Leaderboard
Texas
Rank
Level
1
Kevin Wang
Austin
99
2
User
Houston
93
3
ishaan grewal
Frisco
91
4
User
Dallas
90
5
User
Dallas
86
6
User
Dallas
85
7
Tanisha Grewal
Frisco
83
8
test
Frisco
81
9
CEL-IV
Dallas
79
10
CEL IV
Dallas
79
11
CEL 4
Dallas
78
12
Mike Orr
San Antonio
78
13
User
Dallas
78
14
User
Dallas
77
15
Gertibird
Abilene
77
16
CL
Dallas
76
17
CEL IV
Dallas
76
18
User
Dallas
76
19
User
Dallas
76
20
Jake Bloodsworth
Austin
75
21
User
Dallas
75
22
User
Richardson
75
23
User
Dallas
72
24
User
Dallas
72
25
User
Dallas
72
26
User
Conroe
72
27
Doug Childs
Hallsville
71
28
User
Dallas
70
29
User
Dallas
69
30
Joseph Hernandez
College Station
69
31
User
Austin
67
32
Caleb Strange
Dallas
67
33
User
Dallas
66
34
User
Dallas
66
35
User
Dallas
66
36
Bryan Vu
Austin
66
37
Ed
Missouri City
66
38
User
San Antonio
65
39
User
Dallas
65
40
User
Dallas
65
41
User
Dallas
64
42
User
Houston
64
43
CEL IV
Dallas
64
44
Jeffrey Huang
Austin
64
45
J Hulme
Austin
64
46
User
Austin
64
47
User
Dallas
63
48
User
Mckinney
63
49
User
Dallas
63
50
User
Austin
63
51
CEL-IV
Richardson
62
52
User
Dallas
62
53
User
San Antonio
62
54
Andrew Couser
Weatherford
61
55
User
San Antonio
61
56
CEL IV
Dallas
61
57
User
Houston
61
58
Svante
Pearland
61
59
User
Dallas
61
60
CEL IV
Dallas
59
61
User
Dallas
59
62
User
Dallas
59
63
User
Dallas
58
64
User
Dallas
58
65
User
Dallas
57
66
User
Dallas
57
67
User
Dallas
57
68
Osten Loo
Dallas
57
69
User
Dallas
56
70
User
Wylie
55
71
User
Dallas
55
72
User
Dallas
55
73
User
Dallas
54
74
Michael Rodgers
Houston
54
75
User
Houston
53
76
Spencer Luu
University Park
53
77
User
Dallas
53
78
User
Houston
52
79
User
Houston
52
80
Joseph Cano
Fulshear
52
81
Pi Monster
Farmers Branch
52
82
User
Austin
52
83
User
Houston
52
84
User
Houston
52
85
User
Dallas
52
86
User
Conroe
52
87
Joy F
Westlake
51
88
User
College Station
51
89
User
Conroe
51
90
User
Houston
51
91
User
Houston
51
92
asdfasdfasdf
Rockwall
51
93
User
Dallas
51
94
User
Austin
51
95
User
Houston
51
96
User
Houston
51
97
User
Conroe
51
98
User
Dallas
51
99
Jasmine Pandit
City
51
100
User
Houston
51
101
Math god
San Antonio
50
102
Hwa-Shen Tan
Houston
50
103
Lainey Hamilton
Longview
49
104
User
Austin
49
105
User
Dallas
49
106
User
Dallas
49
107
User
Houston
49
108
User
Beaumont
49
109
KK SS
Houston
48
110
Adam Chen
Cinco Ranch
48
111
User
North Richland Hills
48
112
User
Dallas
48
113
Tarun Mohan
Dallas
48
114
User
Dallas
48
115
User
Dallas
48
116
Robert Silvy
Houston
48
117
User
Houston
47
118
User
Dallas
47
119
User
Dallas
47
120
User
San Antonio
47
121
User
Dallas
47
122
User
Southlake
47
123
User
Plano
46
124
User
Dallas
46
125
User
Dallas
46
126
Michalis Paizanis
Denton
46
127
User
Dallas
46
128
User
Dallas
45
129
User
Frisco
45
130
User
San Antonio
45
131
User
Dallas
45
132
User
San Antonio
45
133
User
Dallas
44
134
Sam
Houston
44
135
YT Dragonlak
Dallas
44
136
User
Houston
44
137
Addison Johnson
Longview
44
138
User
Dallas
44
139
Daniel Cho
Fort Worth
44
140
User
Grapevine
44
141
User
Dallas
43
142
Shaayaan Sayed
Missouri City
43
143
User
Dallas
43
144
User
San Antonio
43
145
User
San Antonio
43
146
User
Fort Worth
43
147
User
Dallas
42
148
User
Dallas
42
149
User
San Antonio
42
150
User
Austin
42
151
User
Austin
42
152
User
Austin
42
153
User
Austin
41
154
User
San Antonio
41
155
User
Rockwall
41
156
User
