Leaderboard
Washington
Rank
Level
1
Detective Pikachu
Redmond
100
2
Debabi
Redmond
100
3
User
Seattle
99
4
imo miklos
Seattle
99
5
timmytanner21
Bellevue
94
6
User
Seattle
93
7
User
Seattle
93
8
ravenclawstrength 7
Seattle
93
9
Sauske
Bellevue
92
10
Grizzly Bear
Bellevue
92
11
Myra Lee
Bellevue
87
12
Michael Li
Mill Creek
86
13
Ben Yu
Seattle
86
14
Luis Sanchez-Ulloa
Seattle
85
15
UserChitti
Bellevue
84
16
Daneo
South Hill
82
17
User
South Hill
75
18
Ignacio Barela
Bellevue
73
19
Katie Lee
Bellevue
72
20
Xx_360NoscopeBigChungusDogLasagna_xX
Bellevue
72
21
Kailey
West Richland
72
22
John Smith
Seattle
71
23
Marcuschau
Bellevue
69
24
Andrea Lim
Bellevue
68
25
Joy Kim
Bellevue
68
26
Estra
Redmond
67
27
User
Seattle
66
28
User
Seattle
65
29
caden tan
Bellevue
63
30
User
Seattle
61
31
Timothy Tan
Bellevue
59
32
Jingyang Guo
Seattle
57
33
Jeff Rubingh
Seattle
57
34
Marina Sol
Seattle
57
35
David Barr
Kent
56
36
User
Seattle
56
37
User
Bainbridge Island
54
38
annsgrqce
Bellevue
54
39
Hayden Kwan♟♟♟
Bellevue
53
40
Caden Tan
Bellevue
51
41
Pam F
Seattle
51
42
Toby Clark
Shoreline
50
43
Andrew Tang
Bellevue
50
44
Señor Vac
Bellevue
50
45
xhaw hann
Seattle
50
46
SoSoSophia
Bellevue
49
47
User
Vancouver
49
48
User
South Hill
49
49
User
Bellevue
49
50
Justin Jradi
Seattle
48
51
User
Bainbridge Island
47
52
User
Seattle
46
53
User
Seattle
45
54
User
Seattle
44
55
Abe Link
Seattle
44
56
Moon
Bellevue
43
57
User
Big Lake
42
58
Nora Rothman
Seattle
42
59
ZappyScroll6732
Sammamish
42
60
JeshKeem
South Hill
42
61
null
Bellevue
42
62
Sofie Luk
Bellevue
41
63
Nathan
Bellevue
41
64
User
Seattle
41
65
Zachary Banken
Wenatchee
41
66
User
Graham
41
67
Brian Todd
Seattle
41
68
Maddie
Bellevue
40
69
NateTheGreat
Bellingham
40
70
User
Redmond
40
71
Jacqueline Hergott
Bellevue
39
72
User
Bellevue
39
73
User
Seattle
39
74
User
Issaquah
38
75
User
Seattle
38
76
User
South Hill
37
77
User
Redmond
36
78
Rahul Chandra
Battle Ground
36
79
User
Seattle
35
80
Jason Lee
Bellevue
35
81
User
Bellevue
35
82
User
Bothell
35
83
Koalalover
Bellevue
34
84
User
Kent
34
85
User
Bellevue
34
86
nathaniel
South Hill
34
87
Jonathan Lai
Sammamish
34
88
KitKatKayda
Bellevue
33
89
I love Zac
Bellevue
33
90
User
Seattle
33
91
User
South Hill
33
92
Nona art
Bellevue
32
93
User
Pullman
32
94
Kyle G
Sammamish
32
95
User
Bellevue
31
96
Sketchez
Bellevue
31
97
Hanah
Bellevue
31
98
User
Bellevue
31
99
User
Kirkland
31
100
User
Seattle
31
101
User
South Hill
31
102
User
Bellevue
31
103
User
Kent
31
104
User
Bellevue
30
105
tv규민
Bellevue
30
106
User
Bellevue
30
107
User
Seattle
30
108
Beastx4 43
Seattle
30
109
User
Bellevue
29
110
Hanzo festive
Bellevue
29
111
User
Bellevue
29
112
User
South Hill
29
113
User
Bellevue
29
114
User
Vancouver
29
115
User
Spokane
28
116
Elizabeth Shin
Kirkland
28
117
User
Bellevue
28
118
User
South Hill
28
119
User
Seattle
28
120
YO YO A PIRATES LIFE FOR ME
Bellevue
27
121
User
Sammamish
27
122
User
Bellevue
27
123
Claire James
Bellevue
27
124
User
Seattle
27
125
Jonathon Jerome
Oyehut-Hogans Corner
27
126
User
Seattle
27
127
User
Seattle
26
128
User
Ellensburg
26
129
User
Seattle
26
130
User
Bellevue
26
131
cenzo
Seattle
26
132
User
Vancouver
26
133
User
South Hill
25
134
User
Renton
25
135
User
Bellevue
25
136
User
Mount Vista
25
137
User
South Hill
