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