Leaderboard
Texas
Rank
Level
1
Jeffrey Huang
Austin
94
2
Michalis Paizanis
Denton
94
3
Aden Smith
Houston
94
4
User
Dallas
92
5
User
Dallas
91
6
User
Dallas
91
7
CEL IV
Dallas
91
8
User
Dallas
90
9
User
Dallas
90
10
User
Richardson
90
11
User
Dallas
90
12
User
Dallas
89
13
User
Dallas
89
14
User
Austin
88
15
CEL-IV
Richardson
88
16
CEL IV
Dallas
88
17
Adam Chen
Cinco Ranch
88
18
User
Dallas
87
19
User
Dallas
87
20
User
Dallas
87
21
CEL IV
Dallas
87
22
User
Dallas
86
23
Andrew Couser
Weatherford
86
24
User
Dallas
86
25
User
Dallas
85
26
User
Dallas
84
27
Mike Orr
Austin
83
28
User
Dallas
83
29
Kevin Copeland
Leander
83
30
User
Houston
82
31
User
Dallas
82
32
User
Dallas
82
33
User
Dallas
81
34
User
Dallas
80
35
justin silver
Denton
80
36
Jordan Wells
Atascocita
80
37
User
Austin
79
38
User
Dallas
79
39
Caleb Strange
Dallas
78
40
Pi Monster
Farmers Branch
78
41
Adrian G Casillas
Austin
77
42
User
Wylie
76
43
Daniel Cho
Fort Worth
76
44
User
Dallas
75
45
Robert Silvy
Houston
75
46
User
Dallas
75
47
sourry
Atascocita
75
48
User
Dallas
74
49
User
Dallas
74
50
User
Dallas
74
51
User
Houston
74
52
Ansun Sujoe
Fort Worth
74
53
User
Plano
74
54
User
Dallas
72
55
User
Austin
72
56
User
Dallas
71
57
Doug Childs
Hallsville
71
58
User
Dallas
70
59
User
Dallas
70
60
User
Dallas
70
61
User
Dallas
69
62
User
Dallas
69
63
User
Dallas
68
64
Bryan Vu
Austin
67
65
User
Dallas
67
66
Derek Cheung
Austin
67
67
User
Houston
66
68
User
Houston
66
69
Joseph Cano
Fulshear
66
70
User
Dallas
66
71
KK SS
Houston
65
72
User
Dallas
65
73
User
Southlake
65
74
User
Katy
65
75
User
Dallas
64
76
Math god
San Antonio
64
77
User
Dallas
64
78
User
Dallas
64
79
User
Houston
64
80
User
Fort Worth
64
81
User
Dallas
63
82
J Hulme
Austin
63
83
User
San Antonio
63
84
User
Dallas
63
85
User
Austin
63
86
User
Dallas
63
87
User
Dallas
63
88
Carl Svahn
Austin
63
89
User
Dallas
62
90
Kenny Huang
Plano
62
91
User
Beaumont
61
92
User
Dallas
61
93
User
San Antonio
61
94
User
Dallas
61
95
Nolan Raimo
Austin
61
96
User
North Richland Hills
60
97
User
Dallas
60
98
User
Dallas
60
99
User
Dallas
60
100
Bob Luo
Plano
60
101
User
Dallas
60
102
User
Missouri City
60
103
User
Abilene
59
104
User
Dallas
59
105
User
Plano
59
106
User
College Station
59
107
Reed Tubbs
Pflugerville
58
108
User
San Antonio
57
109
User
Houston
57
110
User
Allen
57
111
Crespo
Houston
57
112
cool guy
Mission
55
113
User
San Antonio
54
114
Ricky Kirk
Dallas
54
115
User
Irving
53
116
User
Austin
53
117
User
San Antonio
52
118
User
San Antonio
52
119
User
San Antonio
52
120
User
Atascocita
52
121
Katsumi Morales
Dallas
52
122
User
Houston
52
123
Issam
Houston
52
124
User
Dallas
52
125
Mihira Gore
Richmond
52
126
User
Mcallen
52
127
User
Dallas
51
128
User
Richardson
51
129
Robert Silvy
Houston
51
130
Dale Yoon
Houston
51
131
User
Austin
51
132
Ed
Missouri City
50
133
User
Dallas
50
134
User
Dallas
50
135
User
Dallas
50
136
User
Dallas
50
137
User
Dallas
50
138
User
Dallas
50
139
User
Houston
50
140
User
Houston
50
141
User
Dallas
50
142
User
Dallas
50
143
User
Dallas
50
144
User
Dallas
50
145
User
Dallas
50
146
User
San Antonio
49
147
john desmond
Austin
49
148
User
Dallas
49
149
User
Dallas
48
150
Letscher IV
Dallas
48
151
User
Abilene
48
152
User
Austin