25
138
User
Seattle
25
139
User
Seattle
25
140
User
Seattle
24
141
User
Bellevue
24
142
User
South Hill
24
143
User
Seattle
24
144
User
Port Angeles
24
145
User
Bellingham
24
146
User
Federal Way
24
147
User
Federal Way
24
148
User
Bellevue
23
149
User
Bellevue
23
150
Rong Ge
Bellevue
23
151
User
Seattle
23
152
User
South Hill
23
153
User
South Hill
23
154
User
Seattle
23
155
Graceful Grape
Bellevue
22
156
User
Big Lake
22
157
User
Gig Harbor
22
158
User
Seattle
22
159
User
Federal Way
22
160
User
Bellevue
22
161
FG
Pullman
22
162
User
South Hill
22
163
User
South Hill
22
164
User
Bellevue
22
165
BlueHero
Bellevue
22
166
User
Seattle
22
167
User
South Hill
22
168
User
South Hill
22
169
User
South Hill
22
170
User
South Hill
22
171
User
Bellevue
22
172
User
North Creek
22
173
User
West Richland
22
174
User
Tacoma
22
175
User
Seattle
22
176
User
City
21
177
User
Lakewood
21
178
User
Seattle
21
179
User
Bellevue
21
180
User
Bainbridge Island
21
181
User
Bellevue
21
182
User
Redmond
21
183
User
Burien
21
184
User
South Hill
21
185
User
South Hill
21
186
User
South Hill
21
187
Jacqueline Hergott
Bellevue
21
188
User
Bainbridge Island
21
189
User
South Hill
21
190
User
South Hill
21
191
User
South Hill
21
192
User
South Hill
21
193
User
South Hill
21
194
User
Moses Lake
21
195
User
Ephrata
21
196
User
Mercer Island
21
197
User
Bellevue
21
198
User
Seattle
20
199
User
Seattle
20
200
User
Bellevue
20
201
User
South Hill
20
202
Jelly Lau
Bellevue
20
203
User
Seattle
20
204
User
South Hill
20
205
User
Bonney Lake
20
206
User
South Hill
20
207
User
Seattle
20
208
User
South Hill
20
209
User
South Hill
20
210
User
South Hill
20
211
User
West Lake Stevens
20
212
User
Marysville
20
213
User
Bellevue
19
214
Madeline Walker
Bellevue
19
215
User
Bellevue
19
216
User
Seattle
19
217
Dhanya Venkatesh
Redmond
19
218
User
Redmond
19
219
User
Spokane Valley
19
220
User
South Hill
19
221
User
South Hill
19
222
Darth Vader
Bellevue
19
223
User
Seattle
19
224
User
Burien
19
225
User
South Hill
19
226
User
South Hill
19
227
User
South Hill
19
228
User
Everett
19
229
User
Seattle
19
230
Shelby Toy
Bellevue
18
231
User
Seattle
18
232
User
Federal Way
18
233
User
South Hill
18
234
User
South Hill
18
235
User
South Hill
18
236
User
South Hill
18
237
User
Seattle
18
238
User
Vancouver
18
239
User
Birch Bay
18
240
m bruce
Seattle
18
241
User
Seattle
18
242
User
Seattle
17
243
TSM_Cloud 9_Faze_LitJas
Bellevue
17
244
ebrima sambou
Joint Base Lewis-Mcchord
17
245
kathy lee
Bellevue
17
246
AddotheFatto
Bellingham
17
247
Josh
Seattle
17
248
User
South Hill
17
249
Adrean's Airplanes
Bellevue
17
250
User
South Hill
17
251
User
South Hill
17
252
User
South Hill
17
253
User
Bellevue
17
254
User
Tacoma
17
255
User
Richland
17
256
Léo Bruce
Seattle
17
257
User
Seattle
17
258
User
Seattle
16
259
User
Bellevue
16
260
User
Bellevue
16
261
Darth Vader
Bellevue
16
262
User
South Hill
16
263
User
South Hill
16
264
User
Seattle
16
265
User
Seattle
16
266
User
Kent
16
267
Ernest Krumm
Seattle
16
268
User
South Hill
16
269
User
South Hill
16
270
User
Bellevue
15
271
seli nukunya
Bellevue
15
272
User
Redmond
15
273
User
Bellevue
15
274
User
Bellevue
15
275
User
Everett
15
276
User
Bellevue
15
277
User
Bellevue
15
278
Erika
Seattle
15
279
User
Oyehut-Hogans Corner
