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