Leaderboard
Illinois
Rank
Level
1
Siva
Glenview
96
2
Teague Stotlar
Naperville
91
3
Kvothe
Northbrook
90
4
User
Chicago
89
5
User
Chicago
88
6
Michael Lu
Naperville
86
7
User
Northbrook
85
8
User
Aurora
84
9
Chengjian Shi
Chicago
83
10
User
Chicago
83
11
User
Chicago
83
12
User
Evanston
82
13
User
Aurora
82
14
User
Chicago
82
15
John Laboe
Naperville
82
16
Haining Zhou
Chicago
81
17
User
Chicago
81
18
User
Northbrook
81
19
Shin
Evanston
81
20
User
Aurora
81
21
Haining Zhou
Chicago
80
22
User
Evanston
80
23
camden
Aurora
79
24
Jiawei Huang
Champaign
79
25
User
Burr Ridge
78
26
Zach George
Barrington
77
27
Aiman
West Chicago
77
28
Sagar Patel
Chicago
77
29
User
Chicago
77
30
User
Chicago
76
31
User
Naperville
75
32
User
Chicago
73
33
User
Evanston
72
34
Varad Rasalkar
Naperville
71
35
Huidong Ren
Chicago
71
36
User
Chicago
71
37
User
Chicago
71
38
User
Chicago
71
39
3.1415926535897932
Urbana
70
40
Tushar Sharma
Chicago
70
41
User
Chicago
70
42
MelGibson
Chicago
69
43
User
Chicago
67
44
User
Chicago
66
45
User
Evanston
66
46
User
Champaign
66
47
User
Crystal Lake
65
48
User
Chicago
65
49
Arjun Arun
Champaign
63
50
User
Naperville
63
51
User
La Grange Park
62
52
Evgenia Chubata
Chicago
62
53
User
Chicago
62
54
User
Carmi
61
55
User
Chicago
61
56
User
City
60
57
User
Schaumburg
60
58
Yiteng Zhang
Chicago
60
59
User
Mount Zion
59
60
User
Buffalo Grove
59
61
Eric
Chicago
59
62
User
Chicago
58
63
SemperJon B82
Chicago
58
64
User
Chicago
58
65
takyon
Aurora
57
66
User
Chicago
57
67
User
Chicago
57
68
User
Glenview
56
69
User
Chicago
55
70
User
Aurora
55
71
User
Springfield
54
72
AT
Glenview
54
73
Dee Sullivan
Glencoe
54
74
User
Chicago
54
75
Samuel Colunga
Arenzville
53
76
User
Urbana
53
77
User
Champaign
53
78
User
Peoria
53
79
User
Hoffman Estates
52
80
User
Carmi
52
81
User
Naperville
52
82
User
Champaign
52
83
User
Chicago
52
84
T
Evanston
51
85
User
Chicago
51
86
User
Chicago
51
87
User
Springfield
50
88
Ola
Chicago
50
89
Neil Gupta
Naperville
50
90
User
Chicago
50
91
User
Naperville
50
92
Samuel Ko
Skokie
49
93
User
Lake Villa
49
94
Andrew Vallaster
Chicago
49
95
User
Chicago
48
96
User
Chicago
48
97
User
Chicago
48
98
User
Naperville
48
99
User
Chicago
48
100
User
Hillsboro
48
101
Hugo Piña
Chicago
47
102
Daniel Chang
Evanston
47
103
User
Champaign
47
104
Max Liu
Chicago
47
105
User
Evanston
47
106
User
Glenview
47
107
Harshavardhan Baheti
Champaign
47
108
User
Joliet
47
109
User
Aurora
47
110
User
Chicago
46
111
Brad Koszuta
Chicago
46
112
User
Naperville
46
113
harshavardhan
Champaign
46
114
User
Park Ridge
46
115
User
Chicago
46
116
Matt
Springfield
45
117
User
Chicago
45
118
User
Hillsboro
45
119
User
Chicago
44
120
Conner Collins
Evanston
44
121
User
Chicago
44
122
User
Wheaton
44
123
User
Chicago
44
124
Damg22
Northbrook
44
125
User
Chicago
44
126
User
Chicago
43
127
nora c
Chicago
43
128
LSky
Edwardsville
43
129
User
Chicago
43
130
User
Downers Grove
43
131
User
Champaign
43
132
User
Chicago
43
133
User
Chicago
43
134
User
Naperville
43
135
User
Chicago
43
136
User
Champaign
42
137
User
Chicago
42
138
User
Calumet City
42
139
User
Springfield
41
140
User
Mount Zion
41
141
User
Buffalo Grove
41
142
User
Northbrook
41
143
User
Chicago
41
144
User
Chicago
41
145
Rabbi Hashem Shalomovitchbirgenshekelsteinberg
Downers Grove
41
146
User
Lake Summerset
41
147
User
Chicago
41
148
User
Chicago
40
149
User
Naperville
40
150
User
Chicago
40
151
User
Chicago