San Antonio
41
157
User
San Antonio
41
158
User
Dallas
40
159
User
Leander
40
160
User
Houston
40
161
User
San Antonio
40
162
User
Dallas
40
163
User
Dallas
40
164
User
Dallas
40
165
justin silver
Denton
39
166
User
Allen
39
167
User
Austin
39
168
User
University Park
39
169
Crespo
Houston
38
170
user
Frisco
38
171
User
Austin
38
172
Bob Luo
Plano
38
173
User
Friendswood
38
174
Kevin Copeland
Leander
38
175
User
Garland
37
176
Pelke
Round Rock
37
177
User
Dallas
37
178
User
Austin
37
179
Aden Smith
Houston
37
180
User
Austin
37
181
Ricky Kirk
Dallas
37
182
User
San Antonio
37
183
User
Dallas
37
184
User
Pearland
37
185
User
Dallas
36
186
User
San Antonio
36
187
NoThisIsJohn
Austin
36
188
User
Fort Worth
36
189
User
Abilene
36
190
User
Houston
36
191
John Guest
Austin
36
192
myles W
Irving
36
193
User
Atascocita
36
194
User
North Richland Hills
36
195
User
Houston
36
196
User
Garland
36
197
Derek Cheung
Austin
36
198
User
Houston
35
199
User
Frisco
35
200
User
Dallas
35
201
User
Dallas
35
202
User
Dallas
35
203
User
Dallas
35
204
User
Leander
35
205
User
Austin
35
206
User
Wylie
34
207
User
San Antonio
34
208
User
Irving
34
209
User
Dallas
33
210
User
Richardson
33
211
User
San Antonio
33
212
User
Austin
33
213
User
Dallas
33
214
User
San Antonio
33
215
User
San Antonio
33
216
User
Austin
32
217
User
San Antonio
32
218
User
Dallas
32
219
User
Irving
32
220
User
San Antonio
32
221
Robert Silvy
Houston
32
222
User
Leander
32
223
User
Arlington
32
224
User
San Antonio
32
225
User
San Antonio
32
226
User
San Antonio
32
227
User
Wylie
31
228
User
The Woodlands
31
229
User
Irving
31
230
Kenny Huang
Plano
31
231
User
Austin
31
232
Letscher IV
Dallas
30
233
User
Rockwall
30
234
User
Arlington
30
235
User
Plano
30
236
Zhi Huang
Mission
30
237
User
Houston
30
238
User
Houston
30
239
User
Dallas
30
240
Kaitlynn Trinh
Houston
30
241
User
Ennis
30
242
Katsumi Morales
Dallas
30
243
Samagra Jain
Austin
30
244
User
Abilene
30
245
User
Wylie
30
246
User
Frisco
29
247
Nick T
Houston
29
248
Irina
Houston
29
249
User
Austin
29
250
sourry
Atascocita
29
251
User
Dallas
29
252
Jeff T
Houston
29
253
User
Richardson
29
254
User
Abilene
29
255
User
Austin
29
256
User
Dallas
29
257
User
San Antonio
29
258
User
Austin
29
259
User
Austin
28
260
Devan
San Antonio
28
261
Billy Zhao
Flower Mound
28
262
Nik
Dallas
28
263
User
Missouri City
28
264
User
Dallas
28
265
User
San Antonio
28
266
User
Dallas
28
267
User
Houston
28
268
Natalie Stafford
Houston
28
269
User
Houston
28
270
manu gupta
Arlington
28
271
User
Dallas
28
272
User
San Antonio
27
273
Joseph Walker
Lubbock
27
274
User
Houston
27
275
User
Dallas
27
276
Safwan Naviwala
Denton
27
277
Reed Tubbs
Pflugerville
27
278
User
Austin
27
279
Kirsten Collins
Dallas
27
280
User
Houston
27
281
User
Fort Worth
27
282
User
Houston
27
283
User
Houston
26
284
Jordan Wells
Atascocita
26
285
Issam
Houston
26
286
User
El Paso
26
287
Samir Patel
Beaumont
26
288
User
Farmers Branch
26
289
User
Wylie
26
290
User
Garland
26
291
User
San Antonio
26
292
User
Austin
26
293
User
Plano
26
294
User
Round Rock
26
295
User
Coppell
26
296
User
Houston
25
297
EY Lee
Houston
25
298
User
Plano
25
299
User
Houston
25
300
User
Austin
25
301
User
Dallas
25
302
User
Arlington
25
303
User
Houston
25
304
Adrian G Casillas
Austin
25
305
User
Houston
25
306
Jacob Torres
Round Rock
25
307