48
153
User
Dallas
48
154
User
Plano
48
155
User
Austin
47
156
User
Garland
47
157
YT Dragonlak
Dallas
47
158
Kaitlynn Trinh
Houston
47
159
Irina
Houston
47
160
User
Abilene
47
161
User
Dallas
47
162
User
Carrollton
47
163
User
Dallas
46
164
User
Houston
46
165
User
Irving
46
166
User
Austin
46
167
User
Dallas
46
168
User
Dallas
46
169
Ethan Boyd
Dallas
46
170
User
Frisco
46
171
User
San Antonio
45
172
NoThisIsJohn
Austin
45
173
User
North Richland Hills
45
174
User
Wylie
45
175
User
Houston
45
176
User
Houston
45
177
Aryan Padarthi
Allen
45
178
Samagra Jain
Austin
44
179
User
Austin
44
180
User
Austin
44
181
User
Austin
44
182
User
Dallas
44
183
User
Mcallen
44
184
User
Allen
44
185
User
Leander
43
186
User
Dallas
43
187
User
San Antonio
43
188
User
Fort Worth
43
189
User
Plano
43
190
User
Austin
43
191
Hawkins Burns
Bee Cave
43
192
Bethany Disque
Lubbock
43
193
User
Argyle
43
194
User
San Antonio
42
195
User
San Antonio
42
196
User
Arlington
42
197
User
Dallas
42
198
User
Wylie
41
199
User
Wylie
41
200
User
Arlington
41
201
User
Mckinney
41
202
User
Plano
41
203
User
Houston
41
204
User
Longview
41
205
User
Dallas
41
206
User
Houston
40
207
User
Dallas
40
208
Joseph Walker
Lubbock
40
209
User
Farmers Branch
40
210
User
Fort Worth
40
211
User
Dallas
40
212
Matthew Miller
Galveston
40
213
User
Houston
40
214
User
Dallas
40
215
User
Houston
40
216
User
Houston
40
217
User
Houston
40
218
User
Donna
40
219
User
Dallas
40
220
Abhishek Sharma
Plano
40
221
User
Mission
40
222
Patrick Bagge
Houston
40
223
User
Austin
40
224
User
San Antonio
39
225
User
Dallas
39
226
User
Houston
39
227
User
La Porte
39
228
Nikki
El Paso
39
229
User
Houston
39
230
Uzair Akram
Euless
39
231
User
Waco
39
232
Graphing Paper
Atascocita
39
233
User
Austin
39
234
User
Frisco
38
235
Nick T
Houston
38
236
Jeff T
Houston
38
237
Cedric Zhou
Plano
38
238
Rivetrosity
Fort Worth
38
239
User
Atascocita
38
240
User
Austin
38
241
Dhruv Desai
Austin
38
242
User
Fulshear
38
243
User
Dallas
38
244
Amanda Barnwell
Austin
38
245
User
Houston
38
246
User
San Antonio
37
247
User
San Antonio
37
248
Neil Dabholkar
Austin
37
249
User
Dallas
37
250
User
Plano
37
251
User
Weatherford
37
252
User
Fulshear
37
253
User
Donna
37
254
User
Houston
37
255
User
Schertz
37
256
User
Richmond
37
257
User
Austin
37
258
User
Garland
37
259
User
Wylie
36
260
User
Houston
36
261
User
Fort Worth
36
262
Davin Park
Dallas
36
263
User
Corpus Christi
36
264
User
Houston
36
265
Math Test
Frisco
36
266
Billy Zhao
Flower Mound
35
267
User
San Antonio
35
268
User
Houston
35
269
User
Austin
35
270
User
Fulshear
35
271
User
Austin
35
272
User
San Antonio
35
273
User
Richardson
35
274
User
College Station
35
275
User
Katy
35
276
User
Houston
34
277
EY Lee
Houston
34
278
User
Dallas
34
279
User
Houston
34
280
User
Royse City
34
281
User
Houston
34
282
User
Laredo
34
283
User
Sugar Land
34
284
User
Wichita Falls
34
285
User
Houston
34
286
User
Houston
34
287
User
Houston
34
288
User
Houston
33
289
Rhett Marcelo
Houston
33
290
BigBootyBoi
Farmers Branch
33
291
User
Pflugerville
33
292
Aditya Guin
Plano
33
293
User
Katy
33
294
Jacob Torres
Round Rock
32
295
User
Plano
32
296
User
Dallas
32
297
piemanklp
Taylor
32
298
User
Abilene
32
299
User
