Leaderboard
Washington
Rank
Level
1
Grizzly Bear
Bellevue
91
2
Sauske
Bellevue
89
3
timmytanner21
Seattle
87
4
Ben Yu
Seattle
86
5
UserChitti
Bellevue
84
6
Daneo
South Hill
82
7
User
South Hill
75
8
Ignacio Barela
Bellevue
73
9
Kailey
West Richland
72
10
User
Seattle
66
11
Myra Lee
Bellevue
64
12
User
Seattle
61
13
Xx_360NoscopeBigChungusDogLasagna_xX
Bellevue
60
14
Timothy Tan
Bellevue
59
15
David Barr
Kent
56
16
User
Seattle
56
17
Katie Lee
Bellevue
51
18
Caden Tan
Bellevue
51
19
Pam F
Seattle
51
20
xhaw hann
Seattle
50
21
User
Vancouver
49
22
User
South Hill
49
23
User
Bellevue
49
24
caden tan
Bellevue
47
25
Marcuschau
Bellevue
45
26
User
Seattle
45
27
🦋🦋🦋
Bellevue
44
28
Hayden Kwan
Bellevue
44
29
Marina Sol
Bellevue
43
30
Moon
Bellevue
43
31
User
Big Lake
42
32
JeshKeem
South Hill
42
33
null
Bellevue
42
34
User
Seattle
41
35
Zachary Banken
Wenatchee
41
36
User
Graham
41
37
Brian Todd
Seattle
41
38
SoSoSophia
Bellevue
40
39
NateTheGreat
Bellingham
40
40
User
Redmond
40
41
Andrew Tang
Bellevue
39
42
User
Bellevue
39
43
User
Seattle
39
44
Nathan
Bellevue
38
45
User
Seattle
38
46
User
South Hill
37
47
User
Redmond
36
48
Rahul Chandra
Battle Ground
36
49
ravenclawstrength 7
Bellevue
35
50
Andrea Lim
Bellevue
35
51
Jason Lee
Bellevue
35
52
User
Bellevue
35
53
User
Bothell
35
54
nathaniel
South Hill
34
55
Jonathan Lai
Sammamish
34
56
User
Seattle
33
57
User
South Hill
33
58
User
Pullman
32
59
Kyle G
Sammamish
32
60
User
Kirkland
31
61
User
Seattle
31
62
User
South Hill
31
63
User
Bellevue
31
64
User
Kent
31
65
Señor Vac
Bellevue
30
66
User
Seattle
30
67
Beastx4 43
Seattle
30
68
Jeff Rubingh
Seattle
29
69
User
Bellevue
29
70
User
South Hill
29
71
User
Bellevue
29
72
User
Vancouver
29
73
Ryan Martin
Seattle
29
74
User
South Hill
28
75
User
Seattle
28
76
User
Sammamish
27
77
User
Bellevue
27
78
Claire James
Bellevue
27
79
User
Seattle
27
80
Jonathon Jerome
Oyehut-Hogans Corner
27
81
User
Seattle
27
82
User
Seattle
26
83
User
Seattle
26
84
User
Bellevue
26
85
cenzo
Seattle
26
86
User
Vancouver
26
87
User
Renton
25
88
User
Bellevue
25
89
User
Mount Vista
25
90
User
South Hill
25
91
User
Seattle
25
92
User
Seattle
25
93
User
Seattle
24
94
User
Bellevue
24
95
User
South Hill
24
96
User
Seattle
24
97
User
Port Angeles
24
98
User
Bellingham
24
99
User
Federal Way
24
100
User
Federal Way
24
101
User
Bellevue
23
102
Rong Ge
Bellevue
23
103
User
Seattle
23
104
User
South Hill
23
105
User
South Hill
23
106
User
Seattle
23
107
User
Bellevue
22
108
User
Gig Harbor
22
109
User
Seattle
22
110
User
Federal Way
22
111
User
Bellevue
22
112
FG
Pullman
22
113
User
South Hill
22
114
User
South Hill
22
115
User
Bellevue
22
116
BlueHero
Bellevue
22
117
User
Seattle
22
118
User
South Hill
22
119
User
South Hill
22
120
User
South Hill
22
121
User
South Hill
22
122
User
Bellevue
22
123
User
North Creek
22
124
User
West Richland
22
125
User
Tacoma
22
126
User
Seattle
22
127
Koalalover koalas are the best animals in the world
Bellevue
21
128
Spongebob
Bellevue
21
129
Maddie
Bellevue
21
130
User
Seattle
21
131
User
Bellevue
21
132
User
Bainbridge Island
21
133
User
Bellevue
21
134
User
Redmond
21
135
User
Burien
21
136
User
South Hill
21
137
User
South Hill
21
138
User
South Hill
21
139
Jacqueline Hergott
Bellevue
21
140
User
Bainbridge Island
21
141
User