40
152
User
Chicago
40
153
User
Chicago
40
154
User
Chicago
39
155
Jack Ding
Naperville
39
156
User
Niles
39
157
User
Chicago
39
158
User
Chicago
39
159
hugatupe@daabox.com
Chicago
39
160
User
Joliet
39
161
User
Naperville
39
162
User
Chicago
38
163
User
Chicago
38
164
User
Northbrook
38
165
User
Winnebago
38
166
User
Chicago
38
167
User
Chicago
38
168
User
Peoria
38
169
User
Chicago
38
170
User
Chicago
37
171
User
Arlington Heights
37
172
User
Chicago
37
173
User
Mchenry
37
174
User
Tuscola
37
175
realEBN
La Grange
36
176
User
Mount Zion
36
177
Jake Mustain
Evanston
36
178
User
Chicago
36
179
User
Chicago
36
180
User
Chicago
36
181
User
Marion
36
182
Duncan
Chicago
36
183
User
Champaign
35
184
User
Chicago
35
185
User
Chicago
35
186
User
Chicago
35
187
User
Evanston
35
188
Tyler Carlson
Naperville
35
189
Bradley Szoke
Normal
35
190
User
Chicago
35
191
User
Chicago
35
192
User
Rockford
34
193
User
Oregon
34
194
User
Champaign
34
195
SADOR WONDESON
Chicago
34
196
User
Champaign
34
197
User
Chicago
34
198
User
Naperville
33
199
User
Carol Stream
33
200
User
Chicago
33
201
User
Chicago
33
202
User
Chicago
33
203
User
Chicago
33
204
User
Chicago
33
205
User
Chicago
33
206
Valeria Sanabria Guillen
Champaign
32
207
User
Carmi
32
208
User
Beardstown
32
209
Aidan Avakian
Mchenry
32
210
User
Chicago
32
211
User
Romeoville
32
212
User
Chicago
32
213
User
Chicago
32
214
User
Hillsboro
32
215
Andrew Oosterbaan
Lombard
31
216
User
Chicago
31
217
User
Galesburg
31
218
User
Chicago
31
219
User
Chicago
31
220
User
Evanston
31
221
User
Bolingbrook
31
222
User
Champaign
31
223
Lloyd Smith
Chicago
31
224
User
Chicago
31
225
User
Chicago
31
226
User
Chicago
30
227
User
Chicago
30
228
User
Wheaton
30
229
User
Rolling Meadows
30
230
User
Chicago
30
231
User
Chicago
30
232
Ethan L
Chicago
30
233
User
Alsip
30
234
User
Chicago
30
235
User
Chicago
30
236
User
Aurora
30
237
User
Chicago
30
238
User
Evanston
30
239
Someone 17
Chicago
30
240
User
St. Charles
29
241
Barclay Kamaleson
Chicago
29
242
User
Schaumburg
29
243
User
Chicago
29
244
dass
Glasford
29
245
User
Evanston
29
246
User
Chicago
29
247
User
Champaign
29
248
User
Des Plaines
29
249
User
Skokie
29
250
User
Chicago
29
251
User
Champaign
29
252
User
Chicago
29
253
User
Chicago
29
254
User
Greenville
28
255
User
Chicago
28
256
User
Northbrook
28
257
User
Chicago
28
258
User
Elmwood Park
28
259
User
Cicero
28
260
User
Champaign
28
261
User
Evanston
28
262
User
Chicago
28
263
Daniel Ferriss
Champaign
28
264
Phat Quang
Chicago
28
265
User
Chicago
28
266
Hannah Whitehouse
Evanston
28
267
User
Chicago
28
268
User
Naperville
28
269
User
Chicago
28
270
User
Chicago
27
271
User
Chicago
27
272
Wasuji
Woodridge
27
273
User
Glenview
27
274
User
Chicago
27
275
User
Dolton
27
276
Jada Perez
Mchenry
27
277
User
Northbrook
27
278
User
Chicago
27
279
G
Champaign
27
280
User
Minooka
27
281
User
Chicago
27
282
User
Prairie City
27
283
User
Palos Hills
27
284
User
Chicago
27
285
Sharadha Srinivasan
Chicago
27
286
User
Chicago
27
287
User
Chicago
27
288
User
East Saint Louis
27
289
User
Champaign
27
290
User
Burbank
27
291
User
Chicago
27
292
Joe B
Morton
27
293
Abhimanyu Choudhary
Chicago
27
294
User
Chicago
27
295
User
Chicago
27
296
User
Chicago
27
297
User
Chicago
27
298
User
Chicago
26
299
User
Springfield
26
300
User
Buffalo Grove
26
301
User
Chicago
26
302
User
Champaign
26
303
User
Chicago
26
304
User
Northbrook
26
305
User
Chicago
