Leaderboard
Maryland
Rank
Level
1
aperson
Gaithersburg
82
2
User
Ellicott City
76
3
User
City
74
4
User
Frederick
67
5
User
Saint Marys City
65
6
Cameron O'Brien
Frederick
65
7
Sawyer West
Saint Marys City
63
8
User
Baltimore
57
9
Sheikh Selim
Baltimore
55
10
Mikhail Osanov
Baltimore
52
11
R G
Baltimore
52
12
User
North Bethesda
49
13
Akash Marakath
Columbia
45
14
User
Towson
44
15
User
North Bethesda
44
16
User
Severn
43
17
jack b
Ocean City
41
18
User
La Vale
39
19
User
Columbia
36
20
Jonathan Le
Potomac
34
21
Lane Martin
Monkton
34
22
User
North Laurel
33
23
User
Pasadena
32
24
User
College Park
32
25
User
Lutherville-Timonium
32
26
User
Saint Marys City
32
27
User
Baltimore
32
28
Nathan Mozden
Silver Spring
31
29
User
Saint Marys City
31
30
User
Germantown
31
31
User
Frederick
30
32
User
Potomac
30
33
Alex Zhu
College Park
29
34
CameronO
Frederick
29
35
User
College Park
27
36
User
Ellicott City
26
37
AJ Tempura
Columbia
26
38
Ceca
Greenbelt
26
39
User
Clarksville
26
40
User
Elkridge
26
41
akansha sehgal
Baltimore
25
42
Jesus c
Severn
24
43
trent cunningham
Huntingtown
24
44
Caleb Benson
Frederick
24
45
YUNTK
College Park
23
46
Cole Dwiggins
Glenwood
23
47
Alexander Paskal
College Park
23
48
User
Clarksville
23
49
User
Rockville
23
50
User
Baltimore
21
51
User
Rockville
21
52
User
Ellicott City
21
53
Matthew Lee
Gaithersburg
20
54
User
North Bethesda
20
55
Frank Imoehl
Rockville
19
56
User
Rockville
19
57
User
Annapolis
19
58
Micheal Erb
Gaithersburg
19
59
User
Bethesda
19
60
User
Greenbelt
19
61
Momma G
Cape Saint Claire
19
62
User
Rockville
18
63
User
College Park
18
64
User
Baltimore
17
65
User
North Potomac
17
66
User
Germantown
17
67
User
Catonsville
17
68
User
College Park
17
69
User
Ellicott City
17
70
User
Ellicott City
17
71
User
Saint Marys City
17
72
User
Bel Air South
17
73
User
Columbia
16
74
User
Rockville
16
75
User
College Park
16
76
User
Elkton
16
77
User
Fort Meade
16
78
User
Rockville
15
79
User
Baltimore
15
80
User
City
15
81
User
College Park
14
82
User
Parole
14
83
Christopher DeFeo
Bethesda
14
84
Larissa Ault
Bethesda
13
85
User
Baltimore
13
86
User
College Park
13
87
User
Baltimore
13
88
TheGr8TJ
Takoma Park
12
89
User
Baltimore
12
90
User
Baltimore
12
91
User
Baltimore
12
92
Somto Nweke
Baltimore
12
93
User
College Park
12
94
User
Silver Spring
12
95
User
Hyattsville
12
96
User
Baltimore
11
97
User
Fairland
11
98
User
Baltimore
11
99
User
Rockville
11
100
User
Bethesda
11
101
User
North Potomac
11
102
Safwan Sheikh
Baltimore
11
103
User
Columbia
11
104
User
Potomac
11
105
User
Silver Spring
11
106
User
Columbia
11
107
User
Gaithersburg
11
108
Adi Renduchintala
Baltimore
11
109
User
College Park
11
110
User
Owings Mills
11
111
User
Gaithersburg
11
112
User
Fort Meade
11
113
Hunter Rice
Annapolis
11
114
User
Ellicott City
11
115
User
Silver Spring
11
116
User
Elkridge
10
117
User
Crofton
10
118
User
Baltimore
10
119
User
Severn
10
120
User
Ocean Pines
10
121
User
College Park
10
122
User
Taneytown
10
123
User
Severn
10
124
User
Baltimore
10
125
User
Fort Washington
10
126
User
Upper Marlboro
10
127
User
Saint Charles
10
128
User
Seabrook
9
129
User
Reisterstown
9
130
User
Columbia
9
131
User
College Park
9
132
User
Columbia
9
133
User
Columbia
9
134
User
Germantown
9
135
User
Takoma Park
9
136
User
Bethesda
9
137
User
Baltimore
9
138
User
Baltimore
9
139
Raghavendra Murthi
Bethesda
9
140
