Leaderboard
Pennsylvania
Rank
Level
1
User
Stroudsburg
84
2
Shrek Shrotio
Philadelphia
81
3
User
Fountain Hill
80
4
User
Bala Cynwyd
77
5
User
Leacock-Leola-Bareville
74
6
User
Bethlehem
74
7
User
Leacock-Leola-Bareville
73
8
User
Leacock-Leola-Bareville
72
9
User
Stroudsburg
72
10
Ethan Regal
Erie
71
11
User
Leacock-Leola-Bareville
71
12
Joseph Nguyen
Stroudsburg
71
13
User
Leacock-Leola-Bareville
70
14
William Zeng
Pittsburgh
68
15
User
Leacock-Leola-Bareville
65
16
Krit
Harrisburg
64
17
User
Philadelphia
64
18
User
Leacock-Leola-Bareville
62
19
User
Leacock-Leola-Bareville
61
20
User
West Point
59
21
User
Leacock-Leola-Bareville
59
22
User
Philadelphia
59
23
User
Philadelphia
59
24
Edna Castellanos
Easton
58
25
User
Leacock-Leola-Bareville
58
26
Syed Mohammad
Lebanon
56
27
User
Leacock-Leola-Bareville
56
28
User
Bala Cynwyd
55
29
User
Leacock-Leola-Bareville
55
30
ManInJelly
Narberth
54
31
User
Leacock-Leola-Bareville
52
32
Hoyt Gong
Philadelphia
52
33
User
Leacock-Leola-Bareville
51
34
cdatiemo@gmail.com
Philadelphia
51
35
User
West Point
51
36
User
Leacock-Leola-Bareville
51
37
User
Philadelphia
50
38
User
Downingtown
50
39
User
Pittsburgh
50
40
User
Bala Cynwyd
50
41
User
Irwin
49
42
Christopher Atiemo-Obeng
Philadelphia
48
43
User
Pittsburgh
48
44
User
Mechanicsburg
47
45
User
Mcmurray
45
46
User
Stroudsburg
44
47
User
Pittsburgh
44
48
User
Philadelphia
41
49
Xhuliano Brace
Pittsburgh
40
50
User
Pittsburgh
38
51
1. Thou Shant Lewd Kaori
Bethlehem
38
52
User
Philadelphia
38
53
User
Park Forest Village
38
54
Max Arrilus
Phoenixville
38
55
User
Philadelphia
38
56
User
Wynnewood
37
57
User
Philadelphia
35
58
User
Philadelphia
35
59
L. J.
Pittsburgh
35
60
Alexander
Moon
33
61
Zack Lee
Pittsburgh
33
62
User
Pittsburgh
33
63
User
Philadelphia
33
64
User
Oil City
33
65
Josh Jung
Pittsburgh
31
66
User
Stroudsburg
30
67
User
Emmaus
30
68
Pablo Franco
Wilkes-Barre
30
69
Latissimus65
Fountain Hill
29
70
frances
Havertown
29
71
User
Cranberry Township
29
72
User
OHara Township
29
73
User
Wilkes-Barre
29
74
User
Philadelphia
29
75
User
Pittsburgh
28
76
User
Franklin Park
28
77
User
Fawn Grove
28
78
User
Pittsburgh
28
79
User
Pittsburgh
27
80
User
Philadelphia
26
81
User
Springfield
26
82
User
Marlin
26
83
User
Philadelphia
25
84
Michelle Abigan
Philadelphia
24
85
User
Philadelphia
24
86
User
Pittsburgh
24
87
User
Philadelphia
24
88
Luis Magro
Philadelphia
23
89
User
Philadelphia
23
90
User
Dallas
23
91
Kumar
Philadelphia
23
92
User
Doylestown
23
93
User
Fawn Grove
23
94
User
Windsor
23
95
Jack Patrick Briody
Philadelphia
23
96
User
Newtown
23
97
User
Lake Ariel
23
98
User
Conshohocken
22
99
User
Exton
22
100
User
Villanova
22
101
User
Lititz
22
102
User
Philadelphia
22
103
User
Philadelphia
22
104
User
Reading
21
105
User
Mccandless
21
106
User
Pittsburgh
21
107
User
Philadelphia
21
108
Funglun Chan
Pittsburgh
21
109
User
Philadelphia
21
110
User
Birdsboro
21
111
User
Philadelphia
21
112
Lily DeAngelis
Doylestown
21
113
User
Burgettstown
21
114
User
Philadelphia
21
115
User
Devault
20
116
User
Leacock-Leola-Bareville
20
117
User
Philadelphia
20
118
User
Pittsburgh
19
119
User
Philadelphia
