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