Taurian Gibson-Gardner
Takoma Park
9
141
JSB
Bethesda
9
142
User
Grantsville
9
143
User
Fort Meade
9
144
User
Baltimore
9
145
User
Bethesda
9
146
User
Baltimore
9
147
TheGr8TJ
Takoma Park
9
148
User
Rockville
9
149
User
Baltimore
9
150
User
Baltimore
9
151
User
Silver Spring
9
152
User
College Park
9
153
sassy
Silver Spring
9
154
User
Chester
9
155
User
Seabrook
9
156
User
Columbia
8
157
User
Germantown
8
158
User
Rockville
8
159
User
Takoma Park
8
160
User
Takoma Park
8
161
User
Pikesville
8
162
User
Bethesda
8
163
User
Glenelg
8
164
Benedict Wills
Annapolis
8
165
User
Bethesda
8
166
User
Baltimore
8
167
User
Rockville
8
168
Mauricio Apolinar Alberto
Takoma Park
8
169
User
Bethesda
8
170
User
Randallstown
8
171
User
Baltimore
8
172
Syco
Takoma Park
8
173
Rebecca Huynh
Owings Mills
7
174
Fernando Lopez Torres
Takoma Park
7
175
User
Bel Air
7
176
User
Takoma Park
7
177
User
Columbia
7
178
User
Germantown
7
179
User
Takoma Park
7
180
User
Takoma Park
7
181
User
Gaithersburg
7
182
User
Lutherville-Timonium
7
183
User
Urbana
7
184
User
Takoma Park
7
185
User
Silver Spring
7
186
User
Bethesda
7
187
User
Takoma Park
7
188
User
Edgewood
7
189
User
Baltimore
7
190
User
Baltimore
7
191
User
Olney
7
192
User
Fort Washington
7
193
User
Silver Spring
7
194
User
Pasadena
7
195
User
Hollywood
7
196
Nikitha M
Catonsville
7
197
User
College Park
6
198
James Blaser
Saint Marys City
6
199
User
Parkville
6
200
User
Beltsville
6
201
User
Takoma Park
6
202
User
Takoma Park
6
203
Miriam Ortiz Gonzalez
Takoma Park
6
204
User
Catonsville
6
205
User
Rockville
6
206
User
Towson
6
207
User
Fort Washington
6
208
User
Olney
6
209
User
Urbana
6
210
User
Takoma Park
6
211
User
Takoma Park
6
212
User
Takoma Park
6
213
User
Bethesda
6
214
User
Ashton-Sandy Spring
6
215
Dha
Owings Mills
6
216
User
Catonsville
6
217
User
East Riverdale
5
218
User
Frederick
5
219
User
Columbia
5
220
User
College Park
5
221
Berkeley Poulsen
College Park
5
222
User
Gaithersburg
5
223
User
Rockville
5
224
User
Rockville
5
225
User
Hyattsville
5
226
User
Severn
5
227
User
Gaithersburg
5
228
User
Olney
5
229
User
Takoma Park
5
230
User
Frederick
5
231
User
Wheaton-Glenmont
5
232
User
Ellicott City
5
233
User
Baltimore
5
234
User
White Oak
5
235
User
Baltimore
5
236
User
Takoma Park
5
237
User
Bowie
5
238
User
Owings Mills
5
239
User
Frederick
5
240
User
Towson
5
241
User
Takoma Park
5
242
User
Rockville
5
243
User
Carney
5
244
User
Rockville
5
245
User
Rockville
5
246
User
Gaithersburg
4
247
User
Bowleys Quarters
4
248
User
Baltimore
4
249
User
Bethesda
4
250
User
Takoma Park
4
251
User
Takoma Park
4
252
User
Baltimore
4
253
User
Gaithersburg
4
254
User
Hyattsville
4
255
User
Columbia
4
256
User
Fort Washington
4
257
User
Potomac
4
258
User
Gaithersburg
4
259
User
Takoma Park
4
260
User
Edgewood
4
261
Sierra Reid
Towson
4
262
User
Wheaton-Glenmont
4
263
User
Potomac
4
264
User
Takoma Park
4
265
User
Takoma Park
4
266
User
Takoma Park
4
267
User
Takoma Park
4
268
User
Takoma Park
4
269
User
Takoma Park
4
270
User
Westminster
4
271
User
Baltimore
4
272
User
Middletown
4
273
LifeKnight
Bowie
4
274
User
Catonsville
4
275
User
Potomac
4
276
User
Germantown
4
277
User
Bethesda
4
278
User
College Park
4
279
User
Gaithersburg
4
280
User
Fairland
4
281
User
Arnold
4
282
User
Severna Park
4
283
User
Gaithersburg
4
284
User
Baltimore
3
285
User
Takoma Park
3
286
User
Takoma Park
