Leaderboard
North Carolina
North Carolina
United States
Worldwide
Rank
Level
1
Greck
Rocky Point
88
2
User
Durham
85
3
User
Nashville
81
4
Kent A
Cornelius
81
5
User
Chapel Hill
79
6
User
Chapel Hill
79
7
User
Burgaw
77
8
User
China Grove
76
9
Imhotep Truitt
Durham
70
10
sLiNky
Wake Forest
70
11
Angel Escobar
Durham
69
12
Justin Blank
Garner
67
13
Nikhar Patel
Chapel Hill
64
14
Christopher Ortiz
Durham
61
15
Gianni Crivello
Kannapolis
58
16
User
Morrisville
57
17
Jennifer Guerra
Stem
57
18
User
Raleigh
57
19
Owen Smith
Durham
54
20
User
Cary
52
21
CLT DUDE
Charlotte
51
22
User
Durham
49
23
Christian Calderon
Durham
47
24
William Yu
Durham
46
25
MODOZ
Cary
44
26
User
Cary
44
27
User
Chapel Hill
44
28
Paul Finger
Mint Hill
44
29
User
Cary
43
30
Nathan Hales
Winston-Salem
42
31
User
Durham
41
32
User
Winston-Salem
40
33
xd
Durham
40
34
User
Charlotte
40
35
alyssa bandel
Sherrills Ford
40
36
User
Durham
40
37
Jarek Ma
Charlotte
40
38
User
Davidson
40
39
User
Cary
40
40
Aidan Jones
Durham
39
41
User
Greensboro
39
42
User
Charlotte
39
43
Sleevechokes
Kinston
38
44
Kristin Shibu
Durham
38
45
User
Charlotte
37
46
Aashay Arora
Charlotte
37
47
Tanmay Sah
Raleigh
37
48
Kim Li
Raleigh
37
49
Jaime Silva-Flores
Durham
36
50
User
Charlotte
36
51
Oscar Chavarria
Durham
35
52
User
Pittsboro
35
53
User
Pittsboro
35
54
User
Charlotte
35
55
User
Charlotte
35
56
User
Durham
34
57
User
Chapel Hill
34
58
User
Chapel Hill
33
59
User
Durham
33
60
User
Durham
33
61
User
Cary
32
62
Isaiah Thorpe
Durham
32
63
User
Winston-Salem
32
64
User
Chapel Hill
32
65
Ravi Karanam
Durham
32
66
User
Durham
32
67
User
Graham
31
68
User
Chapel Hill
31
69
Newton Yu
Winston-Salem
31
70
User
Raleigh
31
71
User
Carrboro
30
72
User
Durham
30
73
User
Charlotte
30
74
User
Carrboro
30
75
User
Winston-Salem
30
76
User
Durham
30
77
User
Durham
29
78
User
Winston-Salem
29
79
Graham Shunk
Durham
28
80
User
Charlotte
28
81
User
Charlotte
27
82
User
Cary
27
83
Tyree Keith
Greensboro
27
84
User
Durham
27
85
George Rateb
Waxhaw
27
86
Sam Nava Martinez
Durham
26
87
User
Raleigh
26
88
User
Winston-Salem
26
89
Robert Antoine II
Charlotte
26
90
User
Durham
26
91
User
Greensboro
26
92
Carlos Alvarez
Raleigh
26
93
User
Burlington
26
94
User
Charlotte
26
95
Jesse Nance
Durham
25
96
Ahmad Shah
Durham
25
97
User
Durham
25
98
User
Raleigh
25
99
User
Charlotte
25
100
User
Durham
25
101
User
Winterville
25
102
User
China Grove
24
103
Corey Sei Lami Moore
Durham
24
104
User
Charlotte
24
105
User
Chapel Hill
24
106
User
Winston-Salem
24
107
User
Durham
24
108
User
Durham
24
109
FireKitten
Cary
24
110
User
Durham
24
111
User
Charlotte
24
112
User
Durham
23
113
User
Durham
23
114
Christopher Leon Mateo
Durham
23
115
Jose Frausto
Durham
23
116
User
Durham
23
117
User
Chapel Hill
23
118
Austin Tran
Oxford
23
119
User
Greensboro
23
120
User
Buncombe County
23
121
User
Charlotte
22
122
Joseph Rugama
Durham
22
123
User
Cary
22
124
User
Wake Forest
22
125
LMW
Morganton
22
126
User
Durham
22
127
User
Charlotte
22
128
User
China Grove
22
129
User
Wilson
22
130
User
Raleigh
22
131
User
Durham
22
132
User
Charlotte
22
133
Hannah Mendoza
Winston-Salem
21
134
User
Charlotte
21
135
User
Durham
21
136
User
Raleigh
21
137
User
Raleigh
21
138
User
Charlotte
21
139
User
Chapel Hill
21
140
User
Durham
21
141
User
Durham
21
142
Grateb
Waxhaw
21
143
Thomas Freeland
Goldsboro
20
144
User
Pittsboro
20
145
User
Durham
20