South Hill
21
142
User
South Hill
21
143
User
South Hill
21
144
User
South Hill
21
145
User
South Hill
21
146
User
Moses Lake
21
147
User
Ephrata
21
148
User
Mercer Island
21
149
User
Bellevue
21
150
Elizabeth Shin
Bellevue
20
151
Hanzo festive
Bellevue
20
152
User
Bellevue
20
153
User
Bellevue
20
154
User
South Hill
20
155
Jelly Lau
Bellevue
20
156
User
Seattle
20
157
User
South Hill
20
158
User
Bonney Lake
20
159
User
South Hill
20
160
User
Seattle
20
161
User
South Hill
20
162
User
South Hill
20
163
User
South Hill
20
164
User
West Lake Stevens
20
165
User
Marysville
20
166
Sofie Luk
Bellevue
19
167
ZappyScroll6732
Bellevue
19
168
User
Bellevue
19
169
Hanah
Bellevue
19
170
User
Seattle
19
171
Dhanya Venkatesh
Redmond
19
172
User
Redmond
19
173
User
Spokane Valley
19
174
User
South Hill
19
175
User
South Hill
19
176
Darth Vader
Bellevue
19
177
User
Seattle
19
178
User
Burien
19
179
User
South Hill
19
180
User
South Hill
19
181
User
South Hill
19
182
User
Everett
19
183
User
Seattle
19
184
User
South Hill
18
185
User
South Hill
18
186
User
South Hill
18
187
User
South Hill
18
188
User
Seattle
18
189
User
Vancouver
18
190
User
Birch Bay
18
191
m bruce
Seattle
18
192
User
Seattle
18
193
User
Bellevue
17
194
ebrima sambou
Joint Base Lewis-Mcchord
17
195
kathy lee
Bellevue
17
196
AddotheFatto
Bellingham
17
197
Josh
Seattle
17
198
User
South Hill
17
199
Adrean's Airplanes
Bellevue
17
200
User
South Hill
17
201
User
South Hill
17
202
User
South Hill
17
203
User
Bellevue
17
204
User
Tacoma
17
205
User
Richland
17
206
Léo Bruce
Seattle
17
207
User
Seattle
17
208
User
Bellevue
16
209
User
Bellevue
16
210
Darth Vader
Bellevue
16
211
User
South Hill
16
212
User
South Hill
16
213
User
Seattle
16
214
User
Seattle
16
215
User
Kent
16
216
Ernest Krumm
Seattle
16
217
User
South Hill
16
218
User
South Hill
16
219
Sketchez
Bellevue
15
220
User
Bellevue
15
221
User
Seattle
15
222
User
Everett
15
223
User
Bellevue
15
224
User
Bellevue
15
225
Erika
Seattle
15
226
User
Oyehut-Hogans Corner
15
227
User
Seattle
15
228
User
Enumclaw
15
229
User
Seattle
15
230
User
Arlington
15
231
User
South Hill
15
232
User
South Hill
15
233
User
South Hill
15
234
User
Seattle
15
235
User
Lacey
15
236
User
Lynnwood
15
237
User
Gig Harbor
15
238
tv규민
Bellevue
14
239
User
Bellevue
14
240
User
Everett
14
241
User
South Hill
14
242
User
Woodinville
14
243
User
Cheney
14
244
User
South Hill
14
245
User
Seattle
14
246
User
South Hill
14
247
User
Seattle
14
248
User
Castle Rock
14
249
Madeline Walker
Bellevue
13
250
Graceful Grape
Bellevue
13
251
User
Seattle
13
252
User
Bellevue
13
253
User
Spokane Valley
13
254
User
South Hill
13
255
User
South Hill
13
256
User
South Hill
13
257
User
Tacoma
13
258
User
Oyehut-Hogans Corner
13
259
User
Tacoma
13
260
User
Spokane
13
261
User
Roslyn
13
262
User
Seattle
13
263
User
South Hill
13
264
User
South Hill
13
265
User
South Hill
13
266
User
Bellingham
13
267
User
Seattle
13
268
KitKatKayda
Bellevue
12
269
User
Bainbridge Island
12
270
Nona art
Bellevue
12
271
User
Bellevue
12
272
User
Bellevue
12
273
User
South Hill
12
274
User
South Hill
12
275
User
South Hill
12
276
User
South Hill
12
277
User
South Hill
12
278
User
Spokane Valley
12
279
User
Bellingham
12
280
User
Bellingham
12
281
Deepti Anoop
Kirkland
12
282
User
Seattle
12
283