Plano
32
300
User
Dallas
32
301
User
Austin
32
302
User
Temple
32
303
User
Pflugerville
32
304
Minh Nguyen
Houston
32
305
User
El Paso
32
306
User
Dallas
32
307
User
Houston
32
308
User
Houston
31
309
User
Longview
31
310
User
The Woodlands
31
311
User
Dallas
31
312
User
Houston
31
313
User
Carrollton
31
314
Eric Eldridge
Houston
31
315
User
Houston
31
316
User
San Antonio
31
317
User
Houston
31
318
User
Houston
31
319
User
Houston
31
320
User
Bellaire
31
321
Obafemi Ajibola
Arlington
31
322
User
San Antonio
30
323
User
Austin
30
324
User
Houston
30
325
Natalie Stafford
Houston
30
326
User
Houston
30
327
User
Dallas
30
328
Fabian Tarango
Houston
30
329
User
Austin
30
330
Alec Drean
Dallas
30
331
User
Dallas
30
332
User
Houston
30
333
User
Houston
30
334
User
Dallas
30
335
User
Houston
30
336
Angelina Covey
Atascocita
30
337
User
Wylie
30
338
User
Austin
30
339
Morgan Roberts
New Chapel Hill
29
340
User
Dallas
29
341
User
San Antonio
29
342
User
Dallas
29
343
User
Austin
29
344
User
Sugar Land
29
345
User
Fulshear
29
346
User
Dallas
29
347
User
Wichita Falls
29
348
bowdowntothestchair
League City
29
349
User
Fulshear
29
350
User
Dallas
28
351
User
Flower Mound
28
352
User
Austin
28
353
User
Garland
28
354
User
Spring
28
355
User
Houston
28
356
Nicole Hernandez
Houston
28
357
User
Dallas
28
358
User
Coppell
28
359
User
Mansfield
28
360
User
San Antonio
28
361
User
Austin
28
362
Jessica Wright
Missouri City
28
363
Robin
Austin
28
364
User
Wills Point
28
365
User
Dallas
28
366
User
Pearland
28
367
User
Cypress
27
368
User
Houston
27
369
User
Dallas
27
370
User
Dallas
27
371
User
Austin
27
372
User
El Paso
27
373
User
Sugar Land
27
374
Patrick Ly
Houston
27
375
User
Houston
27
376
User
Houston
27
377
User
Spring
27
378
User
College Station
26
379
Zabiah Jones
Houston
26
380
Mary Berkel
Cedar Park
26
381
User
Houston
26
382
User
Corpus Christi
26
383
User
Prosper
26
384
Worf JK
Amarillo
26
385
User
Irving
26
386
User
Dallas
26
387
User
Dallas
26
388
User
Houston
26
389
User
Dallas
26
390
User
Flower Mound
26
391
User
Cross Roads
26
392
User
Irving
26
393
User
Cypress
26
394
User
Dallas
26
395
User
Fulshear
26
396
User
Dallas
26
397
User
Houston
26
398
User
University Park
26
399
User
Houston
26
400
User
Houston
26
401
User
Killeen
26
402
ay nah
Lewisville
26
403
User
Houston
26
404
User
Houston
26
405
User
Arlington
26
406
User
San Antonio
25
407
User
Austin
25
408
User
Prosper
25
409
User
Dallas
25
410
John Young
Austin
25
411
User
Houston
25
412
User
Houston
25
413
User
Austin
25
414
User
Dallas
25
415
User
Lubbock
25
416
User
The Woodlands
25
417
User
El Paso
25
418
User
Plano
25
419
User
Bellaire
25
420
User
Sugar Land
25
421
User
Austin
25
422
User
Houston
25
423
User
Plano
25
424
User
Richmond
25
425
User
New Braunfels
25
426
User
San Antonio
24
427
User
Laredo
24
428
User
Houston
24
429
User
Rowlett
24
430
User
Houston
24
431
User
Houston
24
432
User
Carrollton
24
433
User
Houston
24
434
User
Houston
24
435
User
Cinco Ranch
24
436
Rafael Carrillo
The Woodlands
24
437
Linda Trigg
Houston
24
438
User
Cypress
24
439
User
Lubbock
24
440
User
Frisco
24
441
User
Houston
24
442
User
Dodd City
24
443
User
The Woodlands
24
444
User
Mesquite
24
445
User
Houston
24
446
Blue.
