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