Leaderboard
New Jersey
Rank
Level
1
DoMo
Somerville
93
2
User
Princeton
93
3
Anthony McWilliams
Toms River
91
4
Daniel Kim
Englewood
91
5
Kyle Xiao
Newark
90
6
User
Jersey City
86
7
balsu
Mahwah
83
8
User
Paramus
81
9
User
Princeton
81
10
User
Jersey City
74
11
User
Voorhees Township
71
12
User
Hazlet
68
13
Yohan Ninan
Newark
67
14
User
Newark
66
15
User
Voorhees Township
61
16
User
Hoboken
61
17
User
Harrison
61
18
User
Woodstown
59
19
Prachet Trivedi
Matawan
58
20
User
Voorhees Township
57
21
User
Newark
57
22
Michael B
Middletown
57
23
User
Jersey City
55
24
User
Jersey City
54
25
User
Jersey City
52
26
Christopher Ye
Princeton
50
27
User
Fort Lee
48
28
Dinofloss
Brick
46
29
G
West Windsor Township
46
30
User
Hoboken
46
31
User
Harrison
46
32
User
Newark
45
33
User
Fort Lee
43
34
User
Jersey City
42
35
User
Englewood
42
36
Audrey
Mount Laurel
41
37
User
Old Bridge Township
41
38
User
Jersey City
41
39
User
Old Bridge Township
40
40
User
Franklin Township
40
41
User
Roseland
39
42
User
Edison
39
43
Jesse Pierson
Edison
39
44
User
Edison
38
45
User
Galloway
38
46
User
Somerset
38
47
User
Princeton
37
48
User
Brick
37
49
User
Manchester Township
37
50
Eli Bosnick
Belleville
36
51
User
Jersey City
36
52
User
East Brunswick
35
53
User
Robbinsville
35
54
Todd Solecitto
Washington Township
35
55
User
West Orange
34
56
Amogh Sarangdhar
Montville
33
57
User
Wall Township
31
58
Bob Sanderia
Galloway
31
59
User
New Brunswick
31
60
Cindy Hu
West Orange
31
61
User
Highland Park
31
62
User
Bayonne
31
63
User
Princeton
31
64
User
Jersey City
31
65
Divyendra Patil
Newark
31
66
User
Jersey City
30
67
User
Montclair
30
68
User
Monroe Township
30
69
User
Old Bridge Township
30
70
Elian Polanco
Jersey City
30
71
User
Jersey City
30
72
User
Hanover
30
73
User
Linden
30
74
User
Princeton
30
75
User
Old Bridge Township
29
76
User
Madison
29
77
User
Cliffside Park
28
78
User
West Orange
28
79
User
Old Bridge Township
28
80
Rahul K
Robbinsville
28
81
User
Secaucus
28
82
User
Hoboken
28
83
User
Trenton
28
84
User
Nutley
28
85
Will Spadafora
Montclair
27
86
User
Edison
27
87
User
West Windsor Township
27
88
User
Hackensack
26
89
User
Cherry Hill
26
90
Matthew Grothusen
Hoboken
26
91
User
Morris Township
26
92
User
Old Bridge Township
26
93
User
Old Bridge Township
26
94
User
Harrison
26
95
User
Princeton
26
96
User
South Plainfield
26
97
User
Jersey City
26
98
User
Hoboken
26
99
User
East Hanover
26
100
User
South Plainfield
26
101
User
Teaneck
25
102
User
Bloomfield
25
103
User
Old Bridge Township
25
104
User
Princeton
25
105
appllo
Tenafly
24
106
Sanket Prasad
Edison
24
107
User
Edison
24
108
User
Princeton
24
109
User
Newark
24
110
User
Marlboro Township
23
111
User
Toms River
23
112
User
Cherry Hill
23
113
User
Camden
23
114
User
Morris Plains
23
115
User
Highland Park
23
116
User
Tenafly
23
117
User
Franklin
23
118
User
Lawrence Township
23
119
User
Belleville
23
120
User
Collingswood
23
121
User
Plainfield
23
122
User
Toms River
22
123
User
Clifton
22
124
User
Woodbridge Township
22
125
Myfitness Journey
Montgomery
22
126
User
Mount Laurel
22
127
User
Hammonton
22
128
User
Jersey City
22
129
Sarah Cheng
Princeton
21
130
User
Parsippany-Troy Hills
21
131
User
Fair Lawn
21
132
User
Old Bridge Township
21
133
Neelay
Harrison
21
134
User
Princeton
21
135
User
Livingston
21
136
Vishak Chidam
Jersey City
21
137
