Leaderboard
Ohio
Rank
Level
1
Owen Summers
Mason
100
2
Nathaniel Day
Withamsville
77
3
Tanishq Pauskar
Massillon
65
4
King of the Hill
Blue Ash
63
5
User
Middleburg Heights
60
6
Nathan Johnson
Stow
56
7
User
Columbus
56
8
Puddles Penguin
Columbus
55
9
Scott Billups
Milford
54
10
User
Columbus
53
11
bboi
Chardon
52
12
User
Cincinnati
51
13
User
Cortland
48
14
Joseph Churilla
Westerville
45
15
Cory Huelskamp
Sidney
45
16
Athena
Upper Arlington
44
17
User
Aurora
42
18
User
Beachwood
42
19
QuackDuck
Massillon
40
20
User
Upper Arlington
40
21
Trevor Bischoff
Cleveland
40
22
EASTON SINGER
Solon
39
23
User
North Olmsted
37
24
Lodge Dils
Euclid
37
25
User
Solon
36
26
User
Forest Park
36
27
User
Brookville
36
28
Abhinav Goel
Montgomery
34
29
Justin Schimpf
Centerville
34
30
User
Toledo
33
31
User
Northgate
32
32
User
Columbus
31
33
Alvin Oshiotse
Milford
31
34
User
Blue Ash
31
35
User
Columbus
31
36
User
Solon
30
37
User
Columbus
29
38
User
Cincinnati
29
39
User
Toledo
28
40
User
Columbus
28
41
User
Forest Park
27
42
User
Athens
27
43
User
Solon
27
44
Andrew Zhou
Montgomery
27
45
DANIEL JONES
Columbus
26
46
User
Columbus
26
47
User
Columbus
25
48
User
Dublin
25
49
Maddy simmons
Columbus
24
50
Jon Yoon
Columbus
24
51
User
Columbus
24
52
User
Blue Ash
24
53
User
Cleveland
23
54
ROBERT BATCHLET
Wooster
23
55
Joe Gab nagana
Cleveland
23
56
Tensiechi
Columbus
23
57
User
Columbus
22
58
Mason Greene
Jerry City
22
59
DakotaHale
Ashland
22
60
User
Dublin
22
61
User
Montgomery
22
62
User
Blue Ash
22
63
User
Columbus
22
64
User
Cincinnati
21
65
User
Beckett Ridge
21
66
User
Kent
21
67
User
Fairborn
21
68
User
Saint Clairsville
21
69
User
Lyndhurst
20
70
User
Cincinnati
20
71
Siddharth Vikram
Blue Ash
20
72
Kevin Zhu
Barboursville
20
73
User
Athens
19
74
User
Cincinnati
19
75
Jake V
Toledo
19
76
User
Columbus
19
77
Darren Kaiser
Sidney
19
78
User
Montgomery
19
79
Ediomo Eshett
Gahanna
19
80
HANNAH JOHNSON
Columbus
19
81
User
Mason
18
82
Emanuel Melaku
Reynoldsburg
18
83
User
Dublin
18
84
User
Forest Park
18
85
User
Sidney
18
86
Master Flash
Reynoldsburg
18
87
User
Miamisburg
18
88
User
Cleveland
17
89
User
Withamsville
17
90
User
New Albany
17
91
User
Centerville
16
92
User
Columbus
16
93
Nick M
Columbus
16
94
User
Solon
16
95
Nicholas Johnson
Lewis Center
15
96
User
Sidney
15
97
User
Mason
15
98
Kristen Schutte
Blue Ash
15
99
User
Youngstown
15
100
User
Boardman
15
101
User
Fairfield
15
102
User
Pataskala
15
103
Justin Chen
Richmond Heights
14
104
ANDREW THOMAS
Pataskala
14
105
User
Columbus
14
106
User
Blue Ash
14
107
User
Sidney
14
108
Jeremy Arcus-Goldberg
New Albany
14
109
User
Kent
14
110
User
Columbus
14
111
User
Oxford
14
112
QuackDuck
Massillon
13
113
User
Blue Ash
13
114
User
Westlake
13
115
Josh Kelly
Columbus
13
116
User
Cleveland
13
117
Erice
Finneytown
13
118
User
Defiance
13
119
User
Columbus
13
120
User
Massillon
13
121
User
Sidney
13
122
Zerun Zhao
Blue Ash
13
123
User
Evendale
12
124
User
Westlake
12
125
User
Columbus
12
126
User
Athens
12
127
Divey
Sidney
12
128
User
Cincinnati
12
129
User
Mayfield Heights
12
130
User
Columbus
12
131
User
Pataskala
12
132
User
Morrow
12
133
Chenxi Zhou
Columbus
12
134
User
Columbus
12
135
User
Jerry City
12
136
User
Wooster
11
137
Donggeun Kim
Blue Ash
11
138
User