15
280
User
Seattle
15
281
User
Enumclaw
15
282
User
Seattle
15
283
User
Arlington
15
284
User
South Hill
15
285
User
South Hill
15
286
User
South Hill
15
287
User
Seattle
15
288
User
Lacey
15
289
User
Lynnwood
15
290
User
Gig Harbor
15
291
User
Redmond
14
292
User
Allyn-Grapeview
14
293
Anya
Bellevue
14
294
User
Seattle
14
295
User
Issaquah
14
296
User
Bellevue
14
297
User
Everett
14
298
User
South Hill
14
299
User
Woodinville
14
300
User
South Hill
14
301
User
Seattle
14
302
User
South Hill
14
303
User
Seattle
14
304
User
Castle Rock
14
305
User
Vancouver
13
306
Ryan Martin
Seattle
13
307
User
Tukwila
13
308
Avery Obay
Bellevue
13
309
User
Cheney
13
310
User
Seattle
13
311
User
Bellevue
13
312
User
Spokane Valley
13
313
User
South Hill
13
314
User
South Hill
13
315
User
South Hill
13
316
User
Tacoma
13
317
User
Oyehut-Hogans Corner
13
318
User
Tacoma
13
319
User
Spokane
13
320
User
Roslyn
13
321
User
Seattle
13
322
User
South Hill
13
323
User
South Hill
13
324
User
South Hill
13
325
User
Bellingham
13
326
User
Seattle
13
327
john
Bellevue
12
328
User
Seattle
12
329
User
Seattle
12
330
User
Bellevue
12
331
User
Bellevue
12
332
User
South Hill
12
333
User
South Hill
12
334
User
South Hill
12
335
User
South Hill
12
336
User
South Hill
12
337
User
Spokane Valley
12
338
User
Bellingham
12
339
User
Bellingham
12
340
Deepti Anoop
Kirkland
12
341
User
Seattle
12
342
User
Bellevue
12
343
User
Seattle
12
344
Andrew Hanson
Auburn
11
345
User
Vancouver
11
346
User
Kirkland
11
347
User
Bellevue
11
348
User
South Hill
11
349
User
Spokane Valley
11
350
User
Woodland
11
351
User
Bainbridge Island
11
352
User
Enumclaw
11
353
User
South Hill
11
354
User
Pullman
11
355
User
South Hill
11
356
User
South Hill
11
357
User
South Hill
11
358
User
Everett
11
359
User
Seattle
11
360
User
Tacoma
11
361
User
Seattle
11
362
User
Bellingham
11
363
Penghao Chen
Seattle
11
364
User
Eatonville
11
365
User
Federal Way
10
366
User
Bellevue
10
367
User
Bellevue
10
368
User
Redmond
10
369
User
Seattle
10
370
User
Redmond
10
371
User
Bellevue
10
372
Philip Kim
Bellevue
10
373
User
Seattle
10
374
User
Seattle
10
375
User
Redmond
10
376
User
South Hill
10
377
User
South Hill
10
378
User
South Hill
10
379
User
South Hill
10
380
User
South Hill
10
381
User
Sammamish
10
382
User
Auburn
10
383
User
Seattle
10
384
User
Issaquah
10
385
User
Richland
10
386
User
Ellensburg
9
387
User
Seattle
9
388
User
Allyn-Grapeview
9
389
Konungur Svíþjóðar og Noregs
Bellevue
9
390
User
Everett
9
391
User
Bellevue
9
392
User
Bothell
9
393
User
Seattle
9
394
Makenzi
Seattle
9
395
User
Bellevue
9
396
dr dramatic
South Hill
9
397
User
Seattle
9
398
Zengjian Hu
Bellevue
9
399
User
Seattle
9
400
User
Oyehut-Hogans Corner
9
401
User
Oyehut-Hogans Corner
9
402
User
Enumclaw
9
403
User
South Hill
9
404
User
South Hill
9
405
User
South Hill
9
406
User
South Hill
9
407
User
South Hill
9
408
User
Seattle
9
409
User
Seattle
8
410
User
Seattle
8
411
User
Chimacum
8
412
User
Roslyn
8
413
User
Seattle
8
414
User
Bellevue
8
415
User
Kent
8
416
User
South Hill
8
417
User
Seattle
8
418
User
Redmond
8
419
User
Seattle
8
420
User
Spokane
8
421
User
Enumclaw
8
422
User
South Hill
8
423
User
South Hill
8
424
User
South Hill
8
425
User
Federal Way
8
426
User
Picnic Point-North Lynnwood
8
427
User
