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