Cincinnati
11
139
User
Lorain
11
140
User
Columbus
11
141
User
Columbus
11
142
User
Columbus
11
143
User
Columbus
11
144
User
Springfield
11
145
User
Sharonville
11
146
User
Columbus
11
147
User
Westlake
11
148
Brady Ayers
Orrville
11
149
User
Washington Court House
10
150
Rachelle Hiestand
Sidney
10
151
Sean Griffin
Reynoldsburg
10
152
User
Mason
10
153
User
Chardon
10
154
User
Oxford
10
155
User
Centerville
10
156
User
Loveland
10
157
User
Columbus
10
158
User
Middleburg Heights
10
159
User
Dayton
10
160
User
Casstown
10
161
User
Columbus
10
162
orlando wyman
Ashland
10
163
JEREMY BAILEY
Orrville
10
164
User
Blue Ash
10
165
User
Marion
10
166
Ben
Loveland
10
167
User
Dublin
10
168
User
Akron
10
169
Alex Fazioli
Columbiana
10
170
User
Beavercreek
9
171
Reyna Preza-Ledesma
Painesville
9
172
User
Oxford
9
173
User
Sylvania
9
174
User
Columbus
9
175
User
Columbus
9
176
User
Cincinnati
9
177
User
Columbus
9
178
Sofiya Lozovskiy
Brecksville
9
179
User
Dover
9
180
User
Blue Ash
9
181
Alex P
Cedarville
9
182
7th PVMS Alyson M Dildine
Columbus
9
183
User
Dublin
9
184
User
Reynoldsburg
9
185
User
Perry Heights
9
186
User
Springfield
9
187
User
Columbus
9
188
User
Beachwood
9
189
User
Blue Ash
9
190
User
Strongsville
8
191
User
Wooster
8
192
User
Findlay
8
193
User
Oxford
8
194
User
Columbus
8
195
User
Columbus
8
196
User
Reynoldsburg
8
197
User
Blue Ash
8
198
User
Cincinnati
8
199
User
Oxford
8
200
User
Columbus
8
201
User
Pataskala
8
202
User
Defiance
8
203
User
Oxford
8
204
User
Twinsburg
8
205
User
Norwood
8
206
User
Columbus
8
207
User
Dublin
8
208
EJ
Springdale
8
209
PATRICK MILLS
Newark
8
210
D S
Clyde
8
211
User
Sidney
8
212
User
Cincinnati
8
213
Sabrina Woods
Saint Paris
8
214
Colin Battson
Blue Ash
8
215
User
Kent
8
216
User
Springfield
8
217
User
Toledo
8
218
MJT
New Philadelphia
8
219
User
Indian Hill
8
220
Brandon Engster
Warren
8
221
User
Columbus
7
222
User
Columbus
7
223
User
Centerville
7
224
User
Grove City
7
225
User
Columbus
7
226
User
Fairborn
7
227
Sophia Wolf
Strongsville
7
228
User
Sidney
7
229
User
Minford
7
230
User
Blue Ash
7
231
User
Struthers
7
232
Ethan Ferkins (Student)
Upper Arlington
7
233
Orlando Wyman
Westlake
7
234
User
South Amherst
7
235
User
Columbus
7
236
User
Beavercreek
7
237
User
Mount Repose
7
238
User
Youngstown
7
239
User
Solon
7
240
User
Twinsburg
7
241
User
Sidney
7
242
User
Centerville
7
243
User
City
7
244
User
Cleveland
7
245
User
Sidney
7
246
User
Barnesville
7
247
User
Reynoldsburg
7
248
User
Sidney
7
249
waleed Alnajar
Mason
7
250
User
Cincinnati
7
251
User
Sidney
7
252
User
Columbus
6
253
User
Columbus
6
254
User
Columbus
6
255
User
New Franklin
6
256
User
Cincinnati
6
257
User
Parma
6
258
User
Columbus
6
259
User
Reynoldsburg
6
260
User
Sidney
6
261
User
Middletown
6
262
User
Columbus
6
263
User
Monroe
6
264
User
Akron
6
265
User
Olmsted Falls
6
266
User
Columbus
6
267
User
Cleveland
6
268
User
Cleveland
6
269
User
Columbus
6
270
User
Sidney
6
271
User
Columbus
6
272
User
Upper Sandusky
6
273
User
Columbus
6
274
User
Celina
6
275
Griffin Waddell
Reynoldsburg
6
276
User
Sidney
6
277
User
Cincinnati
6
278
User
Cleveland
6
279
User
Sebring
6
280
User
Perrysburg
6
281
User
Painesville
6
282
User
City
6
283
User
City
6
284
User
Kenwood
6
285
User
Cincinnati
6
286