User
Bellevue
12
284
User
Seattle
12
285
Tony Ayad
Bellevue
11
286
User
Bellevue
11
287
User
South Hill
11
288
User
Spokane Valley
11
289
User
Woodland
11
290
User
Bainbridge Island
11
291
User
Enumclaw
11
292
User
South Hill
11
293
User
Pullman
11
294
User
South Hill
11
295
User
South Hill
11
296
User
South Hill
11
297
User
Everett
11
298
User
Seattle
11
299
User
Tacoma
11
300
User
Seattle
11
301
User
Bellingham
11
302
Penghao Chen
Seattle
11
303
User
Eatonville
11
304
Andrew Hanson
Auburn
10
305
TSM_Cloud 9_Faze_LitJas
Bellevue
10
306
User
Redmond
10
307
User
Bellevue
10
308
Philip Kim
Bellevue
10
309
User
Seattle
10
310
User
Seattle
10
311
User
Redmond
10
312
User
South Hill
10
313
User
South Hill
10
314
User
South Hill
10
315
User
South Hill
10
316
User
South Hill
10
317
User
Sammamish
10
318
User
Auburn
10
319
User
Seattle
10
320
User
Issaquah
10
321
User
Richland
10
322
User
Bellevue
9
323
User
Bothell
9
324
Joy Kim
Bellevue
9
325
User
Seattle
9
326
Makenzi
Seattle
9
327
User
Bellevue
9
328
dr dramatic
South Hill
9
329
User
Seattle
9
330
Zengjian Hu
Bellevue
9
331
User
Seattle
9
332
User
Oyehut-Hogans Corner
9
333
User
Oyehut-Hogans Corner
9
334
User
Enumclaw
9
335
User
South Hill
9
336
User
South Hill
9
337
User
South Hill
9
338
User
South Hill
9
339
User
South Hill
9
340
User
Seattle
9
341
Anya
Bellevue
8
342
User
Roslyn
8
343
User
Seattle
8
344
User
Bellevue
8
345
User
Kent
8
346
User
South Hill
8
347
User
Seattle
8
348
User
Redmond
8
349
User
Seattle
8
350
User
Spokane
8
351
User
Enumclaw
8
352
User
South Hill
8
353
User
South Hill
8
354
User
South Hill
8
355
User
Federal Way
8
356
User
Picnic Point-North Lynnwood
8
357
User
Spokane
8
358
User
Spokane Valley
8
359
Luke Morasch
Spokane
8
360
User
Sunnyside
8
361
User
Federal Way
8
362
monica
Seattle
8
363
User
Bellevue
8
364
Blah Blah
Bellevue
8
365
User
Bellevue
7
366
Shelby Toy
Bellevue
7
367
User
Aberdeen
7
368
seli nukunya
Union Hill-Novelty Hill
7
369
User
Bellevue
7
370
User
Bellevue
7
371
User
Bellevue
7
372
User
Tacoma
7
373
User
Seattle
7
374
User
Shoreline
7
375
User
Seattle
7
376
User
Lacey
7
377
User
Camano Island
7
378
That Hiking Guy
Spokane
7
379
User
Enumclaw
7
380
User
Moses Lake
7
381
User
Wenatchee
7
382
User
Redmond
7
383
User
South Hill
7
384
User
South Hill
7
385
User
South Hill
7
386
User
South Hill
7
387
User
South Hill
7
388
User
South Hill
7
389
User
South Hill
7
390
User
South Hill
7
391
User
Sammamish
7
392
Ariel Naputi
Lakewood
7
393
User
Bellevue
7
394
User
Seattle
7
395
User
Birch Bay
7
396
User
Picnic Point-North Lynnwood
7
397
User
Seattle
7
398
User
Seattle
7
399
User
Puyallup
7
400
User
Tacoma
7
401
User
Bellevue
6
402
User
Federal Way
6
403
User
Vancouver
6
404
User
Bellevue
6
405
User
Mill Creek
6
406
Jaren Lau
Bellevue
6
407
User
Kent
6
408
User
Everett
6
409
User
South Hill
6
410
User
Snoqualmie
6
411
Zoe Zoe
Woodinville
6
412
User
Oyehut-Hogans Corner
6
413
User
Enumclaw
6
414
User
Bellevue
6
415
User
Seattle
6
416
User
South Hill
6
417
User
Kent
6
418
User
Tacoma
6
419
User
Seattle
6
420
User
Central Park
6
421
User
Federal Way
6
422
User
Everett
6
423
User
Seattle
6
424
User
Bothell
6
425
User
Seattle
6
426
User
Seattle
6
427
Konungur Svíþjóðar og Noregs
Bellevue
5
428
User
Redmond
5
429
User
Seattle
5
430
User
Kirkland
5
431
