User
Arlington
6
376
User
Ashburn
6
377
User
Richmond
6
378
User
Gainesville
6
379
User
Fort Lee
6
380
User
Springfield
6
381
User
Tuckahoe
6
382
User
Arlington
6
383
User
Reston
6
384
User
Newport News
6
385
User
Virginia Beach
6
386
User
Harrisonburg
6
387
User
Annandale
6
388
User
Arlington
6
389
User
Culpeper
6
390
User
Culpeper
6
391
User
New Kent
6
392
User
New Kent
6
393
User
Arlington
6
394
User
Culpeper
6
395
User
Culpeper
6
396
User
Fort Belvoir
6
397
User
Woodlake
6
398
User
Vienna
6
399
User
Vienna
6
400
User
Vienna
6
401
User
Lynchburg
6
402
User
Dulles
6
403
User
Tuckahoe
6
404
User
Arlington
6
405
User
Reston
6
406
User
Ashburn
6
407
User
Leesburg
6
408
User
Fredericksburg
5
409
User
Brambleton
5
410
User
Richmond
5
411
User
Chester
5
412
User
Chester
5
413
User
Richmond
5
414
User
Burke
5
415
User
Merrifield
5
416
User
Glen Allen
5
417
User
Tuckahoe
5
418
User
Kilmarnock
5
419
User
Arlington
5
420
User
Herndon
5
421
User
New Kent
5
422
User
New Kent
5
423
Arthur Bowles
Kilmarnock
5
424
User
Blacksburg
5
425
User
Blacksburg
5
426
User
Woodlake
5
427
User
Chesapeake
5
428
User
Abingdon
5
429
User
Tuckahoe
5
430
User
Richmond
5
431
User
Charlottesville
5
432
User
Ashburn
4
433
User
Burke
4
434
User
Sterling
4
435
User
Chesapeake
4
436
User
Chester
4
437
Hunter Griffin
Midlothian
4
438
User
Tysons
4
439
User
Kilmarnock
4
440
Hunter Trylch
Tysons
4
441
User
Tysons
4
442
User
Arlington
4
443
User
Lake Ridge
4
444
User
Fort Lee
4
445
User
Fort Lee
4
446
User
South Riding
4
447
User
Suffolk
4
448
User
Vienna
4
449
User
Arlington
4
450
User
Arlington
4
451
User
Tuckahoe
4
452
User
New Kent
4
453
User
New Kent
4
454
Jacklyn Chen
Charlottesville
4
455
User
Culpeper
4
456
User
Culpeper
4
457
User
Culpeper
4
458
User
Broadway
4
459
User
Vienna
4
460
User
Manassas
4
461
User
Annandale
4
462
User
Chantilly
4
463
User
Arlington
4
464
hutman
Virginia Beach
4
465
User
Gainesville
4
466
User
Annandale
3
467
Clare
Charlottesville
3
468
User
Virginia Beach
3
469
User
Tuckahoe
3
470
User
Tysons
3
471
User
Chantilly
3
472
User
Fairfax
3
473
User
Tuckahoe
3
474
User
Fairfax
3
475
User
Reston
3
476
mymona
Herndon
3
477
User
Lexington
3
478
Dylan J Alliata
Franconia
3
479
User
Richmond
3
480
User
Chesapeake
3
481
User
Lorton
3
482
User
Richmond
3
483
User
Virginia Beach
3
484
User
Williamsburg
3
485
User
Woodlake
3
486
User
Franconia
3
487
User
Woodlake
3
488
User
South Riding
3
489
User
Culpeper
3
490
User
Alexandria
3
491
Nazareous
Alexandria
3
492
User
Leesburg
3
493
User
Potomac Falls
3
494
User
Culpeper
3
495
User
New Kent
3
496
User
New Kent
3
497
User
Virginia Beach
3
498
User
Alexandria
3
499
User
Charlottesville
3
500
User
Vienna
3
501
User
Ashburn
3
502
User
Virginia Beach
3
503
User
Richmond
3
504
User
Charlottesville
3
505
User
Annandale
3
506
User
Glen Allen
3
507
User
Arlington
3
508
User
Chantilly
3
509
User
Richmond
3
510
User
Springfield
3
511
User
Suffolk
3
512
User
Ashburn
3
513
User
Charlottesville
3
514
User
Leesburg
3
515
User
Midlothian
2
516
User
Reston
2
517
User
Pimmit Hills
2
518
User
Newport News
2
519
User
Arlington
2
520
User
Arlington
2
521
User
Chantilly
2
522
Allison
Sterling
2
523
User
Charlottesville
2
524
User
Fairfax
2
525
User
Short Pump
2
526
User
Fairfax
2
527
User
Blacksburg
2
528
User
Roanoke
2
529
User
Blacksburg
2
530
User
Arlington
2
531
User
Richmond
2
532
User
Richmond
2
533
User
Woodlake
2
534
User
Mount Vernon
2
535
User
Richmond
2
536
Ari West
Glen Allen
2
537
User
Tuckahoe
2
538
User
Chantilly
2
539
User
Charlottesville
2
540
User
Blacksburg
2
541
User
Arlington
2
542
User
West Springfield
2
543
User
Virginia Beach
2
544
User
Mclean
2
545
User
West Springfield
2
546
User
Mechanicsville
2
547
User
Centreville
2
548
User
Charlottesville
2
549
Claudia Gentry
Culpeper
2
550
User
Glen Allen
2
551
terryseldon
Kilmarnock
2
552
User
Bristow
2
553
User
New Kent
2
554
User
New Kent
2
555
User
New Kent
2
556
User
New Kent
2
557
User
New Kent
2
558
User
New Kent
2
559
User
Kilmarnock
2
560
User
Culpeper
2
561
User
Lakeside
2
562
Alycia Logan
Culpeper
2
563
User
Arlington
2
564
User
Culpeper
2
565
User
Culpeper
2
566
User
Culpeper
2
567
User
Culpeper
2
568
User
Virginia Beach
2
569
User
Vienna
2
570
User
Vienna
2
571
User
Vienna
2
572
User
Vienna
2
573
User
Vienna
2
574
User
Vienna
2
575
User
Arlington
2
576
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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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