Leaderboard
Ohio
Rank
Level
1
EASTON SINGER
Solon
95
2
User
Solon
90
3
King of the Hill
Blue Ash
87
4
Tanishq Pauskar
Massillon
86
5
Cory Huelskamp
Sidney
84
6
Jon Yoon
Columbus
78
7
Puddles Penguin
Columbus
73
8
User
Cortland
67
9
Nate
Stow
66
10
ROBERT BATCHLET
Wooster
66
11
Joseph Churilla
Westerville
63
12
Athena
Upper Arlington
62
13
Abhinav Goel
Montgomery
60
14
User
Upper Arlington
57
15
RubberDuck
Massillon
56
16
User
North Olmsted
55
17
User
Blue Ash
55
18
QuackDuck
Massillon
54
19
Emanuel Melaku
Reynoldsburg
54
20
User
Columbus
52
21
Mason Greene
Jerry City
51
22
Kevin Zhu
Barboursville
50
23
User
Sidney
49
24
Realism The Future
Boardman
49
25
Andrew Zhou
Montgomery
48
26
User
Beckett Ridge
48
27
Justin Schimpf
Centerville
47
28
User
Northgate
47
29
Nathaniel Day
Cincinnati
46
30
User
Boardman
46
31
User
Cleveland
45
32
Nick M
Columbus
45
33
User
Cleveland
44
34
Trevor Bischoff
Cleveland
43
35
User
Columbus
43
36
User
Solon
42
37
User
Blue Ash
42
38
User
Forest Park
42
39
User
New Albany
42
40
User
Solon
41
41
Rohith Venkataraman
Cincinnati
41
42
User
Cincinnati
40
43
User
Columbus
39
44
User
Montgomery
39
45
User
Blue Ash
39
46
User
Lyndhurst
39
47
User
Cincinnati
37
48
User
Toledo
37
49
DANIEL JONES
Columbus
37
50
Darren Kaiser
Sidney
37
51
User
Cleveland
37
52
User
Cincinnati
36
53
User
Withamsville
36
54
User
Columbus
36
55
Siddharth Vikram
Blue Ash
35
56
User
Pataskala
35
57
Chenxi Zhou
Columbus
35
58
Maddy simmons
Columbus
34
59
Joe Gab nagana
Cleveland
34
60
User
Sidney
34
61
User
Wooster
34
62
User
Mason
34
63
User
Oxford
33
64
Sean Griffin
Reynoldsburg
33
65
User
Dublin
32
66
User
Westlake
32
67
User
Cleveland
31
68
DakotaHale
Ashland
30
69
ANDREW THOMAS
Pataskala
30
70
User
Westlake
30
71
Nicholas Johnson
Lewis Center
29
72
User
Columbus
29
73
JEREMY BAILEY
Orrville
29
74
User
Oxford
29
75
User
Kent
28
76
User
Columbus
28
77
User
New Albany
28
78
Jeremy Arcus-Goldberg
New Albany
28
79
User
Defiance
28
80
User
Twinsburg
28
81
User
Solon
27
82
User
Mason
27
83
Kristen Schutte
Blue Ash
27
84
User
Pataskala
27
85
User
Jerry City
27
86
User
Beachwood
27
87
User
Cleveland
27
88
HANNAH JOHNSON
Columbus
26
89
Alvin Oshiotse
Milford
26
90
Master Flash
Reynoldsburg
26
91
User
Columbus
26
92
User
Columbus
26
93
User
Sharonville
26
94
User
Westlake
26
95
Sofiya Lozovskiy
Brecksville
26
96
User
Reynoldsburg
26
97
User
Oxford
26
98
User
Brooklyn
26
99
User
Cleveland
26
100
Jake V
Toledo
25
101
User
Blue Ash
25
102
User
Springfield
25
103
User
Wooster
25
104
User
Findlay
25
105
User
Oxford
25
106
User
Reynoldsburg
25
107
User
Cincinnati
25
108
User
Brooklyn
25
109
User
Fairfield
24
110
User
Blue Ash
24
111
User
Sidney
24
112
Alex P
Cedarville
24
113
User
Columbus
24
114
User
Dayton
24
115
User
Kent
23
116
Erice
Finneytown
23
117
User
Centerville
23
118
User
Columbus
23
119
User
Akron
23
120
User
Cincinnati
23
121
User
Defiance
23
122
User
Columbus
23
123
User
Brooklyn
23
124
User
Columbus
22
125
User
Sidney
22
126
User
Cincinnati
22
127
User
Sylvania
22
128
User
Columbus
22
129
User
Cincinnati
22
130
User
Beachwood
22
131
User
Sidney
21
132
Josh Kelly
Columbus
21
133
User
Columbus
21
134
orlando wyman
Ashland
21
135
User
Indian Hill
21
136
User
Columbus
21
137
User
Cincinnati
21
138
User
Cleveland
21
139
User
Cleveland
21
140