Spokane
8
428
User
Spokane Valley
8
429
Luke Morasch
Spokane
8
430
User
Sunnyside
8
431
User
Federal Way
8
432
monica
Seattle
8
433
User
Bellevue
8
434
Blah Blah
Bellevue
8
435
User
Bellingham
7
436
Mr.Tim
Bellevue
7
437
User
Aberdeen
7
438
User
Bellevue
7
439
User
Bellevue
7
440
User
Bellevue
7
441
User
Tacoma
7
442
User
Seattle
7
443
User
Shoreline
7
444
User
Seattle
7
445
User
Lacey
7
446
User
Camano Island
7
447
That Hiking Guy
Spokane
7
448
User
Enumclaw
7
449
User
Moses Lake
7
450
User
Wenatchee
7
451
User
Redmond
7
452
User
South Hill
7
453
User
South Hill
7
454
User
South Hill
7
455
User
South Hill
7
456
User
South Hill
7
457
User
South Hill
7
458
User
South Hill
7
459
User
South Hill
7
460
User
Sammamish
7
461
Ariel Naputi
Lakewood
7
462
User
Bellevue
7
463
User
Seattle
7
464
User
Birch Bay
7
465
User
Picnic Point-North Lynnwood
7
466
User
Seattle
7
467
User
Seattle
7
468
User
Puyallup
7
469
User
Tacoma
7
470
User
Richland
6
471
User
Seattle
6
472
User
Lynnwood
6
473
User
Paine Field-Lake Stickney
6
474
User
Seattle
6
475
User
Seattle
6
476
User
Seattle
6
477
User
Seattle
6
478
User
Cascade-Fairwood
6
479
User
Bellevue
6
480
User
Federal Way
6
481
User
Vancouver
6
482
User
Bellevue
6
483
User
Mill Creek
6
484
Jaren Lau
Bellevue
6
485
User
Kent
6
486
User
Everett
6
487
User
South Hill
6
488
User
Snoqualmie
6
489
Zoe Zoe
Woodinville
6
490
User
Oyehut-Hogans Corner
6
491
User
Enumclaw
6
492
User
Bellevue
6
493
User
Seattle
6
494
User
South Hill
6
495
User
Kent
6
496
User
Tacoma
6
497
User
Seattle
6
498
User
Central Park
6
499
User
Federal Way
6
500
User
Everett
6
501
User
Seattle
6
502
User
Bothell
6
503
User
Seattle
6
504
User
Tacoma
5
505
User
City
5
506
Joe Martin
Seattle
5
507
User
Seattle
5
508
Tayyib Sheikh
Seattle
5
509
Navtaj Singh
Pullman
5
510
User
Edmonds
5
511
User
Seattle
5
512
User
Redmond
5
513
User
Seattle
5
514
User
Kirkland
5
515
CHICKEN
Bellevue
5
516
User
Bellevue
5
517
User
Cottage Lake
5
518
User
Redmond
5
519
User
Mount Vernon
5
520
User
Bellevue
5
521
User
Seattle
5
522
User
Bellevue
5
523
User
Enumclaw
5
524
User
Enumclaw
5
525
User
Federal Way
5
526
User
Enumclaw
5
527
User
South Hill
5
528
User
South Hill
5
529
User
South Hill
5
530
User
South Hill
5
531
User
South Hill
5
532
User
South Hill
5
533
User
Loon Lake
5
534
User
Sammamish
5
535
User
Seattle
5
536
User
Seattle
5
537
User
Seattle
4
538
User
Picnic Point-North Lynnwood
4
539
User
Ellensburg
4
540
User
Redmond
4
541
User
Shoreline
4
542
User
Lacey
4
543
User
Artondale
4
544
Haley Ross
Bellevue
4
545
User
Seattle
4
546
User
Lynnwood
4
547
User
Redmond
4
548
User
Longview
4
549
User
Bellevue
4
550
User
Seattle
4
551
User
Eastsound
4
552
User
Seattle
4
553
BIG CHUNGUS BIG CHUNGUS BIG CHUNGUS BIG CHUNGUS BIG CHUNGUS
Bellevue
4
554
User
Bellevue
4
555
User
Everett
4
556
User
Bellevue
4
557
User
Seattle
4
558
User
Seattle
4
559
User
Pullman
4
560
User
Seattle
4
561
User
Poulsbo
4
562
User
Seattle
4
563
User
Redmond
4
564
User
Seattle
4
565
User
North Creek
4
566
User
Bellevue
4
567
Winston Smith
Kirkland
4
568
User
Seattle
4
569
User
Wenatchee
4
570
Steven Wan
Bellevue
4
571
User
Oyehut-Hogans Corner
4
572
User
Bellevue
4
573
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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