User
College Station
25
308
User
Plano
25
309
User
Longview
25
310
User
Fort Worth
25
311
Cara Wakefield
Denton
25
312
User
Arlington
25
313
User
Allen
24
314
User
Dallas
24
315
User
Dallas
24
316
User
San Antonio
24
317
User
Austin
24
318
User
Austin
24
319
User
Austin
24
320
User
Austin
24
321
Ansun Sujoe
Fort Worth
24
322
Marco Varela
Arlington
24
323
User
Houston
24
324
Cedric Zhou
Plano
24
325
User
Allen
24
326
User
Dallas
23
327
User
Austin
23
328
Davin Park
Dallas
23
329
Morgan Roberts
New Chapel Hill
23
330
User
Dallas
23
331
User
Austin
23
332
piemanklp
Taylor
23
333
Melanie Markham
Lake Jackson
23
334
User
Dallas
23
335
User
Fort Worth
23
336
cool guy
Mission
23
337
User
San Antonio
22
338
User
Austin
22
339
User
Houston
22
340
User
Houston
22
341
User
Dallas
22
342
Nolan Raimo
Austin
22
343
User
Dallas
22
344
User
Dallas
22
345
Carl Svahn
Austin
22
346
Rhett Marcelo
Houston
22
347
User
Houston
22
348
User
Houston
22
349
User
Dallas
21
350
User
Dallas
21
351
User
San Antonio
21
352
Rivetrosity
Fort Worth
21
353
User
Dallas
21
354
User
Austin
21
355
User
Houston
21
356
Tammy Sapkota
Houston
21
357
User
Atascocita
21
358
User
Austin
21
359
User
College Station
21
360
User
Corpus Christi
21
361
User
Flower Mound
20
362
Matthew Miller
Galveston
20
363
User
Dallas
20
364
Dale Yoon
Houston
20
365
User
Cypress
20
366
User
Southlake
20
367
User
Dallas
20
368
User
Houston
20
369
User
Houston
20
370
User
Houston
20
371
User
Dallas
20
372
User
Abilene
20
373
yugendra uppalapati
Plano
20
374
User
Dallas
20
375
User
Pharr
20
376
User
Royse City
20
377
User
Austin
20
378
User
Plano
19
379
User
Austin
19
380
User
Austin
19
381
User
Harker Heights
19
382
User
Dallas
19
383
User
Austin
19
384
User
San Antonio
19
385
Zabiah Jones
Houston
19
386
User
The Woodlands
19
387
Sahil Mehta
Austin
19
388
User
Plano
19
389
User
Houston
19
390
john desmond
Austin
19
391
User
Plano
19
392
User
Austin
19
393
Frost Hunter
Houston
19
394
User
Dallas
19
395
User
Dallas
19
396
User
Amarillo
19
397
User
San Antonio
19
398
Serafin G
Richardson
19
399
Mary Berkel
Cedar Park
19
400
User
Frisco
18
401
User
Garland
18
402
User
Plano
18
403
User
Austin
18
404
User
Houston
18
405
User
Laredo
18
406
User
Austin
18
407
Neil Dabholkar
Austin
18
408
User
Sugar Land
18
409
User
Dallas
18
410
User
Dallas
18
411
User
Dallas
18
412
User
Fulshear
18
413
User
Dallas
18
414
User
Prosper
18
415
User
Dallas
18
416
User
Conroe
18
417
User
San Antonio
18
418
User
Austin
18
419
User
Dallas
18
420
User
Houston
18
421
User
San Antonio
18
422
User
Dallas
18
423
User
Houston
18
424
User
San Antonio
18
425
User
San Antonio
18
426
User
Carrollton
18
427
User
La Porte
18
428
User
Austin
18
429
User
Pasadena
18
430
User
Killeen
18
431
User
Houston
18
432
User
Wylie
18
433
User
Houston
18
434
User
Houston
17
435
User
Austin
17
436
User
Dallas
17
437
User
Houston
17
438
User
Austin
17
439
Eli
Houston
17
440
Nikki
El Paso
17
441
User
Houston
17
442
User
Plano
17
443
User
Dallas
17
444
User
Plano
17
445
User
Austin
17
446
Ethan Boyd
Dallas
17
447
User
Weatherford
17
448
User
Houston
17
449
User
Houston
17
450
User
Dallas
17
451
John Young
Austin
17
452
User
Laredo
17
453
User
Spring
17
454
User
Round Rock
17
455
User
Houston
17
456
User
Austin
17
457
User
Houston
17
458
User
Dallas
17
459
User
Austin
16
460
User
Austin
16
461
Advait Joshi
