Leaderboard
Florida
Rank
Level
1
TJ Cubing
Orlando
85
2
User
Gainesville
82
3
Chris Rodriguez
Tamiami
78
4
Adub
Orlando
72
5
User
Tallahassee
70
6
User
Plantation
68
7
Gabriel Diraviam
Pembroke Pines
64
8
User
Plantation
63
9
Lou
Miami
62
10
Sabrina Noborikawa
Miami
61
11
bhaskar mishra
Pembroke Pines
59
12
Hector Orea
Weston
58
13
User
Orlando
56
14
Steven Maradiaga
Miami
55
15
User
Melbourne
55
16
User
Orlando
54
17
Christian Tang
North Naples
53
18
User
Oviedo
53
19
User
Villas
52
20
User
North Miami Beach
51
21
Kiki Patel
Celebration
51
22
User
Wellington
50
23
User
North Miami Beach
49
24
User
Miami Beach
49
25
User
Broward County
49
26
User
Weston
47
27
User
Weston
47
28
Arianna Lopez
North Miami Beach
47
29
User
Pinellas Park
45
30
User
Gainesville
45
31
User
Fort Myers
44
32
User
North Miami
44
33
User
Gainesville
44
34
User
Belleair
43
35
User
Belleair
43
36
Daniel Gomez
Miami
43
37
Tien
Orlando
43
38
User
Bay Lake
41
39
User
Miramar Beach
40
40
User
Miami
39
41
Eugene Li
Cutler Bay
39
42
User
Saint Petersburg
39
43
User
Orlando
39
44
User
Tampa
39
45
User
Orlando
39
46
User
Boca Raton
39
47
User
St. Petersburg
39
48
Emilie Schmidt
Gainesville
38
49
User
Clermont
37
50
Michael Marcotte
Largo
36
51
User
Cape Coral
36
52
User
Pinellas Park
36
53
User
Clermont
36
54
User
Jensen Beach
36
55
User
Orlando
35
56
Starmer
Orlando
35
57
Laís Atilano
Aventura
35
58
David Zhou
Kendall
34
59
User
Sunny Isles Beach
34
60
User
Miami
34
61
User
Fort Myers
34
62
User
Miami
34
63
User
Vero Beach
34
64
User
Bay Lake
33
65
User
Kendall
32
66
User
Gainesville
32
67
User
Ellenton
31
68
User
Golf
31
69
User
Orlando
31
70
User
Miami
31
71
User
Miami
31
72
User
Plantation
31
73
User
Plantation
31
74
User
Plantation
31
75
User
Orlando
31
76
User
Fort Lauderdale
31
77
User
Vineyards
30
78
User
Sanford
30
79
User
Miramar Beach
30
80
Michael Irias
Miami
30
81
User
Gainesville
30
82
travis marciano
Pembroke Pines
30
83
Elizabeth Baldor
Tamiami
29
84
User
Broward County
29
85
User
St. Augustine
28
86
Romina Doubnia
North Miami
28
87
User
Orlando
28
88
User
Tampa
27
89
User
Kendall
27
90
User
Kendall
27
91
User
Sanford
27
92
User
Miami
27
93
User
Bithlo
27
94
Michael
Boca Raton
27
95
User
Aventura
27
96
User
Orlando
27
97
User
Gainesville
27
98
Pablo
Miami
27
99
User
Kings Point
27
100
User
Tallahassee
27
101
User
Union Park
26
102
User
Tallahassee
26
103
Andy Chan
Oviedo
26
104
User
Sanford
26
105
User
Fruit Cove
26
106
User
North Naples
26
107
User
Kendall
26
108
InterTalks 1908
Saint Petersburg
26
109
User
Celebration
26
110
Scott Farwell
Leesburg
26
111
User
Tallahassee
26
112
Lucas Serrott
Homestead
26
113
User
Tallahassee
26
114
Larry Johnson
Safety Harbor
25
115
Rio Torres
Winter Garden
25
116
A Evans
Oviedo
25
117
Alexander Diraviam
Pembroke Pines
25
118
User
Deland
25
119
User
Wellington
24
120
User
Miramar Beach
24
121
User
Plantation
24
122
User
Orlando
24
123
User
Bay Lake
24
124
User
Yulee
24
125
Victor Manuel Bellera
Panama City Beach
24
126
Javier Barajas
Key Biscayne
24
127
User
Boca Raton
24
128
User
Miami
24
129
User
Jacksonville
23
130
User
Lakewood Ranch
23
131
User
Orlando
23
132
User
Bay Lake
23
133
User
Bay Lake
23
134
User
Bay Lake
23
135
User
Pinecrest
23
136
User
Melbourne
23
137
User
Orlando
23
138
User
Lakewood Ranch
22
139
marco peric
Wellington
22
140
User
Sanford
22
