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