CHICKEN
Bellevue
5
432
User
Bellevue
5
433
User
Cottage Lake
5
434
User
Redmond
5
435
User
Mount Vernon
5
436
User
Bellevue
5
437
User
Seattle
5
438
User
Bellevue
5
439
User
Enumclaw
5
440
User
Enumclaw
5
441
User
Federal Way
5
442
User
Enumclaw
5
443
User
South Hill
5
444
User
South Hill
5
445
User
South Hill
5
446
User
South Hill
5
447
User
South Hill
5
448
User
South Hill
5
449
User
Loon Lake
5
450
User
Sammamish
5
451
User
Seattle
5
452
User
Seattle
5
453
User
Spokane
5
454
User
Bellevue
4
455
Addie Kuehner
Bellevue
4
456
User
Everett
4
457
User
Bellevue
4
458
User
Seattle
4
459
User
Seattle
4
460
User
Pullman
4
461
User
Seattle
4
462
User
Poulsbo
4
463
User
Seattle
4
464
User
Redmond
4
465
User
Seattle
4
466
User
North Creek
4
467
User
Bellevue
4
468
Winston Smith
Kirkland
4
469
User
Seattle
4
470
User
Wenatchee
4
471
Steven Wan
Bellevue
4
472
User
Oyehut-Hogans Corner
4
473
User
Bellevue
4
474
User
Aberdeen
4
475
User
Enumclaw
4
476
User
South Hill
4
477
User
Seattle
4
478
User
South Hill
4
479
User
South Hill
4
480
User
South Hill
4
481
User
South Hill
4
482
User
South Hill
4
483
User
South Hill
4
484
User
South Hill
4
485
User
South Hill
4
486
User
South Hill
4
487
User
Seattle
4
488
User
Poulsbo
4
489
User
Manchester
4
490
User
Bellingham
4
491
User
Redmond
4
492
User
Lynnwood
4
493
User
Sammamish
4
494
User
Spokane Valley
4
495
User
Lake Stevens
4
496
User
Lake Stevens
4
497
User
Bellevue
3
498
User
Seattle
3
499
Navtaj Singh
Pullman
3
500
User
Seattle
3
501
User
Everett
3
502
User
Lacey
3
503
User
Seattle
3
504
User
Seattle
3
505
User
Seattle
3
506
Jason Lee
Bellevue
3
507
User
Bellevue
3
508
User
Redmond
3
509
User
Bellevue
3
510
User
Bellevue
3
511
User
South Hill
3
512
User
Seattle
3
513
User
Bellevue
3
514
User
Bellingham
3
515
Jaren Lau
Bellevue
3
516
User
Bellevue
3
517
christina mocan
Bellevue
3
518
User
Oyehut-Hogans Corner
3
519
User
Oyehut-Hogans Corner
3
520
User
Oyehut-Hogans Corner
3
521
User
Central Park
3
522
User
Kalama
3
523
User
Shoreline
3
524
User
Seattle
3
525
the main sandwich
South Hill
3
526
User
South Hill
3
527
User
South Hill
3
528
User
South Hill
3
529
User
Seattle
3
530
User
Kent
3
531
User
South Hill
3
532
User
South Hill
3
533
User
South Hill
3
534
User
South Hill
3
535
User
South Hill
3
536
User
South Hill
3
537
Ron Burns
South Hill
3
538
User
Bellevue
3
539
User
Ellensburg
3
540
User
Bellevue
3
541
Jonathan Morasch
Spokane
3
542
User
Seattle
3
543
User
Steilacoom
3
544
User
South Hill
3
545
User
Seattle
3
546
User
Kent
3
547
User
Mukilteo
3
548
User
Seattle
3
549
User
Bellevue
3
550
User
Bellingham
3
551
User
Seattle
3
552
User
Seattle
3
553
User
Seattle
3
554
User
Seattle
2
555
BIG CHUNGUS BIG CHUNGUS BIG CHUNGUS BIG CHUNGUS BIG CHUNGUS
Bellevue
2
556
Avery Obay
Bellevue
2
557
User
Seattle
2
558
Mr.Tim
Bellevue
2
559
User
Country Homes
2
560
User
Lacey
2
561
User
Auburn
2
562
User
Seattle
2
563
User
Issaquah
2
564
User
White Center
2
565
User
Poulsbo
2
566
User
Bellevue
2
567
User
Bellevue
2
568
User
Othello
2
569
User
Seattle
2
570
User
Bellevue
2
571
User
Camano Island
2
572
User
Bothell
2
573
rayraythecheese
Bellevue
2
574
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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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