146
User
Durham
20
147
User
Durham
20
148
User
Greensboro
20
149
User
Cary
20
150
User
Durham
20
151
User
Rolesville
20
152
Arvin Singh
Roxboro
20
153
User
Chapel Hill
20
154
User
Winston-Salem
20
155
User
Durham
20
156
User
Cullowhee
20
157
User
Charlotte
20
158
User
Louisburg
20
159
Timothy Manuel
Asheville
20
160
User
Spring Lake
20
161
User
Raleigh
20
162
Kelsey Hamilton
Concord
19
163
Kelvin Soto
Durham
19
164
Kevin Nelson
Charlotte
19
165
User
Raleigh
19
166
User
Eden
19
167
User
Kings Mountain
19
168
User
Burlington
19
169
User
Durham
19
170
Tipster Trickster
Matthews
19
171
User
Charlotte
19
172
User
Chapel Hill
19
173
Emeline Van Deman
Durham
19
174
User
Stem
19
175
User
Durham
19
176
User
Apex
19
177
User
Cary
19
178
User
Raleigh
18
179
User
Winterville
18
180
User
Concord
18
181
User
Durham
18
182
Caiden Smallwood
Durham
18
183
CoolstarDJ
Mooresville
18
184
User
China Grove
18
185
User
Winston-Salem
18
186
Dylan Woodlief
Oxford
18
187
Ethan Ruderman
Creedmoor
18
188
User
Stem
18
189
Raya Hope
Asheville
18
190
User
Charlotte
18
191
User
Charlotte
18
192
User
Kill Devil Hills
17
193
User
China Grove
17
194
User
Graham
17
195
User
Durham
17
196
User
Durham
17
197
Easha Kuber
Oxford
17
198
Zean Dy-Cok
Durham
17
199
Jasmine Lassiter
Raleigh
17
200
User
Cary
17
201
User
Durham
17
202
User
Greensboro
16
203
User
Durham
16
204
User
Durham
16
205
User
Apex
16
206
Lil XxSnipezXx
Durham
16
207
User
Durham
16
208
User
Burlington
16
209
User
Burlington
16
210
User
Chapel Hill
16
211
User
Asheville
16
212
User
Durham
16
213
Chris Desmond
Raleigh
16
214
User
Charlotte
16
215
User
Durham
16
216
User
China Grove
15
217
Alexis Velazques
Durham
15
218
User
Sanford
15
219
User
Graham
15
220
User
High Point
15
221
User
Greensboro
15
222
User
Creedmoor
15
223
alex varela
Sanford
15
224
User
Durham
15
225
Ian Salazar
Durham
14
226
Messiah Brandon
Durham
14
227
Sergio Campos
Durham
14
228
User
Charlotte
14
229
User
China Grove
14
230
Hans Schoppe
Oxford
14
231
User
Charlotte
14
232
User
Kings Mountain
14
233
User
Durham
13
234
User
Durham
13
235
Jaime Chavez
Durham
13
236
User
Burlington
13
237
Xander Ashcraft
Greenville
13
238
User
Cary
13
239
User
China Grove
13
240
William Johnson
Oxford
13
241
User
Creedmoor
13
242
User
Durham
13
243
User
Onslow County
13
244
Cristian Jimenez
Durham
12
245
Omar Ramos
Durham
12
246
User
Lumberton
12
247
Elian Vargas Hernandez
Durham
12
248
User
Burlington
12
249
Vasilis Gkoulioumis-Mantzoukas
Durham
12
250
User
Raleigh
12
251
User
Charlotte
12
252
User
Raleigh
12
253
Annie Hughes
Stem
12
254
User
Charlotte
12
255
User
Raleigh
12
256
User
Burlington
12
257
User
Huntersville
12
258
rjmeaney
Chapel Hill
12
259
User
China Grove
12
260
lemere jones
Durham
12
261
User
Creedmoor
12
262
User
Charlotte
12
263
Aaron Manuel
Asheville
12
264
User
China Grove
11
265
User
Hendersonville
11
266
User
Chapel Hill
11
267
User
Charlotte
11
268
User
Durham
11
269
User
China Grove
11
270
User
Raleigh
11
271
Elliot Umbarger
Creedmoor
11
272
Eli Edds
Oxford
11
273
User
Raleigh
11
274
Esau Cunningham II
Sanford
11
275
User
Greensboro
11
276
User
Franklin
11
277
User
Durham
11
278
User
Greensboro
10
279
User
China Grove
10
280
User
Durham
10
281
Alexander Thomas
Mint Hill
10
282
User
Oxford
10
283
User
Durham
10
284
User
Cary
10
285
User
Pittsboro
10
286
Jenny Parks
Chapel Hill
10
287
User
Charlotte
10
288
User
Charlotte
10
289
