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