User
Cleveland
21
141
Rachelle Hiestand
Sidney
20
142
User
Springfield
20
143
Brandon Engster
Warren
20
144
User
Reynoldsburg
20
145
User
Cleveland
20
146
Justin Chen
Richmond Heights
19
147
User
Monroe
19
148
User
Casstown
19
149
User
City
19
150
User
Columbus
19
151
User
Columbus
19
152
User
Reynoldsburg
19
153
User
Olmsted Falls
19
154
User
Upper Sandusky
19
155
QuackDuck
Massillon
18
156
User
Columbus
18
157
User
Columbus
18
158
Brady Ayers
Orrville
18
159
User
Loveland
18
160
User
Cincinnati
18
161
User
Columbus
18
162
User
Pataskala
18
163
User
Parma
18
164
User
Columbus
18
165
Griffin Waddell
Reynoldsburg
18
166
User
Cincinnati
18
167
kate
Cleveland
18
168
User
Cleveland
18
169
User
North Olmsted
18
170
User
Massillon
17
171
User
Evendale
17
172
Divey
Sidney
17
173
User
Blue Ash
17
174
User
Marion
17
175
Alex Fazioli
Columbiana
17
176
D S
Clyde
17
177
Ethan Ferkins (Student)
Upper Arlington
17
178
Orlando Wyman
Westlake
17
179
User
Athens
17
180
User
Columbus
17
181
User
Pickerington
17
182
User
Morrow
16
183
Donggeun Kim
Blue Ash
16
184
User
Columbus
16
185
User
Washington Court House
16
186
User
Dayton
16
187
User
Blue Ash
16
188
User
Strongsville
16
189
User
Fairborn
16
190
User
Barnesville
16
191
User
Columbus
16
192
User
New Franklin
16
193
User
Monroe
16
194
User
City
16
195
User
City
16
196
User
Reynoldsburg
16
197
User
Fairfield
16
198
User
Cleveland
16
199
User
Columbus
15
200
User
Columbus
15
201
User
Columbus
15
202
User
Blue Ash
15
203
User
Blue Ash
15
204
User
Sidney
15
205
User
Ravenna
15
206
User
Oxford
15
207
User
Cedarville
15
208
User
Columbus
15
209
User
Beaverdam
15
210
User
Columbus
14
211
Colin Battson
Blue Ash
14
212
User
Cincinnati
14
213
User
Shaker Heights
14
214
User
Beckett Ridge
14
215
User
Columbus
14
216
User
Reynoldsburg
14
217
User
Bluffton
14
218
User
Toledo
14
219
User
Columbus
14
220
User
Columbus
13
221
User
Middleburg Heights
13
222
Ben
Loveland
13
223
User
South Amherst
13
224
User
Sidney
13
225
User
Cleveland
13
226
Sabrina Woods
Saint Paris
13
227
User
Columbus
13
228
User
Painesville
13
229
User
Grove City
13
230
User
Centerville
13
231
User
Reynoldsburg
13
232
sunnysideup
Cleveland
13
233
User
Dublin
13
234
User
Chardon
12
235
User
Oxford
12
236
User
Beavercreek
12
237
User
Columbus
12
238
User
Columbus
12
239
User
Perry Heights
12
240
User
Norwood
12
241
PATRICK MILLS
Newark
12
242
User
Kent
12
243
User
Sidney
12
244
User
Columbus
12
245
User
Dublin
12
246
User
Columbus
12
247
Christian Landers
Columbus
12
248
User
Chardon
12
249
User
Cleveland
12
250
User
Oxford
11
251
MJT
New Philadelphia
11
252
User
Blue Ash
11
253
User
Sidney
11
254
User
Cincinnati
11
255
User
Sidney
11
256
User
Parma
11
257
User
Akron
11
258
User
Cuyahoga Falls
11
259
User
Columbus
11
260
User
Mayfield Heights
11
261
User
Cleveland
11
262
User
Cleveland
11
263
User
Columbus
11
264
User
Columbus
10
265
User
Columbus
10
266
Reyna Preza-Ledesma
Painesville
10
267
EJ
Springdale
10
268
User
Toledo
10
269
User
Grove City
10
270
Sophia Wolf
Strongsville
10
271
User
Middletown
10
272
User
Sidney
10
273
User
Sidney
10
274
User
Perrysburg
10
275
User
Lima
10
276
User
City
10
277
User
City
10
278
User
Reynoldsburg
10
279
User
Beachwood
10
280
User
Columbus
10
281
User
Springfield
9
282
User
Centerville
9
283
User
Minford