Shamika Brower
Boone
10
290
User
China Grove
10
291
User
China Grove
10
292
User
China Grove
10
293
User
Winston-Salem
10
294
User
Charlotte
10
295
User
Raleigh
10
296
Zahviair Grace-Thomas
Sanford
10
297
Liliana Centeno-Angel _ Student - LigonMS
Raleigh
10
298
User
Weaverville
10
299
User
Charlotte
10
300
User
Havelock
10
301
User
Raleigh
10
302
Nicholas Miller
China Grove
9
303
User
China Grove
9
304
User
Charlotte
9
305
User
Durham
9
306
User
Davidson
9
307
User
Greenville
9
308
User
Morganton
9
309
User
Charlotte
9
310
User
Charlotte
9
311
User
Charlotte
9
312
User
Pittsboro
9
313
User
Cary
9
314
User
China Grove
9
315
User
China Grove
9
316
User
China Grove
9
317
User
China Grove
9
318
Jully Marban
Oxford
9
319
josie sanfelici
Oxford
9
320
User
Raleigh
9
321
tykale-lawrene
Sanford
9
322
Trinity Yarborough _ Student - LigonMS
Raleigh
9
323
User
Cary
9
324
User
Charlotte
9
325
User
Chapel Hill
9
326
User
Asheville
9
327
User
China Grove
8
328
Dylan Carter
China Grove
8
329
User
China Grove
8
330
Kamauri Hinton
Durham
8
331
User
Durham
8
332
User
Durham
8
333
User
Durham
8
334
User
Sanford
8
335
User
Durham
8
336
User
Avon
8
337
BatRJMan
Mooresville
8
338
User
Charlotte
8
339
User
Charlotte
8
340
User
Greensboro
8
341
User
Cornelius
8
342
User
Creedmoor
8
343
User
Charlotte
8
344
Frankie Cruz
Siler City
8
345
User
China Grove
8
346
User
China Grove
8
347
User
Cary
8
348
User
Burlington
8
349
User
Charlotte
8
350
Martin Castillo-Hermoza
Oxford
8
351
User
Butner
8
352
Kaire Ryland
Sanford
8
353
User
Raleigh
8
354
User
Raleigh
8
355
User
Monroe
8
356
User
Burlington
8
357
User
Charlotte
8
358
User
China Grove
7
359
User
China Grove
7
360
Kevin Robles
Durham
7
361
Cedric Streeter
Charlotte
7
362
Madi
China Grove
7
363
User
China Grove
7
364
Jamir Richardson
Durham
7
365
User
Concord
7
366
User
Charlotte
7
367
User
Sanford
7
368
User
High Point
7
369
User
High Point
7
370
User
Charlotte
7
371
User
Mooresville
7
372
User
Raleigh
7
373
User
Rolesville
7
374
User
Charlotte
7
375
User
Charlotte
7
376
User
Charlotte
7
377
User
Greenville
7
378
Angelo Giron-Dominguez
Fearrington
7
379
Jaxon Globuschutz
Fearrington
7
380
Alexis Maldonado Reyes
Pittsboro
7
381
User
Williamston
7
382
User
Cary
7
383
Rihanna Carrington _ Student - LigonMS
Cary
7
384
User
China Grove
7
385
User
China Grove
7
386
User
Charlotte
7
387
Rossember Merlos Campos
Oxford
7
388
Thomas Alling
Durham
7
389
Blue Majors
Oxford
7
390
Jai Walker
Sanford
7
391
Yordy Vasquez Hernandez
Sanford
7
392
User
Raleigh
7
393
User
Raleigh
7
394
sharmyn
Raleigh
7
395
User
Raleigh
7
396
User
Burlington
7
397
User
Burlington
7
398
User
Burlington
7
399
User
Durham
6
400
User
China Grove
6
401
Ditaevon Doakes
Durham
6
402
User
China Grove
6
403
User
China Grove
6
404
User
China Grove
6
405
User
Cary
6
406
User
Durham
6
407
Jason Bonilla
Durham
6
408
User
Reidsville
6
409
User
Wake Forest
6
410
User
Sanford
6
411
User
Durham
6
412
User
Roxboro
6
413
User
Roxboro
6
414
User
Graham
6
415
User
Durham
6
416
User
Raleigh
6
417
User
Graham
6
418
User
Winston-Salem
6
419
User
Cary
6
420
User
Chapel Hill
6
421
User
Asheville
6
422
User
City
6
423
User
Chapel Hill
6
424
User
Raleigh
6
425
User
Winston-Salem
6
426
Dalila
Pittsboro
6
427
Zariya Edwards
Pittsboro
6
428
Scarlett Munoz Ortiz
Fearrington
6
429
Laisha Zagada Amador
Fearrington
6
430
User
Siler City
6
431
User
Greensboro
6
432
User
China Grove