26
306
User
Chicago
26
307
User
Chicago
26
308
Matthew Sanders
Chicago
26
309
User
Chicago
26
310
User
Niles
26
311
User
Chicago
26
312
User
Palos Hills
26
313
User
Shorewood
25
314
User
Chicago
25
315
User
Lombard
25
316
User
Aurora
25
317
User
Nashville
25
318
User
Chicago
25
319
User
Chicago
25
320
User
Bolingbrook
25
321
User
Plainfield
25
322
User
Chicago
25
323
User
Romeoville
25
324
User
Chicago
25
325
User
St. Charles
25
326
OLIVIA YOUNG
Romeoville
25
327
ANGEL BRACHO
Romeoville
25
328
User
Chicago
25
329
User
Chicago
25
330
User
East Peoria
24
331
User
Lindenhurst
24
332
User
Tinley Park
24
333
User
Carmi
24
334
User
Mchenry
24
335
Lloyd Smith
Chicago
24
336
User
Chicago
24
337
User
Chicago
24
338
User
Chicago
24
339
User
Chicago
24
340
User
Chicago
24
341
Endre Triantafillou
Chicago
24
342
User
Chicago
24
343
User
Chicago
24
344
User
Chicago
23
345
Alexandra A
Oak Park
23
346
User
Lemont
23
347
User
Chicago
23
348
User
Chicago
23
349
User
Chicago
23
350
User
Naperville
23
351
User
Mchenry
23
352
User
Chicago
23
353
User
Chicago
23
354
User
Chicago
23
355
User
Chicago
23
356
User
City
23
357
User
Romeoville
23
358
User
Chicago
23
359
User
Wheaton
23
360
User
Chicago
23
361
User
Urbana
23
362
User
Chicago
23
363
User
Chicago
23
364
Youngan Austin
Chicago
23
365
User
Freeport
23
366
User
Chicago
23
367
User
Chicago
23
368
User
Chicago
23
369
User
Chicago
22
370
Ethan Rasmussen
Mchenry
22
371
User
Chicago
22
372
User
Naperville
22
373
User
Galesburg
22
374
Adam Textor
Mchenry
22
375
User
Mokena
22
376
Mohsin Rizvi
Woodstock
22
377
Gabriela Dago
Chicago
22
378
User
Chicago
22
379
User
Downers Grove
22
380
Zaine Trent
Galesburg
22
381
User
Addison
22
382
User
Bolingbrook
22
383
User
Urbana
22
384
User
Skokie
22
385
User
Chicago
22
386
User
Chicago
22
387
Gaurav vaghmar
Chicago
22
388
Bryan Dominguez
Chicago
22
389
Hooriya Rathuer
Chicago
22
390
User
Chicago
22
391
User
Chicago
22
392
User
Chicago
22
393
User
City
21
394
User
Chicago
21
395
User
Lisle
21
396
User
Chicago
21
397
User
Chicago
21
398
Gavin Rehberg
Mchenry
21
399
User
Champaign
21
400
User
Chicago
21
401
User
Chicago
21
402
Jason Abrams
Chicago
21
403
User
Chicago
21
404
User
Mchenry
21
405
User
Chicago
21
406
Deidra Shannon
Homewood
21
407
User
Chicago
21
408
User
Schaumburg
21
409
User
Chicago
21
410
User
Chicago
21
411
User
Wheaton
21
412
User
Chicago
21
413
User
Lindenhurst
21
414
Nicholas Ramos
Hinsdale
21
415
User
Chicago
21
416
Mary Bakus
Galesburg
21
417
User
Waukegan
21
418
User
Beardstown
21
419
User
Dekalb
21
420
User
Chicago
21
421
User
Chicago
21
422
User
Chicago
21
423
User
Chicago
21
424
User
Chicago
21
425
User
Chicago
21
426
User
Chicago
21
427
User
Chicago
21
428
User
Chicago
21
429
Saadia Abdi
Chicago
21
430
Aden Ramirez
Chicago
21
431
Callista Hansen
Chicago
21
432
Roger Borrell Anglada
Chicago
21
433
User
Wilmette
21
434
User
Chicago
21
435
User
Chicago
21
436
User
Chicago
21
437
User
Naperville
21
438
Sebastian Rivera
Chicago
21
439
User
Downers Grove
21
440
User
Chicago
20
441
User
Tinley Park
20
442
User
Naperville
20
443
Dan
Belvidere
20
444
User
East Peoria
20
445
User
Chicago
20
446
User
Chicago
20
447
User
Niles
20
448
Bishop Kyser
Galesburg
20
449
User
Chicago
20
450
User
Chicago
20
451
User
Chicago
20
452
User
Champaign
20
453
Jia Shah
Mchenry
20
454
Rafael Pena
Mchenry
20
455
User
Chicago
20
456
User
Plainfield
20
457
User
Chicago
20
458
User
Chicago
20