9
284
User
Beavercreek
9
285
User
Sidney
9
286
User
Columbus
9
287
Ediomo Eshett
Gahanna
9
288
User
Celina
9
289
User
Cincinnati
9
290
User
Oxford
9
291
User
Toledo
9
292
User
Sidney
9
293
User
Sidney
9
294
User
Sidney
9
295
User
Reynoldsburg
9
296
User
Delta
9
297
User
Athens
9
298
User
Dayton
9
299
User
Gahanna
9
300
User
Reynoldsburg
9
301
User
Cincinnati
9
302
waleed Alnajar
Mason
8
303
7th PVMS Alyson M Dildine
Columbus
8
304
User
Cleveland
8
305
User
Cleveland
8
306
User
Withamsville
8
307
User
Delaware
8
308
User
Sidney
8
309
User
Sidney
8
310
User
Sidney
8
311
User
Cincinnati
8
312
User
Columbus
8
313
User
Reynoldsburg
8
314
User
Columbus
8
315
User
Pickerington
8
316
User
Centerville
8
317
Carla Dearth
New London
8
318
User
Archbold
8
319
User
Wooster
8
320
User
Brunswick
8
321
User
Sebring
7
322
User
Cincinnati
7
323
User
Beavercreek
7
324
User
Columbus
7
325
User
Columbus
7
326
User
Strongsville
7
327
User
Strongsville
7
328
User
Greenville
7
329
User
Columbus
7
330
Avery
Finneytown
7
331
User
Blue Ash
7
332
User
Blue Ash
7
333
User
Stow
7
334
User
Sidney
7
335
User
Beckett Ridge
7
336
User
Perrysburg
7
337
User
Toledo
7
338
User
Cincinnati
7
339
User
Franklin
7
340
User
Clyde
7
341
User
Sylvania
7
342
User
Cleveland
7
343
User
Reynoldsburg
7
344
User
Reynoldsburg
7
345
User
Reynoldsburg
7
346
User
Columbus
7
347
User
Avon Lake
7
348
User
London
7
349
User
Athens
7
350
User
Twinsburg
7
351
User
Lebanon
7
352
User
Columbus
7
353
User
Hudson
6
354
User
Beavercreek
6
355
User
Columbus
6
356
User
Fairview Park
6
357
User
Hilliard
6
358
User
Sidney
6
359
User
Sidney
6
360
User
Steubenville
6
361
User
New Richmond
6
362
User
Sidney
6
363
User
Sidney
6
364
User
Columbus
6
365
User
Struthers
6
366
User
City
6
367
User
City
6
368
User
Reynoldsburg
6
369
User
Columbus
6
370
User
Oxford
6
371
User
Columbus
6
372
User
Beckett Ridge
6
373
User
Cleveland
6
374
User
Reynoldsburg
6
375
User
Cleveland
5
376
User
Beavercreek
5
377
User
Cleveland
5
378
User
Loveland
5
379
Edward Deng
Cleveland Heights
5
380
Alyson Dildine
Columbus
5
381
User
Massillon
5
382
User
Beachwood
5
383
User
Westerville
5
384
User
Elyria
5
385
User
Brewster
5
386
User
Sidney
5
387
User
Sidney
5
388
User
Sidney
5
389
User
Sidney
5
390
User
Toledo
5
391
User
Reynoldsburg
5
392
User
Reynoldsburg
5
393
User
Delaware
5
394
User
City
5
395
User
City
5
396
User
City
5
397
User
Columbus
5
398
User
Millville
5
399
User
Powell
5
400
User
Shaker Heights
5
401
User
Reynoldsburg
5
402
User
Gahanna
5
403
User
Reynoldsburg
5
404
User
Reynoldsburg
5
405
User
Monroe
4
406
User
Beavercreek
4
407
User
Beavercreek
4
408
User
Westerville
4
409
User
Toledo
4
410
John August Faber Brase
Columbus
4
411
User
Columbus
4
412
User
Sidney
4
413
User
Columbus
4
414
User
Newark
4
415
User
Lakewood
4
416
User
Middleburg Heights
4
417
User
Dayton
4
418
Shamini Gopinath
Sharonville
4
419
User
Lima
4
420
User
Columbus
4
421
User
Sidney
4
422
User
Columbus
4
423
User
Sidney
4
424
User
Sidney
4
425
User
Pataskala
4
426
User
Wetherington
4
427
User
Strongsville
4
428
User
Cleveland
4
429
User
Columbus
4
430
User
Cleveland Heights
4
431
User
Toledo
4
432
User
Shaker Heights
4
433
User
City
4
434
User
Reynoldsburg
4
435
User
Athens
4
436
User
Cincinnati
4
437
User
Mount Vernon
4
438
User
Mentor
4
439
