Leaderboard
New Jersey
Rank
Level
1
Gianni Manginelli
Raritan
96
2
Alan Shenkerman
Marlboro Township
93
3
Michael Bosworth
Rumson
93
4
Sharon
West Windsor Township
89
5
door glass
Closter
88
6
Hannah Chu
Hopewell Township
88
7
Rahul Jain
Woodbridge Township
87
8
Kara Korussi
East Brunswick
74
9
User
Hopewell Township
72
10
User
Princeton
72
11
Mohammed Namazi
Hazlet
71
12
User
Newark
70
13
User
East Brunswick
69
14
Oops Oops
Paterson
64
15
III
Edison
64
16
User
Jersey City
64
17
Cory Sternin
Moorestown
64
18
Dirk Noom
Secaucus
62
19
Epic Bhangra
Old Bridge Township
58
20
Rawlison Zhang
Princeton
58
21
User
Voorhees Township
57
22
User
Old Bridge Township
57
23
User
South Brunswick Township
53
24
Anitya Gupta
Harrison
53
25
User
Franklin Township
53
26
User
Montclair
49
27
William Qiu
Jersey City
48
28
User
South Brunswick Township
47
29
User
Teaneck
46
30
User
Jackson
46
31
Juan
Newark
45
32
User
Teaneck
45
33
User
Bridgewater Township
44
34
User
Jersey City
44
35
User
Jersey City
43
36
Junjie Lin
Englewood
43
37
User
Mount Laurel Township
42
38
Katelyn D'Ambrosio
Westwood
42
39
JD
Washington Township
41
40
User
New Providence
41
41
User
Montgomery
40
42
User
Bernards
40
43
DoMo
Somerville
40
44
User
Voorhees Township
40
45
Andrew Song
Bridgewater Township
40
46
Anthony McWilliams
Toms River
39
47
User
Teaneck
38
48
Ru Poissen
Fort Lee
38
49
User
Montgomery
37
50
Falgoun Paatil
Hoboken
35
51
User
Franklin Township
35
52
User
Mount Holly
35
53
User
Teaneck
34
54
User
Montgomery
34
55
User
Tewksbury
33
56
User
Paramus
32
57
Daniel Kim
Hackensack
32
58
User
Voorhees Township
32
59
User
North Brunswick Township
31
60
User
Fort Lee
31
61
User
Princeton
31
62
User
Voorhees Township
31
63
User
Fort Lee
31
64
User
Upper Saddle River
30
65
User
Princeton
30
66
Prachet Trivedi
Matawan
30
67
User
Harrison
30
68
User
West New York
29
69
User
Franklin Township
29
70
User
Atlantic City
29
71
User
Newark
29
72
User
Englewood
29
73
User
Frankford
29
74
Dinofloss
Brick
28
75
User
Woodbridge Township
28
76
User
Westwood
28
77
User
Clifton
28
78
User
Ridgewood
28
79
User
Mount Laurel Township
27
80
G
West Windsor Township
27
81
Kyle Xiao
Newark
27
82
Of Rio
Keyport
27
83
MangoZoha BlueberryArifJam
Union
27
84
User
West Windsor Township
27
85
User
Keyport
27
86
User
Hoboken
26
87
Shiroe Chin
Livingston
26
88
User
Newark
26
89
User
Toms River
26
90
User
Princeton
26
91
User
Newark
25
92
Akhilesh Mahajan
Kearny
25
93
User
Newark
25
94
User
Marlboro Township
25
95
User
Jersey City
25
96
User
Bernards
25
97
User
Old Bridge Township
25
98
Justin Frazee
Lacey Township
24
99
User
Roseland
24
100
User
Hoboken
24
101
Mrgrothusen
Jersey City
24
102
User
Edison
24
103
User
Montgomery
24
104
User
Teaneck
23
105
Audrey
Mount Laurel
23
106
User
Roseland
23
107
User
Piscataway
23
108
User
Southampton Township
23
109
User
Jersey City
22
110
User
Edison
22
111
User
Old Bridge Township
22
112
User
Ridgewood
22
113
User
Bordentown Township
22
114
User
Jersey City
22
115
Joseph Bond MacDonald
Washington Township
21
116
User
Newark
21
117
Sohum Kshirsagar
West Windsor Township
21
118
User
Woodstown
21
119
User
Hopewell Township
21
120
User
Englewood
20
121
User
Montgomery
20
122
User
Edison
20
123
User
Jersey City
20
124
User
Newark
20
125
User
Montclair
20
126
Mahesh Kshirsagar
West Windsor Township
20
127
User
Franklin Township
20
128
xJames
West Windsor Township
19
129
User
East Orange
19
130
User
Old Bridge Township
19
131
User
Marlboro Township
19
132
User
Princeton
19
133
User
Old Bridge Township
19
134
User
Jersey City
19
135
User
East Brunswick
19
136
Bodhi Nguyen
Jersey City
18
137
User
Bellmawr
18
138
User
Franklin Township
18
139
Cristian Torres
Newark
18
140
User
Caldwell
18
141
User
Clifton
18
142
User
Princeton
18
143
User
Hoboken
18
144
User
Jersey City
17
145
User
Newark
17
146
User
Newark
17
147
User
Fair Lawn
17
148
User
Old Bridge Township
17
149
User
Teaneck
17
150
User
Westwood
17
151
Will Spadafora
Montclair
17
152
User
West Orange