Round Rock
16
462
User
Austin
16
463
User
Wichita Falls
16
464
User
Dallas
16
465
User
Mckinney
16
466
User
Houston
16
467
User
Dallas
16
468
egrg
Houston
16
469
User
Houston
16
470
Fabian Tarango
Houston
16
471
User
Dallas
16
472
User
Austin
16
473
User
Alice
16
474
User
Katy
16
475
User
Austin
16
476
User
Sugar Land
16
477
User
Austin
16
478
User
Fulshear
16
479
User
Austin
16
480
User
The Woodlands
16
481
User
Temple
16
482
User
Austin
16
483
User
San Antonio
16
484
User
Houston
16
485
User
Houston
16
486
User
Irving
16
487
Alec Drean
Dallas
16
488
User
Mesquite
16
489
BigBootyBoi
Farmers Branch
15
490
User
Dallas
15
491
User
Mcallen
15
492
Charles Laison
Dallas
15
493
User
Pflugerville
15
494
User
Dallas
15
495
User
Dallas
15
496
User
Dallas
15
497
User
Austin
15
498
User
Dallas
15
499
User
Denton
15
500
User
Dallas
15
501
User
Houston
15
502
User
Corpus Christi
15
503
Hawkins Burns
Bee Cave
15
504
User
Houston
15
505
User
Plano
15
506
User
Prosper
15
507
User
Dallas
15
508
User
San Antonio
15
509
User
Pflugerville
15
510
User
Dallas
15
511
User
Denton
15
512
User
Nacogdoches
15
513
User
San Antonio
15
514
User
San Antonio
15
515
User
Houston
15
516
User
Irving
15
517
User
San Antonio
15
518
Nidhi Mishra
Austin
15
519
User
Southlake
14
520
User
Dallas
14
521
Dhruv Desai
Austin
14
522
Uzair Akram
Euless
14
523
User
Houston
14
524
User
Willow Park
14
525
User
Allen
14
526
Math Test
Frisco
14
527
User
San Antonio
14
528
User
Dallas
14
529
User
Houston
14
530
User
Austin
14
531
User
Plano
14
532
User
Round Rock
14
533
User
Houston
14
534
User
Wichita Falls
14
535
User
Cypress
14
536
User
New Braunfels
14
537
User
Irving
14
538
User
Plano
14
539
User
Dallas
14
540
User
Dallas
14
541
User
Dallas
14
542
User
Atascocita
14
543
User
Dallas
14
544
Nicole Hernandez
Houston
14
545
User
Houston
14
546
User
Fulshear
14
547
User
Austin
14
548
User
Dallas
14
549
User
Donna
14
550
Nolan Wu
Houston
14
551
User
Austin
14
552
User
Houston
14
553
User
Dallas
14
554
User
Plano
14
555
Minh Nguyen
Houston
14
556
User
Dallas
14
557
User
El Paso
14
558
User
Rowlett
14
559
User
Dallas
14
560
User
Longview
14
561
User
Carrollton
14
562
User
Dallas
14
563
User
Austin
14
564
User
Fulshear
14
565
User
Houston
14
566
User
Houston
14
567
User
Waco
13
568
User
Houston
13
569
User
Houston
13
570
User
Austin
13
571
Bethany Disque
Lubbock
13
572
User
Houston
13
573
User
College Station
13
574
User
Austin
13
575
Eric Eldridge
Houston
13
576
User
Houston
13
577
User
Argyle
13
578
User
Abilene
13
579
bowdowntothestchair
League City
13
580
User
Fulshear
13
581
User
Garland
13
582
Hollow
Dallas
13
583
User
Houston
13
584
User
Carrollton
13
585
User
Dallas
13
586
User
Austin
13
587
User
Dallas
13
588
RawLawson
Cypress
13
589
User
El Paso
13
590
User
Donna
13
591
User
Houston
13
592
Ace
Houston
13
593
User
San Antonio
12
594
User
Richardson
12
595
User
Houston
12
596
User
Prosper
12
597
KaruBachha
Austin
12
598
User
Denton
12
599
User
Temple
12
600
User
Austin
12
601
Graphing Paper
Atascocita
12
602
User
Houston
12
603
Aryan Padarthi
Allen
12
604
User
Cedar Park
12
605
User
Dallas
12
606
User
Houston
12
607
User
Houston
12
608
Luke Weaver
Austin
12
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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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