459
Makenzie Leopold
Princeton
20
460
User
Chicago
20
461
User
Chicago
20
462
User
Darien
20
463
User
City
20
464
User
Minooka
20
465
Arianna Masi
Chicago
20
466
User
Princeton
20
467
Ana Adan
Cicero
20
468
Selmir Kosut
Chicago
20
469
Angel Rhone
Chicago
20
470
Emmett Calto
Chicago
20
471
User
Arlington Heights
20
472
User
Palatine
20
473
User
Chicago
20
474
User
Evanston
20
475
User
Downers Grove
20
476
User
Chicago
20
477
User
Chicago
20
478
User
Chicago
19
479
User
East Peoria
19
480
Owen Czarny
Mchenry
19
481
Jamison Shea
Mchenry
19
482
User
Buffalo Grove
19
483
User
Champaign
19
484
Grant Z
Lake Zurich
19
485
Landon Sloan
Galesburg
19
486
User
Chicago
19
487
David Hoguet
Mchenry
19
488
User
Aurora
19
489
User
Aurora
19
490
User
Chicago
19
491
User
Arlington Heights
19
492
User
Chicago
19
493
User
Bolingbrook
19
494
User
Chicago
19
495
User
Glenview
19
496
User
Northbrook
19
497
User
Chicago
19
498
User
Chicago
19
499
User
Champaign
19
500
User
Evanston
19
501
User
Batavia
19
502
User
Chicago
19
503
User
Romeoville
19
504
User
Chicago
19
505
User
Decatur
18
506
User
Elgin
18
507
User
East Peoria
18
508
User
Mchenry
18
509
Azam Ahmed
Skokie
18
510
User
Chicago
18
511
User
Westchester
18
512
User
Mchenry
18
513
User
Chicago
18
514
User
Chicago
18
515
User
Chicago
18
516
User
Bolingbrook
18
517
User
Aurora
18
518
Salomon Rodriguez
Chicago
18
519
User
Chicago
18
520
User
Chicago
18
521
User
Chicago
18
522
User
Glen Ellyn
18
523
User
Chicago
18
524
User
Hillsboro
18
525
User
Chicago
18
526
User
Chicago
18
527
User
Chicago
17
528
User
East Peoria
17
529
User
Richton Park
17
530
User
East Peoria
17
531
User
Elgin
17
532
User
East Peoria
17
533
Derek M. Lough
Bloomington
17
534
ben
Mchenry
17
535
User
Mchenry
17
536
User
Chicago
17
537
User
Galesburg
17
538
User
Carpentersville
17
539
User
Galesburg
17
540
User
Mchenry
17
541
User
Chicago
17
542
User
Urbana
17
543
User
Champaign
17
544
User
Chicago
17
545
User
Mchenry
17
546
User
Chicago
17
547
K B
Chicago
17
548
User
Chicago
17
549
User
Chicago
17
550
User
Naperville
17
551
User
New Lenox
17
552
User
Plainfield
17
553
Tony
Romeoville
17
554
User
Chicago
17
555
User
Des Plaines
17
556
User
Fairview Heights
17
557
User
Chicago
17
558
User
Chicago
17
559
User
Palatine
17
560
Blake Campbell
Chicago
17
561
User
Bourbonnais
17
562
User
Chicago
17
563
User
Chicago
17
564
User
Chicago
17
565
User
Chicago
17
566
User
Chicago
17
567
User
Quincy
16
568
User
East Peoria
16
569
User
Pekin
16
570
Benjamin Ward
Chicago
16
571
User
East Peoria
16
572
User
Chicago
16
573
User
Chicago
16
574
User
Downers Grove
16
575
User
East Peoria
16
576
User
Skokie
16
577
Megan Crow
Mchenry
16
578
User
Mchenry
16
579
User
Dekalb
16
580
User
Chicago
16
581
User
Chicago
16
582
User
Litchfield
16
583
User
Chicago
16
584
User
Chicago
16
585
User
Wheaton
16
586
User
Oak Park
16
587
User
Deerfield
16
588
User
Aurora
16
589
User
Peoria
16
590
User
Chicago
16
591
User
Plainfield
16
592
User
Chicago
16
593
User
City
16
594
User
Bolingbrook
16
595
User
Chicago
16
596
User
Champaign
16
597
User
Bolingbrook
16
598
User
Chicago
16
599
anna
Chicago
16
600
User
Aurora
16
601
Mahdi Sheikh
Chicago
16
602
Madalyn Hernandez
Chicago
16
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I AM THE ONE
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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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