Wylie
24
447
Lucie Pyle
Houston
24
448
User
Plano
24
449
User
Cinco Ranch
24
450
User
Conroe
24
451
User
Fort Worth
24
452
User
Houston
24
453
User
Richardson
24
454
User
Dallas
24
455
User
Missouri City
24
456
User
San Antonio
23
457
User
San Antonio
23
458
Eli
Houston
23
459
User
The Woodlands
23
460
User
Houston
23
461
User
Friendswood
23
462
User
Austin
23
463
User
Houston
23
464
Eli Rowe
Fairfield
23
465
User
Dallas
23
466
User
Lewisville
23
467
User
Houston
23
468
User
Plano
23
469
User
Pflugerville
23
470
User
Euless
23
471
User
Fort Worth
23
472
User
Houston
23
473
User
Mesquite
23
474
User
Dallas
23
475
User
Burleson
23
476
User
Fulshear
23
477
User
Grapevine
23
478
Miriam Ballard
Austin
23
479
Gabrielle Parrish
Barton Creek
23
480
User
Allen
23
481
sebastian ramirez
Laredo
23
482
User
Denton
23
483
Amanda J
Allen
23
484
User
Sugar Land
23
485
User
Katy
23
486
User
Irving
23
487
User
Plano
23
488
User
Plano
23
489
Mithil Kansara
Plano
23
490
Lainey Hamilton
Longview
22
491
User
Houston
22
492
egrg
Houston
22
493
User
Atascocita
22
494
User
Houston
22
495
User
Fulshear
22
496
User
College Station
22
497
User
Abilene
22
498
User
Houston
22
499
User
Prosper
22
500
User
Denton
22
501
User
Temple
22
502
User
Flower Mound
22
503
User
New Braunfels
22
504
User
Round Rock
22
505
Phillip Ferguson
Austin
22
506
User
Houston
22
507
User
Dallas
22
508
User
San Antonio
22
509
User
Houston
22
510
Edward Garmon
Richardson
22
511
User
Dodd City
22
512
User
Houston
22
513
User
University Park
22
514
User
Houston
22
515
Lauren Douglas
Houston
22
516
S. Diaz
Oak Valley
22
517
User
Atascocita
22
518
User
Dallas
22
519
User
Leander
21
520
User
Austin
21
521
User
Dallas
21
522
User
Plano
21
523
User
Houston
21
524
User
Dallas
21
525
User
Houston
21
526
User
Bedford
21
527
User
Plano
21
528
Alejandro Saldaña Gamiño
Arlington
21
529
Josalyn Lievers
Dallas
21
530
Sue Ellen Fernandez
San Antonio
21
531
User
Dallas
21
532
User
Houston
21
533
Samuel Mason
Lubbock
21
534
User
Dallas
21
535
User
Leander
21
536
User
Irving
21
537
User
Plano
21
538
User
Round Rock
21
539
User
Austin
21
540
User
Mansfield
21
541
User
Dallas
21
542
User
Houston
21
543
User
Houston
21
544
User
Wylie
21
545
User
Houston
21
546
User
Houston
21
547
User
Houston
20
548
User
Dallas
20
549
User
San Antonio
20
550
User
Denton
20
551
User
Houston
20
552
User
Allen
20
553
User
Dallas
20
554
User
Houston
20
555
User
Houston
20
556
User
Brownsville
20
557
User
Mount Enterprise
20
558
User
Austin
20
559
User
Dallas
20
560
User
Houston
20
561
User
Arlington
20
562
User
Dallas
20
563
User
Houston
20
564
User
Castroville
20
565
User
El Paso
20
566
User
Austin
20
567
User
Dallas
20
568
User
Austin
20
569
User
Houston
20
570
User
Austin
20
571
User
Forney
20
572
User
City
20
573
User
Wichita Falls
20
574
User
Houston
20
575
User
La Porte
20
576
User
Sugar Land
20
577
User
Plano
20
578
User
Irving
20
579
Srinivas Pulipaka
Waco
20
580
User
Dallas
19
581
User
Dallas
19
582
User
Houston
19
583
User
Dallas
19
584
User
San Antonio
19
585
User
San Antonio
19
586
User
Irving
19
587
User
Cypress
19
588
User
New Braunfels
19
589
Nolan Wu
Houston
19
590
User
Austin
19
591
User
Dallas
19
592
User
Austin
19
593
User
Austin
19
594
User
Houston
19
595
User
Houston
19
596
Adam Chen
Fulshear
19
597
User
Houston
19
598
Elias Harrison
Odessa
19
599
User
Dallas
19
600
User
Houston
19
601
User
Plano
19
602
User
College Station
19
603
User
El Paso
19
604
User
Houston
19
605
User
Denton
19
606
User
Grapevine
19
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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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