141
User
Orlando
22
142
User
Davie
22
143
User
Miami
22
144
User
Oviedo
22
145
User
Tampa
22
146
User
Fort Myers
22
147
User
Navarre
22
148
User
Tampa
22
149
User
Fort Myers
22
150
User
Gainesville
22
151
User
Broward County
22
152
SOPHIE LI
Oviedo
21
153
User
Sanford
21
154
User
Cutler Bay
21
155
User
Bay Lake
21
156
User
Union Park
21
157
User
Fort Lauderdale
21
158
User
Jacksonville
21
159
User
Richmond West
21
160
User
West Palm Beach
21
161
User
St. Petersburg
21
162
User
St. Petersburg
21
163
Joanne
Sanibel
21
164
User
Temple Terrace
20
165
User
Miami
20
166
Samuel Norey
Pompano Beach
20
167
User
Sanford
20
168
User
Tampa
20
169
User
Miami
20
170
Obviouslyniyah
Richmond West
20
171
User
Largo
20
172
User
Jacksonville
20
173
User
Miami
20
174
Erica Hill
St. Petersburg
20
175
User
Miami
20
176
User
Winter Garden
20
177
User
Palm Beach Gardens
19
178
MRMFade
Sanford
19
179
User
Fort Myers
19
180
User
Bay Lake
19
181
User
Bradenton
19
182
User
Port St. Lucie
19
183
User
Cutler Bay
19
184
User
Saint Cloud
19
185
User
San Carlos Park
19
186
User
Orlando
19
187
User
Tallahassee
19
188
User
Orlando
19
189
Firas Zeidan
Jacksonville
19
190
User
Orlando
19
191
User
Leesburg
19
192
User
Lynn Haven
19
193
User
Pensacola
18
194
User
Oviedo
18
195
User
Oviedo
18
196
User
Gainesville
18
197
User
Miami
18
198
User
Union Park
18
199
overloadcode
Boynton Beach
18
200
User
Orlando
18
201
User
Temple Terrace
18
202
User
Tamarac
18
203
User
Destin
18
204
User
Orlando
18
205
User
Bartow
18
206
User
Orlando
18
207
User
Orlando
18
208
User
Orlando
18
209
Yoevelyn Rodriguez
North Miami Beach
18
210
User
Weston
17
211
User
Wellington
17
212
User
Palm Beach Gardens
17
213
User
Davie
17
214
User
Miami Beach
17
215
User
Hialeah Gardens
17
216
User
Orlando
17
217
User
Sanford
17
218
User
Palm Coast
17
219
User
Miami
17
220
User
Miami
17
221
User
Eglin Air Force Base
17
222
User
Fort Lauderdale
17
223
User
Orlando
17
224
User
Jacksonville
17
225
Aqueel Majied
Jacksonville
16
226
User
Sanford
16
227
Brian Loftus
Brandon
16
228
User
Fort Lauderdale
16
229
User
Oviedo
16
230
User
Pompano Beach
16
231
User
Largo
16
232
User
Maitland
16
233
User
Palm Coast
16
234
User
Tampa
16
235
User
University
16
236
asdf
Winter Garden
16
237
User
Orlando
16
238
User
Orlando
16
239
User
Orlando
16
240
User
Orlando
16
241
User
Palm Beach Gardens
15
242
User
Sanford
15
243
User
Sanford
15
244
User
Goldenrod
15
245
User
Union Park
15
246
Aggressivemonkey
Spring Hill
15
247
User
Bay Lake
15
248
User
Miami
15
249
User
Union Park
15
250
User
Miami
15
251
User
Union Park
15
252
User
Miami
15
253
User
City
15
254
User
Union Park
15
255
User
Bayshore Gardens
15
256
User
Orlando
15
257
User
Lake Mary
15
258
Nicole Ferguson
Spring Hill
15
259
User
Naples
15
260
User
Orlando
15
261
User
Orlando
15
262
User
Orlando
15
263
User
Miami
15
264
User
Broward County
15
265
User
Orlando
15
266
User
Myrtle Grove
14
267
Krista
Myakka City
14
268
User
Pinellas Park
14
269
User
Sanford
14
270
User
Miramar Beach
14
271
User
Union Park
14
272
User
Gainesville
14
273
User
Clearwater
14
274
User
Hialeah
14
275
User
Bay Lake
14
276
User
Doral
14
277
User
Alachua
14
278
User
Gainesville
14
279
User
Orange Park
14
280
User
North Fort Myers
14
281
David Nagli
Hallandale Beach
14
282
User
Fort Lauderdale
14
283
User
Union Park
14
284
User
Tampa
14
285