User
Columbiana
6
287
User
Gahanna
6
288
User
Parma
6
289
User
Columbus
6
290
User
Oxford
6
291
User
Beavercreek
6
292
User
Cincinnati
5
293
User
Toledo
5
294
User
Sidney
5
295
User
Sidney
5
296
sunnysideup
Cleveland
5
297
User
Sidney
5
298
User
Akron
5
299
User
Cleveland
5
300
User
Reynoldsburg
5
301
User
Cleveland
5
302
User
Columbus
5
303
User
Cuyahoga Falls
5
304
User
Marysville
5
305
User
Cleveland Heights
5
306
User
Ravenna
5
307
User
Columbus
5
308
User
Strongsville
5
309
User
Strongsville
5
310
User
Dublin
5
311
User
Grove City
5
312
User
Withamsville
5
313
User
Cincinnati
5
314
User
Columbus
5
315
Christian Landers
Columbus
5
316
User
Oxford
5
317
User
Centerville
5
318
User
Delaware
5
319
User
Hudson
5
320
User
Sidney
5
321
User
Columbus
5
322
User
Columbus
5
323
User
Cedarville
5
324
User
Beavercreek
5
325
User
Greenville
5
326
User
Fairfield
5
327
User
Sidney
5
328
User
Sidney
5
329
User
Mayfield Heights
5
330
User
Columbus
5
331
User
Cincinnati
5
332
User
Cleveland
5
333
User
Reynoldsburg
5
334
Avery
Finneytown
5
335
User
Grove City
5
336
User
Columbus
5
337
User
Columbus
5
338
User
Chardon
5
339
User
Columbus
4
340
User
City
4
341
User
City
4
342
User
Blue Ash
4
343
User
Blue Ash
4
344
User
Columbus
4
345
User
Fairview Park
4
346
User
Stow
4
347
User
Beaverdam
4
348
User
Columbiana
4
349
User
City
4
350
User
Reynoldsburg
4
351
User
Cincinnati
4
352
User
Cleveland
4
353
User
Reynoldsburg
4
354
User
Sidney
4
355
User
Columbus
4
356
User
Beckett Ridge
4
357
User
Perrysburg
4
358
User
Toledo
4
359
User
Reynoldsburg
4
360
User
Cleveland
4
361
User
Delta
4
362
User
Cincinnati
4
363
User
Franklin
4
364
User
Columbus
4
365
User
Clyde
4
366
User
Mentor
4
367
User
Hilliard
4
368
User
Athens
4
369
User
Dayton
4
370
User
Beavercreek
4
371
User
Pickerington
4
372
User
Cleveland
4
373
User
Sidney
4
374
User
Sidney
4
375
User
Steubenville
4
376
User
New Richmond
4
377
User
Massillon
4
378
User
Sidney
4
379
User
Sidney
4
380
User
Loveland
4
381
User
Gahanna
4
382
User
Canton
4
383
User
Centerville
4
384
User
Columbus
4
385
Jacob Auvil
Cleveland
3
386
User
Reynoldsburg
3
387
User
Columbiana
3
388
Zachary Hadaway
Kettering
3
389
User
Beachwood
3
390
User
Sylvania
3
391
User
Cleveland
3
392
User
Springboro
3
393
Carla Dearth
New London
3
394
Edward Deng
Cleveland Heights
3
395
User
Westerville
3
396
Alyson Dildine
Columbus
3
397
User
Columbus
3
398
User
Elyria
3
399
User
Wadsworth
3
400
User
Grove City
3
401
User
Brewster
3
402
User
Monroe
3
403
User
Struthers
3
404
User
Sidney
3
405
User
Sidney
3
406
User
Sidney
3
407
User
Sidney
3
408
User
Reynoldsburg
3
409
User
Beavercreek
3
410
User
Beavercreek
3
411
User
Reynoldsburg
3
412
User
Reynoldsburg
3
413
User
Columbus
3
414
User
Toledo
3
415
User
Columbus
3
416
User
Reynoldsburg
3
417
User
Reynoldsburg
3
418
User
Westerville
3
419
User
Toledo
3
420
User
Avon Lake
3
421
John August Faber Brase
Columbus
3
422
User
London
3
423
User
Columbus
3
424
User
Sidney
3
425
User
City
3
426
User
City
3
427
User
Athens
3
428
User
Columbus
3
429
Crystal
Cleveland
3
430
User
Dublin
3
431
User
Cincinnati
3
432
User
Reynoldsburg
3
433
User
Fairborn
3
434
User
Columbus
2
435
User
Newark
2
436
User
Lakewood
2
437
User
Middleburg Heights
2
438
User
Columbus
2
439
User
Dayton
2