17
153
User
Wall Township
17
154
User
New Milford
17
155
User
Franklin Township
17
156
User
Edison
17
157
User
Elmwood Park
17
158
User
Old Bridge Township
17
159
Jose Hernandez-Morel
Newark
17
160
Bob Sanderia
Galloway
16
161
User
Piscataway Township
16
162
User
Hackensack
16
163
User
Edison
16
164
User
Harrison
16
165
User
Edison
16
166
User
Jersey City
16
167
User
Atlantic City
16
168
Eli Bosnick
Belleville
16
169
User
Jersey City
16
170
Cristopher Flores
Newark
16
171
User
Millville
16
172
Amogh Sarangdhar
Montville
16
173
User
Montclair
16
174
User
Wall Township
16
175
User
Southampton Township
16
176
appllo
Tenafly
15
177
User
Newark
15
178
User
Jersey City
15
179
User
Hackensack
15
180
User
Cliffside Park
15
181
User
Brick
15
182
Michael Sorkin
Maplewood
15
183
User
New Brunswick
15
184
User
Roselle
15
185
Skyly Ramirez
Newark
15
186
User
Montgomery
14
187
User
Newark
14
188
User
Bloomfield
14
189
User
Jersey City
14
190
User
West Orange
14
191
User
Marlboro Township
14
192
User
Lodi
14
193
User
Jersey City
14
194
User
East Hanover
14
195
User
Waldwick
13
196
User
Franklin Township
13
197
User
Franklin Township
13
198
User
West Orange
13
199
User
Paramus
13
200
Sarah Cheng
Princeton
13
201
User
Monroe Township
13
202
User
Cherry Hill
13
203
User
Paramus
13
204
User
Newark
13
205
User
Parsippany-Troy Hills
13
206
Elian Polanco
Jersey City
13
207
User
Newark
13
208
elena
Bridgewater Township
13
209
User
Montgomery
13
210
User
Jersey City
12
211
User
Old Bridge Township
12
212
Jesse Pierson
Edison
12
213
User
Keyport
12
214
User
Toms River
12
215
User
Franklin Township
12
216
Mauricio Marin
Newark
12
217
Sarah Velez
Newark
12
218
User
Newark
12
219
User
Toms River
12
220
User
Newark
12
221
User
Jersey City
12
222
User
Maplewood
12
223
User
Old Bridge Township
12
224
User
Clifton
12
225
Cindy Hu
West Orange
12
226
User
Princeton
12
227
User
Roselle
12
228
Sanket Prasad
Edison
12
229
User
Howell
12
230
User
Highland Park
12
231
User
Atlantic City
12
232
User
Edison
12
233
User
Hanover
12
234
User
Cherry Hill
12
235
User
Montgomery
12
236
User
Camden
12
237
User
Bayonne
12
238
User
Galloway
12
239
Subal Bhattarai
Mahwah
11
240
Nick H
Princeton
11
241
User
Morris Plains
11
242
User
Old Bridge Township
11
243
Rahul K
Robbinsville
11
244
kevin lu
Park Ridge
11
245
User
Parsippany-Troy Hills
11
246
User
Robbinsville
11
247
User
Paterson
11
248
Todd Solecitto
Washington Township
11
249
User
Villas
11
250
User
Rahway
11
251
User
Milford
11
252
User
Morris Township
11
253
User
Edison
11
254
User
Westfield
11
255
User
South Brunswick Township
11
256
pablo chavez
Hoboken
11
257
User
Manchester Township
11
258
User
Newark
11
259
User
West Windsor Township
11
260
User
Parsippany-Troy Hills
11
261
User
East Windsor
11
262
Tamanna Rajpoot
Monroe Township
11
263
User
Franklin Township
11
264
User
Newark
11
265
User
Old Bridge Township
11
266
User
Newark
11
267
User
East Orange
11
268
User
Newark
11
269
User
Newark
11
270
User
Newark
10
271
User
South Brunswick Township
10
272
User
Princeton
10
273
User
Franklin Township
10
274
User
New Brunswick
10
275
User
East Brunswick
10
276
User
Jersey City
10
277
User
Old Bridge Township
10
278
User
Old Bridge Township
10
279
User
Jersey City
10
280
User
Jersey City
10
281
User
Ocean City
10
282
User
Newark
10
283
User
Passaic
10
284
User
Secaucus
10
285
User
South Brunswick Township
10
286
User
Lakewood Township
10
287
User
Randolph
10
288
User
Jersey City
10
289
User
Newark
10
290
User
Jersey City
10
291
User
North Bergen
10
292
Divyendra Patil
Newark
10
293
User
East Brunswick
10
294
User
Fair Lawn
10
295
User
Atlantic City
10
296
User
Springfield Township
10
297
User
Teaneck
9
298
User
Atlantic City
9
299
User
Montgomery
9
300
User
Roselle
9
301
User
East Brunswick
9
302
User
West Windsor Township
9
303
User
Hanover
9
304
User
Hackensack
9
305
User
Linden
9
306
User
Closter
9
307
User
Phillipsburg
9
308
User
Ridgewood
9
309
User
Perth Amboy
9
310
User
Franklin Township
9
311
User