3
287
User
Goddard
3
288
User
Takoma Park
3
289
User
Takoma Park
3
290
User
Takoma Park
3
291
User
Frederick
3
292
User
Bethesda
3
293
Katie
Baltimore
3
294
User
Colesville
3
295
User
Takoma Park
3
296
User
Takoma Park
3
297
User
Travilah
3
298
User
Bethesda
3
299
User
Montgomery Village
3
300
User
Baltimore
3
301
User
Parkton
3
302
User
North Bethesda
3
303
User
Bethesda
3
304
User
Baltimore
3
305
User
Saint Marys City
3
306
User
Takoma Park
3
307
User
Towson
3
308
User
College Park
3
309
User
Gaithersburg
3
310
User
Owings Mills
3
311
User
Annapolis
3
312
User
Takoma Park
3
313
User
Takoma Park
3
314
User
Takoma Park
3
315
User
Ellicott City
3
316
User
Takoma Park
3
317
User
Takoma Park
3
318
User
Takoma Park
3
319
User
Takoma Park
3
320
User
Takoma Park
3
321
User
Takoma Park
3
322
User
Bethesda
3
323
User
Rockville
3
324
User
Greater Landover
3
325
User
Perry Hall
3
326
User
Takoma Park
3
327
User
Catonsville
3
328
User
Baltimore
3
329
User
South Laurel
3
330
User
Takoma Park
3
331
User
Cockeysville
3
332
User
Takoma Park
3
333
User
Hagerstown
3
334
User
Towson
3
335
User
Rockville
3
336
User
Baltimore
3
337
User
Baltimore
3
338
User
Owings Mills
3
339
User
Severna Park
3
340
David Cerna Martir
Takoma Park
3
341
User
Silver Spring
2
342
User
Baltimore
2
343
User
Silver Spring
2
344
Eleshady Gebreyohannes
Takoma Park
2
345
User
Catonsville
2
346
Ariel Dickison
Takoma Park
2
347
User
Glenelg
2
348
User
Ellicott City
2
349
User
Montgomery Village
2
350
User
Baltimore
2
351
User
Severn
2
352
User
Dundalk
2
353
Cristina Tous
College Park
2
354
User
Lutherville-Timonium
2
355
User
Laurel
2
356
User
Annapolis
2
357
User
Takoma Park
2
358
User
Takoma Park
2
359
User
Takoma Park
2
360
User
Takoma Park
2
361
User
Takoma Park
2
362
User
Takoma Park
2
363
User
Takoma Park
2
364
User
Woodbine
2
365
User
Parkville
2
366
User
Pasadena
2
367
User
Baltimore
2
368
User
Takoma Park
2
369
User
Takoma Park
2
370
User
Takoma Park
2
371
User
Takoma Park
2
372
User
College Park
2
373
User
Fort Washington
2
374
User
Frederick
2
375
User
Pikesville
2
376
User
Baltimore
2
377
User
Cumberland
2
378
User
Hanover
2
379
User
Catonsville
2
380
User
Frederick
2
381
User
Owings Mills
2
382
User
Ballenger Creek
2
383
User
Laurel
2
384
User
Reisterstown
2
385
User
Baltimore
2
386
User
Upper Marlboro
2
387
User
Bel Air
2
388
User
Largo
2
389
User
Frederick
2
390
User
Severna Park
2
391
User
Takoma Park
2
392
User
Takoma Park
2
393
User
Takoma Park
2
394
User
Takoma Park
2
395
User
Baltimore
2
396
User
Reisterstown
2
397
User
Saint Marys City
2
398
User
Bethesda
2
399
User
Baltimore
2
400
User
Glenwood
2
401
User
Takoma Park
1
402
User
Takoma Park
1
403
User
Takoma Park
1
404
User
Takoma Park
1
405
Amate Ayivi Koutodjo
Takoma Park
1
406
Ozy Silverstone
Takoma Park
1
407
User
Takoma Park
1
408
User
Middle River
1
409
User
Rockville
1
410
User
Takoma Park
1
411
User
Takoma Park
1
412
User
Takoma Park
1
413
Ranashia
Takoma Park
1
414
User
Takoma Park
1
415
User
Edgewood
1
416
User
Rockville
1
417
User
Takoma Park
1
418
User
Takoma Park
1
419
User
Takoma Park
1
420
User
Takoma Park
1
421
User
Takoma Park
1
422
User
Rockville
1
423
User
North Bethesda
1
424
User
Takoma Park
1
425
User
Bethesda
1
426
User
Catonsville
1
427
User
Eldersburg
1
428
User
Rockville
1
429
User
Sparks Glencoe
1
430
User
Baltimore
1
431
User
Edgewood
1
432
Matthew Lee
Gaithersburg