440
User
Delaware
2
441
User
Oxford
2
442
User
Columbus
2
443
User
Beckett Ridge
2
444
Shamini Gopinath
Sharonville
2
445
User
Lima
2
446
User
City
2
447
User
City
2
448
User
City
2
449
User
Pepper Pike
2
450
User
Columbus
2
451
User
Columbus
2
452
User
Sidney
2
453
User
Columbus
2
454
User
Columbus
2
455
User
Sidney
2
456
User
Sidney
2
457
User
Pataskala
2
458
User
Forest Park
2
459
User
Wetherington
2
460
Thomas Clark
Strongsville
2
461
User
Strongsville
2
462
User
Lorain
2
463
User
Columbus
2
464
User
Millville
2
465
User
Akron
2
466
User
Strongsville
2
467
User
Richfield
2
468
User
Powell
2
469
User
Cleveland
2
470
Zariah Mitchell
Cleveland
2
471
User
Shaker Heights
2
472
User
Reynoldsburg
2
473
User
Gahanna
2
474
User
Reynoldsburg
2
475
User
Reynoldsburg
2
476
User
Medina
2
477
User
Cincinnati
2
478
User
Sylvania
2
479
Timofey Bessonov
Blue Ash
2
480
User
Columbiana
2
481
User
Columbiana
2
482
User
Strongsville
2
483
User
Columbus
2
484
User
Massillon
2
485
User
Cleveland Heights
2
486
User
Columbus
2
487
User
City
2
488
User
Columbus
2
489
User
Toledo
2
490
User
Lakewood
2
491
User
Columbus
2
492
User
Sidney
2
493
User
Toledo
2
494
User
Sidney
2
495
Addie Jenkins
North Canton
2
496
User
Cleveland
2
497
User
Sidney
2
498
User
Shaker Heights
2
499
User
Sharonville
2
500
User
Columbus
2
501
User
Sidney
1
502
User
Shaker Heights
1
503
User
Reynoldsburg
1
504
User
Cincinnati
1
505
Brianna Jones
Columbus
1
506
User
City
1
507
User
Beckett Ridge
1
508
User
Oxford
1
509
User
Reynoldsburg
1
510
User
Beckett Ridge
1
511
User
Athens
1
512
User
Columbus
1
513
User
Eaton
1
514
User
Columbus
1
515
User
Cincinnati
1
516
User
Mount Vernon
1
517
User
Norwood
1
518
User
Shelby
1
519
User
Mansfield
1
520
User
Columbus
1
521
User
Parma
1
522
User
Oxford
1
523
User
Columbus
1
524
User
Stow
1
525
User
Mentor
1
526
User
North Royalton
1
527
User
Newark
1
528
User
Struthers
1
529
User
Broadview Heights
1
530
User
Broadview Heights
1
531
User
Broadview Heights
1
532
User
Perrysburg
1
533
User
Avon Lake
1
534
User
Columbus
1
535
User
Mount Carmel
1
536
User
Newark
1
537
User
Reynoldsburg
1
538
User
Reynoldsburg
1
539
User
Reynoldsburg
1
540
User
Akron
1
541
User
Toledo
1
542
User
Cincinnati
1
543
User
Shaker Heights
1
544
User
Toledo
1
545
User
Montgomery
1
546
User
Blue Ash
1
547
User
Blue Ash
1
548
User
Columbus
1
549
Priscilla Cianelli
Strongsville
1
550
User
Broadview Heights
1
551
User
North Canton
1
552
User
City
1
553
User
City
1
554
User
Coldstream
1
555
User
Columbus
1
556
User
Reynoldsburg
1
557
User
Sidney
1
558
User
New Carlisle
1
559
User
Reynoldsburg
1
560
Realism The Future
Boardman
1
561
User
Pataskala
1
562
User
Columbus
1
563
User
Sylvania
1
564
User
Sidney
1
565
User
Sidney
1
566
User
Sidney
1
567
User
Sidney
1
568
User
Sidney
1
569
User
Solon
1
570
User
Cleveland
1
571
User
Hamilton
1
572
User
North Olmsted
1
573
User
Columbus
1
574
User
Upper Sandusky
1
575
User
Columbus
1
576
User
Upper Arlington
1
577
User
Cleveland
1
578
User
Cleveland
1
579
User
Franklin
1
580
User
Columbus
1
581
User
Reynoldsburg
1
582
User
Columbus
1
583
User
Westerville
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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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