Paterson
9
312
User
North Bergen
9
313
User
Harrison
9
314
User
Harrison
9
315
User
Jersey City
9
316
Kaira Vazquez
Newark
9
317
User
Old Bridge Township
9
318
User
Princeton
9
319
Sandy Bonilla
Newark
9
320
User
Highland Park
9
321
User
Newark
9
322
User
Fair Lawn
9
323
User
Secaucus
9
324
User
Princeton
9
325
User
Newark
9
326
User
Bridgewater
9
327
User
Newark
9
328
User
Elizabeth
9
329
User
Newark
9
330
User
Old Bridge Township
8
331
User
South Plainfield
8
332
Daniel Mata Murillo
Newark
8
333
User
Hoboken
8
334
User
Sparta Township
8
335
User
Princeton
8
336
User
Toms River
8
337
User
Paterson
8
338
User
South Brunswick Township
8
339
User
Princeton
8
340
User
Hazlet
8
341
User
Princeton
8
342
User
Newark
8
343
User
Toms River
8
344
User
Edison
8
345
User
Piscataway
8
346
CIRA
Newark
8
347
User
Fort Lee
8
348
User
Jersey City
8
349
User
Jersey City
8
350
User
Closter
8
351
User
Newark
8
352
User
Tenafly
8
353
User
Newark
8
354
User
Montgomery
8
355
User
Bloomfield
8
356
User
Newark
8
357
Lewis Lantigua
Newark
8
358
User
Jersey City
8
359
User
Hoboken
8
360
User
Livingston
8
361
Shahil Shrestha
Jersey City
8
362
User
Parsippany-Troy Hills
8
363
User
Jersey City
8
364
User
Camden
8
365
User
Newark
8
366
User
Asbury Park
8
367
User
Asbury Park
8
368
User
City
8
369
User
Woodbridge Township
8
370
User
Princeton
8
371
User
Cherry Hill
8
372
User
Princeton Meadows
8
373
User
Jersey City
8
374
User
East Hanover
8
375
Flavio
North Bergen
8
376
User
Flemington
7
377
User
Bayonne
7
378
User
Weehawken
7
379
User
Collingswood
7
380
User
Raritan
7
381
User
Jersey City
7
382
User
Newark
7
383
Erick Romano
Newark
7
384
Oscar Recinos-Giron
Newark
7
385
User
Montgomery
7
386
A J
Newark
7
387
User
Franklin Township
7
388
User
Kearny
7
389
andress etindele
Secaucus
7
390
User
Newark
7
391
Jack F
Newark
7
392
Isiah Medina
Newark
7
393
User
Jersey City
7
394
User
Kearny
7
395
Neelay
Harrison
7
396
User
Lawrence Township
7
397
User
Westwood
7
398
User
Jersey City
7
399
User
Raritan
7
400
User
East Brunswick
7
401
User
Newark
7
402
User
Newark
7
403
User
Belleville
7
404
User
Bloomfield
7
405
User
Princeton
7
406
User
Ridgefield Park
7
407
User
Clifton
7
408
User
Ridgefield
7
409
User
City
7
410
User
Woodbridge Township
7
411
User
Monroe Township
7
412
User
Newark
7
413
User
Highland Park
7
414
User
West Orange
7
415
User
Edison
7
416
Carol Lorac
City
7
417
Marbely Martinez
Newark
7
418
User
Princeton
7
419
User
Newark
7
420
User
Newark
7
421
User
Newark
7
422
User
Raritan
7
423
Myfitness Journey
Montgomery
7
424
User
Mount Laurel
7
425
User
Westfield
7
426
User
Paterson
7
427
User
Newark
6
428
User
Newark
6
429
Josef Kariuki
Union
6
430
User
Ridgewood
6
431
User
Newark
6
432
User
Newark
6
433
User
Edison
6
434
User
Livingston
6
435
User
North Bergen
6
436
User
Montvale
6
437
User
Oakland
6
438
User
Sparta Township
6
439
Vishak Chidam
Jersey City
6
440
Vishy C
Jersey City
6
441
User
Newark
6
442
User
Wyckoff
6
443
User
Newark
6
444
Cira Robinson
Newark
6
445
User
Paterson
6
446
User
West Windsor Township
6
447
User
Toms River
6
448
User
Edison
6
449
User
Fair Lawn
6
450
User
Newark
6
451
polly
Bridgewater
6
452
User
Newark
6
453
Victor Jhoan Bonilla
Newark
6
454
User
Passaic
6
455
Teoeuroi
Jersey City
6
456
User
Newark
6
457
User
Princeton
6
458
User
North Bergen
6
459
User
Old Bridge Township
6
460
User
Newark
6
461
User
Newark
6
462
User
Maplewood
6
463
User
Freehold Township
6
464
User
Livingston
6
465
Debbus Q
Evesham Township
6
466
User
Jersey City
6
467
User
Summit
6
468
User
Newark
6
469
User
Newark
6
470
User
Egg Harbor Township
6
471
User
Oakland
6
472
User
Bloomfield
6
473
User
Jersey City
6
474
User
Monroe Township
6
475
User
Old Bridge Township
6
476
User
Bridgewater
6
477
User
Brick
6
478
User
Holmdel
6
479
User
Holmdel
6
480
User
West Milford
6
481
User
Secaucus
6
482
User
Franklin Township
6
483
User
Montgomery
6
484
User
Newark
6
485
User
Jersey City
6
486
User
Woolwich Township
6
487
User
West Long Branch
6
488
User