Vishy C
Jersey City
21
138
User
Pennsauken Township
21
139
User
Mount Laurel
21
140
User
Old Bridge
21
141
User
Paramus
20
142
Nick H
Princeton
20
143
User
South Brunswick Township
20
144
User
Ridgewood
20
145
User
North Bergen
20
146
User
Newark
20
147
User
Old Bridge Township
20
148
User
Princeton
19
149
User
Hanover
19
150
User
Princeton
19
151
User
Clifton
19
152
User
Princeton
19
153
User
North Bergen
19
154
User
East Windsor
19
155
User
Ridgewood
19
156
User
Clifton
19
157
User
Monroe Township
19
158
User
Mahwah
19
159
User
Paterson
18
160
User
Newark
18
161
User
Ocean City
18
162
User
Hackensack
18
163
User
Bloomfield
18
164
User
Camden
18
165
User
Highland Park
18
166
User
Edgewater
18
167
User
Jersey City
18
168
User
Paramus
18
169
Skyly Ramirez
Newark
17
170
User
Passaic
17
171
User
Bridgewater
17
172
User
Secaucus
17
173
User
Ridgewood
17
174
User
North Bergen
17
175
User
Jersey City
17
176
User
Jersey City
17
177
User
Jersey City
16
178
User
Jersey City
16
179
User
North Bergen
16
180
User
Jersey City
16
181
User
Collingswood
16
182
User
Jersey City
16
183
User
Cherry Hill
16
184
User
Beachwood
16
185
User
Vineland
16
186
Sandy Bonilla
Newark
15
187
User
Princeton
15
188
User
Parsippany-Troy Hills
15
189
User
Raritan
15
190
User
Ridgefield
15
191
User
Monroe Township
15
192
User
East Brunswick
15
193
User
Cranford
15
194
User
Elmwood Park
15
195
Katherine Laurens
Jersey City
15
196
User
Princeton
15
197
User
Harrison
15
198
User
Vineland
15
199
User
Trenton
15
200
ACE FACE
Edison
15
201
Daniel Mata Murillo
Newark
14
202
User
Toms River
14
203
User
West Windsor Township
14
204
User
Newark
14
205
User
Ridgefield Park
14
206
User
Hackensack
14
207
User
Linden
14
208
User
Fort Lee
14
209
User
Madison
14
210
User
Highland Park
14
211
User
Piscataway Township
14
212
User
Newark
14
213
User
Princeton
14
214
User
Tabernacle
14
215
User
New Brunswick
14
216
Michael Sorkin
Maplewood
14
217
User
Piscataway Township
14
218
User
Monroe Township
14
219
User
Piscataway Township
14
220
User
Perth Amboy
13
221
User
Paterson
13
222
Mahesh Kshirsagar
West Windsor Township
13
223
Kaira Vazquez
Newark
13
224
User
Newark
13
225
User
Newark
13
226
User
Princeton
13
227
User
Woodbridge Township
13
228
User
Princeton Meadows
13
229
User
Flemington
13
230
User
Jersey City
13
231
User
Maplewood
13
232
User
Summit
13
233
User
New Hanover Township
13
234
User
Highland Park
13
235
User
East Brunswick
13
236
User
Jersey City
13
237
User
Brick
13
238
User
New Providence
13
239
User
Voorhees Township
13
240
User
Sparta Township
12
241
User
Paterson
12
242
User
Toms River
12
243
User
Fort Lee
12
244
User
Newark
12
245
User
Newark
12
246
User
Newark
12
247
Lewis Lantigua
Newark
12
248
User
Asbury Park
12
249
A J
Newark
12
250
User
Jersey City
12
251
User
Edison
12
252
User
Princeton
12
253
User
Newark
12
254
User
Highland Park
12
255
User
Piscataway Township
12
256
User
Edison
12
257
User
Mount Laurel
12
258
User
Kearny
12
259
User
Elizabeth
12
260
User
Ridgewood
12
261
User
Livingston
11
262
User
Jersey City
11
263
User
Weehawken
11
264
Erick Romano
Newark
11
265
User
Newark
11
266
Oscar Recinos-Giron
Newark
11
267
Teoeuroi
Jersey City
11
268
User
Oakland
11
269
User
Bloomfield
11
270
User
Cliffside Park
11
271
User
Jackson
11
272
User
City
11
273
User
Newark
11
274
Subharthi Sen
Princeton
11
275
User
Princeton