User
Gainesville
14
286
User
Miami
14
287
User
Tampa
14
288
User
Miami
14
289
DanielaF
Coral Springs
13
290
User
Greenacres
13
291
User
Apopka
13
292
User
Union Park
13
293
User
Brandon
13
294
Hoxie Ackerman
Fort Lauderdale
13
295
User
Tampa
13
296
User
Greenacres
13
297
User
Tampa
13
298
User
Tampa
13
299
User
North Naples
13
300
User
Country Club
13
301
User
Orlando
13
302
User
Casselberry
13
303
User
Gainesville
13
304
User
Miami
13
305
User
Riverview
12
306
Chambers
Quincy
12
307
User
Tallahassee
12
308
User
Lake Magdalene
12
309
User
Sanford
12
310
User
Orlando
12
311
User
University
12
312
User
Union Park
12
313
User
Royal Palm Beach
12
314
User
Ormond Beach
12
315
User
Orlando
12
316
User
Saint Petersburg
11
317
User
Union Park
11
318
User
Bithlo
11
319
User
Ferry Pass
11
320
User
Nocatee
11
321
User
Bay Lake
11
322
User
Dundee
11
323
User
Gainesville
11
324
User
Casselberry
11
325
User
Hollywood
11
326
User
Fort Lauderdale
11
327
User
Orlando
11
328
User
Miami
11
329
User
Orlando
10
330
michael kadow
Vero Beach South
10
331
User
Saint Petersburg
10
332
User
Boynton Beach
10
333
User
Miami Beach
10
334
User
Tarpon Springs
10
335
User
Daytona Beach
10
336
User
Sanford
10
337
User
Orlando
10
338
User
Lauderhill
10
339
User
Orlando
10
340
User
Miami
10
341
Nickelle Vil
Buena Ventura Lakes
10
342
User
Orlando
10
343
User
Saint Petersburg
10
344
Cardell Michael
Orlando
10
345
User
Orlando
10
346
User
Miami
10
347
Miki Hirama
Gainesville
10
348
User
Orlando
10
349
User
St. Petersburg
10
350
Fátima
Miami
10
351
User
Winter Park
10
352
User
Coral Gables
9
353
User
Tampa
9
354
User
Sanford
9
355
User
Sanford
9
356
User
Sanford
9
357
User
Jacksonville
9
358
User
Union Park
9
359
User
Orlando
9
360
User
Four Corners
9
361
User
Jacksonville
9
362
User
Tallahassee
9
363
User
Miami
9
364
Kieffer James
Miami
9
365
User
Jacksonville
9
366
User
Doral
9
367
User
Miami
9
368
User
Plantation
9
369
User
Orlando
9
370
User
Orange City
9
371
User
Juno Beach
9
372
User
Hialeah
9
373
User
Miami
9
374
User
Winter Garden
9
375
User
Largo
9
376
User
Orlando
9
377
User
Fort Myers
9
378
User
Miami
9
379
User
Riverview
9
380
User
Orlando
8
381
User
Florida Ridge
8
382
User
Hurlburt Field
8
383
User
Tampa
8
384
User
Miami
8
385
User
West Palm Beach
8
386
Shannon
Orlando
8
387
User
Orlando
8
388
Logan
Sanford
8
389
User
Jacksonville
8
390
User
Tallahassee
8
391
User
Union Park
8
392
User
Fort Lauderdale
8
393
User
New Port Richey East
8
394
User
Miami
8
395
User
Orlando
8
396
User
Miami
8
397
Moe Jimenez
Palmetto Bay
8
398
User
Clermont
8
399
User
Saint Petersburg
8
400
User
Lutz
8
401
User
Weston
8
402
User
Doral
8
403
User
Doral
8
404
User
Miami Beach
8
405
User
Jacksonville
8
406
User
Weston
8
407
User
St. Augustine
8
408
User
Miami
8
409
User
Tallahassee
8
410
User
Lake Worth
8
411
User
West Palm Beach
8
412
User
Lauderdale-By-The-Sea
7
413
User
Fort Lauderdale
7
414
User
Safety Harbor
7
415
User
Miami
7
416
User
University
7
417
User
Boca Raton
7
418
HannahRie VDH
Brent
7
419
User
Sanford
7
420
User
Sanford
7
421
User
Sanford
7
422
Max Buchanan
Sanford
7
423
User
Sanford
7
424
User
Tampa
7
425
User
Cypress Lake
7
426
User
Kissimmee
7
427
User
Ormond Beach
7
428
User
Immokalee
7
429
User
North Naples
7
430
User
Villas
7
431
User
Fort Pierce
7
432
User
Union Park
7
433
User
Doral
7
434
User
Princeton