1
433
User
Bel Air North
1
434
User
Stevensville
1
435
User
Clarksville
1
436
User
Lochearn
1
437
Toddy Ison
Gaithersburg
1
438
User
Baltimore
1
439
User
Salisbury
1
440
User
Takoma Park
1
441
User
Takoma Park
1
442
User
Bowie
1
443
User
Bethesda
1
444
User
Baltimore
1
445
User
Baltimore
1
446
User
Annapolis
1
447
Lane Martin
Lutherville-Timonium
1
448
User
Takoma Park
1
449
User
North Laurel
1
450
User
North Bethesda
1
451
User
Rockville
1
452
User
Rockville
1
453
User
Rockville
1
454
Elijah Peaks
Takoma Park
1
455
User
Columbia
1
456
User
Takoma Park
1
457
User
Takoma Park
1
458
User
Takoma Park
1
459
User
Takoma Park
1
460
User
Takoma Park
1
461
User
Takoma Park
1
462
User
Takoma Park
1
463
User
Takoma Park
1
464
User
Takoma Park
1
465
User
Takoma Park
1
466
User
Takoma Park
1
467
User
Takoma Park
1
468
User
Rockville
1
469
User
Rockville
1
470
User
Takoma Park
1
471
User
Takoma Park
1
472
User
Takoma Park
1
473
User
Takoma Park
1
474
User
Silver Spring
1
475
User
Rockville
1
476
User
Fairland
1
477
User
Glen Arm
1
478
User
Takoma Park
1
479
User
Takoma Park
1
480
User
Takoma Park
1
481
User
Takoma Park
1
482
User
Takoma Park
1
483
User
Beltsville
1
484
User
Takoma Park
1
485
User
Takoma Park
1
486
User
Takoma Park
1
487
User
Takoma Park
1
488
User
Takoma Park
1
489
User
Takoma Park
1
490
User
Takoma Park
1
491
User
Takoma Park
1
492
User
Takoma Park
1
493
User
Crofton
1
494
User
Towson
1
495
User
Takoma Park
1
496
User
Takoma Park
1
497
Gary Einhorn
Takoma Park
1
498
User
Aspen Hill
1
499
User
Montgomery Village
1
500
User
Prince GeorgeS County
1
501
User
Rockville
1
502
User
Eldersburg
1
503
User
North Potomac
1
504
User
Aberdeen
1
505
User
Parole
1
506
User
Damascus
1
507
User
Edgewood
1
508
User
Rockville
1
509
User
Redland
1
510
User
Catonsville
1
511
User
Baltimore
1
512
User
Rockville
1
513
User
Rockville
1
514
User
Rockville
1
515
User
Rockville
1
516
User
Gaithersburg
1
517
User
Gaithersburg
1
518
User
Rockville
1
519
User
Rockville
1
520
User
Rockville
1
521
User
Rockville
1
522
User
Potomac
1
523
User
Rockville
1
524
User
Rockville
1
525
User
Rockville
1
526
User
Columbia
1
527
User
Rockville
1
528
User
Rockville
1
529
User
Aspen Hill
1
530
User
Rockville
1
531
User
Rockville
1
532
User
Baltimore
1
533
User
Rockville
1
534
User
Rockville
1
535
User
Damascus
1
536
User
Elkridge
1
537
User
Rockville
1
538
User
Rockville
1
539
User
North Bethesda
1
540
User
Rockville
1
541
User
Rockville
1
542
User
Rockville
1
543
User
Rockville
1
544
User
Rockville
1
545
User
Bowie
1
546
User
Rockville
1
547
User
Potomac
1
548
User
Rockville
1
549
User
Rockville
1
550
User
Owings Mills
1
551
User
Rockville
1
552
User
Rockville
1
553
User
Rockville
1
554
User
Rockville
1
555
User
Rockville
1
556
User
Oxon Hill-Glassmanor
1
557
User
Rockville
1
558
User
Rockville
1
559
User
Germantown
1
560
User
Gaithersburg
1
561
User
Baltimore
1
562
User
Leonardtown
1
563
User
Upper Marlboro
1
564
User
Hyattsville
1
565
User
Bowie
1
566
User
Baltimore
1
567
User
Chevy Chase
1
568
User
Wheaton-Glenmont
1
569
User
Baltimore
1
570
User
Pikesville
1
571
User
Baltimore
1
572
User
Baltimore
1
573
User
Green Valley
1
574
User
Savage
1
575
User
Greenbelt
1
576
User
Clarksburg
1
577
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 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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