11
276
User
South Plainfield
11
277
User
Ridgefield Park
10
278
Jose Hernandez-Morel
Newark
10
279
User
Newark
10
280
User
Newark
10
281
User
Newark
10
282
User
Westfield
10
283
User
Toms River
10
284
User
Jersey City
10
285
User
Clifton
10
286
User
North Caldwell
10
287
User
Marlboro Township
10
288
User
Mahwah
10
289
User
Highland Park
10
290
User
Highland Park
10
291
User
Highland Park
10
292
User
Highland Park
10
293
User
Tinton Falls
10
294
User
River Edge
10
295
User
Princeton
10
296
User
Westwood
10
297
User
Middletown
10
298
Nyce
Trenton
10
299
User
Wayne
10
300
User
Jersey City
9
301
User
Oakland
9
302
User
Paterson
9
303
User
Edison
9
304
User
Newark
9
305
Victor Jhoan Bonilla
Newark
9
306
User
Newark
9
307
User
Newark
9
308
User
Freehold Township
9
309
User
West Long Branch
9
310
User
West Windsor Township
9
311
User
Monroe Township
9
312
User
Woodbridge Township
9
313
User
Jersey City
9
314
User
Bergenfield
9
315
User
Lopatcong
9
316
User
Ho-Ho-Kus
9
317
User
Union
9
318
User
Jersey City
9
319
User
Cresskill
9
320
User
Livingston
9
321
User
Monroe Township
9
322
User
Bayonne
9
323
User
Morristown
9
324
User
Jackson
9
325
User
Clifton
9
326
User
Carteret
9
327
User
Brick
8
328
User
Old Bridge Township
8
329
User
Jersey City
8
330
User
Ridgefield Park
8
331
User
Newark
8
332
Camilo Aguilar
Newark
8
333
Sai Villegas
Newark
8
334
User
North Bergen
8
335
User
Newark
8
336
User
Newark
8
337
User
Newark
8
338
User
Flemington
8
339
User
Monroe Township
8
340
Julinel Rodriguez
Newark
8
341
Carlos Gomez
Newark
8
342
Jack Ferrufino
Newark
8
343
User
Newark
8
344
User
Newark
8
345
Camilo Aguilar Espitia
Newark
8
346
User
Newark
8
347
User
Newark
8
348
User
Newark
8
349
User
Jersey City
8
350
User
Bergenfield
8
351
User
Glassboro
8
352
User
Holmdel
8
353
User
Hawthorne
8
354
User
Newark
8
355
User
Parsippany-Troy Hills
8
356
User
Highland Park
8
357
User
Irvington
8
358
User
Montgomery
8
359
User
Princeton
8
360
User
Ridgewood
8
361
User
Voorhees Township
8
362
User
Monroe Township
8
363
User
South Plainfield
8
364
User
Ramsey
8
365
User
Piscataway Township
7
366
User
Jersey City
7
367
Sal Cohen
Egg Harbor Township
7
368
Wendy Cedillos
Newark
7
369
User
Newark
7
370
User
Newark
7
371
User
Jersey City
7
372
User
Newark
7
373
User
Toms River
7
374
Katia Nieves
Newark
7
375
User
Newark
7
376
User
Newark
7
377
User
Newark
7
378
User
Newark
7
379
User
Jackson
7
380
User
Jersey City
7
381
User
Edison
7
382
User
Fair Lawn
7
383
User
Westfield
7
384
User
Summit
7
385
User
Highland Park
7
386
User
Highland Park
7
387
User
Galloway
7
388
User
Newark
7
389
User
Newark
7
390
User
Burlington Township
7
391
User
Bellmawr
7
392
User
Lambertville
7
393
User
Sayreville
7
394
User
Bridgewater
7
395
User
Edison
7
396
User
Nutley
7
397
User
Vineland
7
398
User
Parsippany-Troy Hills
7
399
User
Paterson
6
400
Keycy Montano🇸🇻🎮⚽️
Newark
6
401
ALESSANDRO CERVERA
Paterson
6
402
Cira Robinson
Newark
6
403
User
Franklin Township
6
404
Angelin Castro
Newark
6
405
User
Newark
6
406
User
Newark
6
407
User
Newark
6
408
User
Newark
6
409
User
Newark
6
410
User
Newark
6
411
User
Newark
6
412
User
Newark
6
413
User
Newark
6
414
User
Clifton
6
415
User
North Bergen
6
416
User
West Long Branch
6
417
User
New Brunswick
6
418
User
Montclair
6
419
User
Paterson
6
420
User
Montgomery
6
421
User
Woodbridge Township
6
422
Sesilja ?