7
435
User
Tampa
7
436
User
Miami
7
437
User
Orlando
7
438
User
Miami
7
439
User
Saint Petersburg
7
440
User
Miami
7
441
User
Interlachen
7
442
Juan Carlos
Miami
7
443
User
Miami
7
444
User
Orlando
7
445
User
Pembroke Pines
7
446
User
Orlando
7
447
User
Largo
7
448
User
Vero Beach
7
449
User
Spring Hill
7
450
User
Port St. Lucie
7
451
User
Miami
7
452
User
Miami
7
453
User
Orlando
7
454
anonymous user
Jacksonville
6
455
ARC Trooper
North Miami Beach
6
456
User
Miami
6
457
User
Hurlburt Field
6
458
User
Sanford
6
459
User
Miami Gardens
6
460
User
Deltona
6
461
User
Port St. Lucie
6
462
User
Union Park
6
463
User
Sanford
6
464
User
Sanford
6
465
PewDiePie
Sanford
6
466
User
Palm Beach Gardens
6
467
User
Brandon
6
468
User
Pembroke Pines
6
469
User
Miami
6
470
User
Crestview
6
471
User
Union Park
6
472
User
Tallahassee
6
473
User
St. Augustine
6
474
User
Jacksonville
6
475
User
Greenacres
6
476
User
Auburndale
6
477
User
Kendall
6
478
User
Union Park
6
479
User
Saint Petersburg
6
480
User
Ives Estates
6
481
User
Temple Terrace
6
482
User
Union Park
6
483
User
Poinciana
6
484
User
Orlando
6
485
User
Miami
6
486
User
Naples
6
487
User
Pensacola
6
488
User
Valrico
6
489
User
Sarasota
6
490
User
Orlando
6
491
User
Rockledge
6
492
User
Fort Lauderdale
5
493
User
Palm City
5
494
User
Miami
5
495
User
West Melbourne
5
496
User
Seminole
5
497
User
Miami
5
498
User
Oviedo
5
499
Goddard
Miami
5
500
User
Saint Petersburg
5
501
User
Niceville
5
502
User
Sanford
5
503
User
Sanford
5
504
User
Sanford
5
505
User
Sanford
5
506
User
Pinellas Park
5
507
User
Oviedo
5
508
User
Sarasota Springs
5
509
User
Fort Lauderdale
5
510
User
Pine Hills
5
511
User
Orangetree
5
512
User
Orlando
5
513
User
Florida Ridge
5
514
User
Gainesville
5
515
User
Cape Coral
5
516
User
Tarpon Springs
5
517
User
Hollywood
5
518
User
Miami
5
519
User
Melbourne
5
520
User
Brandon
5
521
User
Winter Haven
5
522
User
Kings Point
5
523
User
Apopka
5
524
User
Lakewood Ranch
5
525
User
Tallahassee
5
526
User
Cape Coral
5
527
User
Orlando
5
528
User
Boca Raton
5
529
User
Weston
5
530
User
Tampa
5
531
User
Miami
5
532
User
Orlando
5
533
User
Jacksonville
5
534
User
Yulee
5
535
User
Naples
5
536
User
Orlando
5
537
User
Orlando
5
538
User
Miami
5
539
User
Miami
5
540
User
Miami
5
541
Christopher Cobb
Lake City
4
542
User
North Lauderdale
4
543
User
Ocala
4
544
User
Pensacola
4
545
User
Port St. Lucie
4
546
User
Saint Petersburg
4
547
User
Port Orange
4
548
User
Miami
4
549
User
Miami
4
550
User
Davie
4
551
Daniel Lopez
Tamiami
4
552
User
Goldenrod
4
553
User
Spring Hill
4
554
User
Wesley Chapel
4
555
User
Miramar
4
556
User
City
4
557
User
Fort Lauderdale
4
558
User
Pompano Beach
4
559
User
Naples
4
560
User
Miami Beach
4
561
Madison Spencer
Gainesville
4
562
User
City
4
563
User
Miami
4
564
User
Bellview
4
565
User
Sebring
4
566
User
Osprey
4
567
User
Miami
4
568
Harry Padilla
Doral
4
569
User
Miami
4
570
User
Orlando
4
571
User
Hialeah
4
572
User
Melbourne
4
573
User
Port St. Lucie
4
574
User
Miami
4
575
User
Warrington
4
576
User
Tampa
4
577
User
Coral Springs
4
578
User
City
4
579
User
Miami
4
580
User
Panama City Beach
4
581
User
North Naples
4
582
User
Gainesville
4
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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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