Highland Park
6
489
User
Ridgefield Park
6
490
User
West Windsor Township
6
491
User
Chester Township
6
492
User
Clifton
6
493
User
Pennsauken Township
6
494
User
North Caldwell
5
495
User
Newark
5
496
User
Princeton
5
497
User
Cherry Hill
5
498
User
Newark
5
499
Camilo Aguilar
Newark
5
500
Sai Villegas
Newark
5
501
User
Monroe Township
5
502
User
New Hanover Township
5
503
User
North Bergen
5
504
User
Newark
5
505
User
Newark
5
506
User
Piscataway Township
5
507
User
Collingswood
5
508
User
Newark
5
509
User
Marlboro Township
5
510
User
Flemington
5
511
User
Irvington
5
512
User
Monroe Township
5
513
User
Newark
5
514
Julinel Rodriguez
Newark
5
515
Carlos Gomez
Newark
5
516
User
Newark
5
517
User
Newark
5
518
User
Newark
5
519
Camilo Aguilar Espitia
Newark
5
520
User
Clifton
5
521
User
Sayreville
5
522
User
Edgewater
5
523
User
Woodbridge Township
5
524
User
Hamilton Township
5
525
User
Manalapan Township
5
526
User
North Caldwell
5
527
User
Edison
5
528
User
South Brunswick Township
5
529
User
Paterson
5
530
User
Mount Laurel
5
531
User
Mahwah
5
532
User
Cranford
5
533
User
Elizabeth
5
534
User
North Brunswick Township
5
535
User
Kearny
5
536
User
Newark
5
537
User
Jersey City
5
538
User
Beachwood
5
539
User
Elizabeth
5
540
User
Randolph
5
541
User
Elmwood Park
5
542
User
Franklin
5
543
Michael Yang
Jersey City
5
544
User
Newark
5
545
User
Newark
5
546
User
Montgomery
5
547
User
Clifton
5
548
Jefry G
Newark
5
549
User
North Bergen
5
550
User
Mahwah
5
551
Pac-Man
Holmdel
5
552
User
Holmdel
5
553
User
Highland Park
5
554
User
Belleville
5
555
User
Ridgewood
5
556
User
Newark
5
557
Ismail Richardson
Newark
5
558
Katherine Laurens
Jersey City
5
559
User
Princeton
5
560
User
Ridgewood
5
561
User
Jersey City
5
562
User
Cliffside Park
5
563
User
Newark
5
564
User
Newark
5
565
User
Jackson
5
566
Sal Cohen
Egg Harbor Township
5
567
User
Chester Township
5
568
Wendy Cedillos
Newark
5
569
User
Newark
5
570
User
Newark
5
571
User
Jersey City
5
572
User
Newark
5
573
User
Jersey City
5
574
User
Jersey City
5
575
User
Newark
5
576
User
Toms River
5
577
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5
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584
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5
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5
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5
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4
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4
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4
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4
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4
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4
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4
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3
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3
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3
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3
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3
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3
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888
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1
983
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1
984
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1
985
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1
986
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987
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988
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1
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Newark
1
991
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1
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1
993
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Newark
1
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1000
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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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