User
North Royalton
4
440
User
Avon Lake
4
441
User
Reynoldsburg
4
442
User
Reynoldsburg
4
443
User
Reynoldsburg
4
444
User
Columbus
4
445
User
Grove City
4
446
User
Columbus
4
447
User
Centerville
3
448
User
Columbus
3
449
User
Forest Park
3
450
Thomas Clark
Strongsville
3
451
User
Lorain
3
452
User
Akron
3
453
User
Strongsville
3
454
User
Richfield
3
455
User
Sylvania
3
456
Timofey Bessonov
Blue Ash
3
457
User
Strongsville
3
458
User
Massillon
3
459
User
Columbus
3
460
User
Sidney
3
461
User
Sidney
3
462
User
Sidney
3
463
User
Sidney
3
464
User
Reynoldsburg
3
465
User
Beckett Ridge
3
466
User
Oxford
3
467
User
Norwood
3
468
User
Mansfield
3
469
User
Parma
3
470
User
Struthers
3
471
User
Broadview Heights
3
472
User
Broadview Heights
3
473
User
Broadview Heights
3
474
User
Mount Carmel
3
475
User
Broadview Heights
3
476
User
City
3
477
User
City
3
478
User
Reynoldsburg
3
479
User
Reynoldsburg
3
480
User
Cleveland
3
481
User
North Olmsted
3
482
User
Columbus
3
483
User
Upper Sandusky
3
484
User
Cleveland
3
485
User
Cleveland
3
486
User
Columbus
3
487
User
Reynoldsburg
3
488
User
Wadsworth
3
489
User
Reynoldsburg
3
490
User
Reynoldsburg
3
491
User
Reynoldsburg
3
492
User
Columbus
3
493
User
Columbus
3
494
User
Columbus
2
495
Addie Jenkins
North Canton
2
496
User
Columbus
2
497
User
Cincinnati
2
498
Brianna Jones
Columbus
2
499
Zerun Zhao
Blue Ash
2
500
User
Eaton
2
501
User
Columbus
2
502
User
Shelby
2
503
User
Columbus
2
504
User
Oxford
2
505
User
Newark
2
506
User
Perrysburg
2
507
User
Columbus
2
508
User
Toledo
2
509
User
Cincinnati
2
510
User
Shaker Heights
2
511
User
Toledo
2
512
User
Montgomery
2
513
User
Blue Ash
2
514
User
Blue Ash
2
515
Priscilla Cianelli
Strongsville
2
516
User
North Canton
2
517
User
Coldstream
2
518
User
Columbus
2
519
User
Sidney
2
520
User
New Carlisle
2
521
User
Pataskala
2
522
User
Columbus
2
523
User
Sidney
2
524
User
Sidney
2
525
User
Sidney
2
526
User
Sidney
2
527
User
Sidney
2
528
User
Solon
2
529
User
Hamilton
2
530
User
Columbus
2
531
User
Upper Arlington
2
532
User
Franklin
2
533
User
Columbus
2
534
User
Newark
2
535
User
Delaware
2
536
User
Lakewood
2
537
User
Cedarville
2
538
User
Reynoldsburg
2
539
User
Massillon
2
540
User
Broadview Heights
2
541
User
Akron
2
542
User
Cleveland
2
543
User
Struthers
2
544
User
Columbus
2
545
User
Lordstown
2
546
User
Toledo
2
547
User
Cincinnati
2
548
User
North Jackson
2
549
User
Reynoldsburg
2
550
Andrey Ganin
Sylvania
2
551
User
Reynoldsburg
2
552
User
Milford
2
553
User
Reynoldsburg
2
554
User
Reynoldsburg
2
555
User
Cleveland Heights
2
556
User
Cleveland
2
557
User
Cincinnati
2
558
User
Hudson
2
559
User
Springfield
2
560
User
Alliance
2
561
User
Fairfield
2
562
User
Reynoldsburg
2
563
User
Reynoldsburg
2
564
User
Mason
2
565
User
Reynoldsburg
2
566
User
Reynoldsburg
2
567
User
Willow Wood
2
568
User
Aurora
2
569
User
Cincinnati
2
570
User
Columbus
2
571
User
Barberton
2
572
User
Columbus
1
573
User
Mentor
1
574
User
Avon Lake
1
575
User
Brunswick
1
576
QuackDuck
Massillon
1
577
User
Cincinnati
1
578
Andrew Chamoun
Mayfield Heights
1
579
User
Cleveland
1
580
User
Baltimore
1
581
User
Akron
1
582
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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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