6
433
User
China Grove
6
434
User
Cary
6
435
User
China Grove
6
436
User
China Grove
6
437
User
China Grove
6
438
User
China Grove
6
439
User
Raleigh
6
440
User
Charlotte
6
441
Ekomobong Ikpatt
Stem
6
442
User
Charlotte
6
443
User
Charlotte
6
444
Tyler Hendricks
Durham
6
445
User
Durham
6
446
User
Durham
6
447
User
Creedmoor
6
448
User
Raleigh
6
449
DaOna Thomas-Martin
Sanford
6
450
User
Sanford
6
451
Samuel Hernandez Cruz
Sanford
6
452
Nixonosorio
Sanford
6
453
hotoniel cruz martinez
Sanford
6
454
anthony castro
Sanford
6
455
User
Sanford
6
456
User
Raleigh
6
457
Cannon Brown _ Student - LigonMS
Raleigh
6
458
User
Charlotte
6
459
User
Raleigh
6
460
John Buchser
Raleigh
6
461
User
Raleigh
6
462
User
Carrboro
6
463
User
Greensboro
6
464
User
Cary
6
465
User
Charlotte
5
466
User
Greenville
5
467
Carlos Gonzalez
Durham
5
468
User
China Grove
5
469
User
China Grove
5
470
User
China Grove
5
471
User
China Grove
5
472
User
China Grove
5
473
Immanuel Holland
Durham
5
474
User
Pittsboro
5
475
User
Charlotte
5
476
Brycen Morrow
Wake Forest
5
477
User
Cary
5
478
User
Sanford
5
479
User
Durham
5
480
User
Durham
5
481
User
Charlotte
5
482
User
Chapel Hill
5
483
User
Charlotte
5
484
User
Graham
5
485
User
Graham
5
486
User
Dunn
5
487
User
Huntersville
5
488
User
Charlotte
5
489
User
Matthews
5
490
Allen Coin
Raleigh
5
491
User
Roxboro
5
492
User
Sanford
5
493
User
Charlotte
5
494
Chelsea Prieto
Fearrington
5
495
User
Fearrington
5
496
User
Pittsboro
5
497
User
Pittsboro
5
498
User
Pittsboro
5
499
User
Pittsboro
5
500
User
Siler City
5
501
User
Fearrington
5
502
User
China Grove
5
503
User
China Grove
5
504
User
China Grove
5
505
User
China Grove
5
506
User
China Grove
5
507
User
Cary
5
508
User
China Grove
5
509
User
China Grove
5
510
User
China Grove
5
511
User
China Grove
5
512
User
China Grove
5
513
User
Cary
5
514
User
Winston-Salem
5
515
User
Chapel Hill
5
516
User
Mint Hill
5
517
Antoinette Hawley
Oxford
5
518
User
Charlotte
5
519
User
Charlotte
5
520
User
Oxford
5
521
Cyrus Chamgbe
Sanford
5
522
Abdulilah Altadmouri
Sanford
5
523
Joan Garcia
Sanford
5
524
User
Sanford
5
525
User
Charlotte
5
526
User
Raleigh
5
527
Jeremy Genao-Arevalo _ Student - LigonMS
Raleigh
5
528
User
Raleigh
5
529
User
Raleigh
5
530
User
Burlington
5
531
User
Kings Mountain
5
532
User
Winston-Salem
5
533
User
Kings Mountain
5
534
User
Charlotte
5
535
User
Marvin
4
536
User
Raleigh
4
537
User
Burlington
4
538
User
China Grove
4
539
User
China Grove
4
540
User
China Grove
4
541
User
China Grove
4
542
User
China Grove
4
543
User
Black Mountain
4
544
User
Sanford
4
545
User
Sanford
4
546
User
Spring Lake
4
547
User
Sanford
4
548
User
Jacksonville
4
549
User
Chapel Hill
4
550
User
Charlotte
4
551
User
Durham
4
552
User
Asheboro
4
553
User
Charlotte
4
554
User
Graham
4
555
User
Greenville
4
556
User
Graham
4
557
User
Mooresville
4
558
User
Cary
4
559
User
Durham
4
560
User
Township Of Taylorsville
4
561
User
Durham
4
562
User
Durham
4
563
User
Durham
4
564
User
Rocky Mount
4
565
David Francisco Hernandez
Pittsboro
4
566
User
Asheville
4
567
User
Fearrington
4
568
User
Siler City
4
569
User
Pittsboro
4
570
User
Siler City
4
571
User
Siler City
4
572
User
Pittsboro
4
573
User
Siler City
4
574
User
Pittsboro
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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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