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