Plumsted Township
6
423
User
Newark
6
424
User
Mount Laurel
6
425
User
Morris Plains
6
426
User
East Rutherford
6
427
User
Matawan
6
428
User
Princeton
6
429
User
Moorestown
6
430
User
Newark
6
431
User
Collingswood
6
432
User
North Bergen
6
433
User
Ringwood
6
434
User
Jersey City
6
435
User
Princeton
6
436
User
Union
5
437
User
Newark
5
438
User
Jersey City
5
439
User
Paterson
5
440
User
Newark
5
441
User
Newark
5
442
User
Newark
5
443
User
Newark
5
444
User
Newark
5
445
User
Newark
5
446
Jose Matias
Newark
5
447
Cristopher Flores
Newark
5
448
Abraham Regalado
Newark
5
449
User
Jersey City
5
450
User
Newark
5
451
User
Newark
5
452
User
Newark
5
453
Denys Salazar-Galdamez
Newark
5
454
Jeydrick Cruz
Newark
5
455
ismail
Newark
5
456
Matthew Irizarry
Newark
5
457
Mauricio Marin
Newark
5
458
User
Newark
5
459
User
Newark
5
460
User
Newark
5
461
User
Newark
5
462
User
Newark
5
463
User
Newark
5
464
User
Newark
5
465
User
Newark
5
466
User
Newark
5
467
User
Newark
5
468
User
Toms River
5
469
Jack Skorupka
Middletown
5
470
User
Jersey City
5
471
User
Piscataway Township
5
472
User
Scotch Plains
5
473
User
West Orange
5
474
User
Monroe Township
5
475
Fyttyn
Secaucus
5
476
User
Egg Harbor Township
5
477
User
Burlington Township
5
478
User
Princeton
5
479
User
East Amwell Township
5
480
User
Hoboken
5
481
User
Highland Park
5
482
User
Toms River
5
483
User
Piscataway Township
5
484
User
Secaucus
5
485
User
West Windsor Township
5
486
User
Franklin Lakes
5
487
User
Jersey City
5
488
User
North Bergen
5
489
User
Bedminster Township
5
490
User
Englewood
5
491
User
Mount Arlington
5
492
Veda Kunamneni
Hightstown
5
493
User
Dayton
5
494
User
South Plainfield
5
495
User
Fort Lee
4
496
User
Parsippany-Troy Hills
4
497
User
New Brunswick
4
498
User
Jersey City
4
499
User
Paterson
4
500
User
Paterson
4
501
User
Paterson
4
502
User
South Brunswick Township
4
503
User
Newark
4
504
User
Newark
4
505
Jefry Salazar Galdamez
Newark
4
506
User
Newark
4
507
Zaniaya
Newark
4
508
User
Delanco
4
509
User
Hoboken
4
510
Quener Lopez Cruz
Newark
4
511
User
Newark
4
512
User
Newark
4
513
Arieliz Nazario
Newark
4
514
User
Newark
4
515
User
Newark
4
516
User
Newark
4
517
User
Newark
4
518
User
Newark
4
519
User
Newark
4
520
User
Newark
4
521
User
Newark
4
522
User
Newark
4
523
User
Newark
4
524
Himany Santos Quezada
Newark
4
525
User
Newark
4
526
john peepee
Newark
4
527
User
Newark
4
528
User
Newark
4
529
User
Newark
4
530
User
Newark
4
531
User
Newark
4
532
User
Newark
4
533
User
Newark
4
534
User
Paterson
4
535
User
Clifton
4
536
User
Jersey City
4
537
User
Nutley
4
538
User
Elizabeth
4
539
User
North Bergen
4
540
User
Westfield
4
541
User
Edison
4
542
User
Wayne
4
543
User
Cherry Hill
4
544
User
Edison
4
545
User
New Brunswick
4
546
User
Newark
4
547
User
South Orange
4
548
User
West Windsor Township
4
549
User
Summit
4
550
Mario
Pennsauken Township
4
551
Dewen liu
Evesham Township
4
552
User
Edison
4
553
User
Princeton
4
554
User
Metuchen
4
555
User
Edison
4
556
User
Piscataway Township
4
557
User
North Bergen
4
558
User
Tenafly
4
559
User
Belleville
4
560
User
Bayonne
4
561
User
Kendall Park
4
562
User
Secaucus
3
563
User
Parsippany-Troy Hills
3
564
User
Fort Lee
3
565
User
Little Ferry
3
566
User
Paterson
3
567
User
Paterson
3
568
User
Newark
3
569
User
Newark
3
570
User
Newark
3
571
Robin Rosario
Newark
3
572
User
Newark
3
573
User
Newark
3
574
User
Newark
3
575
User
Newark
3
576
User
Linden
3
577
User
Newark
3
578
User
Newark
3
579
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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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