19
120
Cody
Bala Cynwyd
19
121
User
Haverford
19
122
User
Allentown
19
123
User
State College
19
124
User
Wilkinsburg
19
125
User
West Chester
19
126
User
Pittsburgh
18
127
User
Leacock-Leola-Bareville
18
128
User
Moon
18
129
User
Concordville
18
130
User
Philadelphia
18
131
User
Bradford
18
132
User
Villanova
17
133
User
Pittsburgh
17
134
User
Boothwyn
17
135
User
Hummelstown
17
136
User
State College
17
137
User
Bloomsburg
17
138
User
City
17
139
User
Easton
17
140
User
Bala Cynwyd
17
141
User
Philadelphia
17
142
User
Philadelphia
17
143
User
Bethlehem
17
144
User
Bala Cynwyd
16
145
User
Pittsburgh
16
146
Euclid1023
Philadelphia
16
147
Eric Hoofler
Pittsburgh
16
148
User
Philadelphia
16
149
User
Bensalem
16
150
User
Plymouth Meeting
16
151
User
Philadelphia
16
152
User
Paoli
16
153
User
Philadelphia
16
154
User
Selinsgrove
15
155
User
Leacock-Leola-Bareville
15
156
User
Erie
15
157
User
East Norriton
15
158
User
Philadelphia
15
159
User
Pittsburgh
15
160
User
Philadelphia
15
161
User
Bensalem
15
162
Jamie Sollinger
Oil City
15
163
User
Doylestown
14
164
User
Philadelphia
14
165
User
Philadelphia
14
166
User
Pittsburgh
14
167
User
Harrisburg
14
168
User
Bala Cynwyd
14
169
User
Philadelphia
14
170
User
Philadelphia
14
171
Christian Asumadu
Philadelphia
14
172
User
Pittsburgh
14
173
User
Mechanicsburg
14
174
User
Philadelphia
14
175
User
Feasterville-Trevose
14
176
User
Doylestown
14
177
User
Burgettstown
14
178
Bethann Dolbow
York
14
179
User
Aston
13
180
Chris Doms
Bethlehem
13
181
User
Pittsburgh
13
182
User
Bethlehem
13
183
User
Pittsburgh
13
184
User
Allentown
13
185
User
Philadelphia
13
186
User
Philadelphia
13
187
User
Malvern
13
188
User
Philadelphia
13
189
User
New Holland
13
190
User
Philadelphia
13
191
User
Pittsburgh
13
192
User
Boothwyn
13
193
seth chizmar
Sandy Lake
13
194
User
Sewickley Hills
13
195
User
Leith-Hatfield
13
196
User
Easton
13
197
User
Pittsburgh
13
198
User
Philadelphia
13
199
User
Philadelphia
13
200
Kyle D'Arcy
Chadds Ford
13
201
User
Pittsburgh
13
202
User
Philadelphia
13
203
User
Pittsburgh
12
204
User
State College
12
205
User
Upper Darby
12
206
User
Philadelphia
12
207
Dana Yi
Philadelphia
12
208
User
Acme
12
209
User
Gettysburg
12
210
User
Philadelphia
12
211
Mike Molaro
Penn Hills
12
212
User
Philadelphia
12
213
User
Pittsburgh
12
214
User
Stroudsburg
12
215
User
Havertown
12
216
User
City
12
217
User
Philadelphia
12
218
User
East Stroudsburg
11
219
User
Villanova
11
220
User
Philadelphia
11
221
William Pratt
Broomall
11
222
User
Stockertown
11
223
User
New Kensington
11
224
User
Ebensburg
11
225
User
Sandy Lake
11
226
User
Bradford
11
227
Anna Hayes
Wilson
11
228
User
Pittsburgh
11
229
User
Selinsgrove
11
230
User
Pittsburgh
11
231
User
Willow Hill
11
232
User
Pottsgrove
11
233
User
Pittsburgh
11
234
User
Jersey Shore
11
235
User
Philadelphia
11
236
User
Shippensburg
11
237
User
York
11
238
User
Philadelphia
11
239
User
Philadelphia
11
240
User
King Of Prussia
11
241
User
Penn Hills
11
242
User
City
11
243
User
Philadelphia
11
244
User
Pittsburgh
10
245
Sandra
Bethlehem
10
246
User
Pittsburgh
10
247
User
Pittsburgh
10
248
User
Pittsburgh
10
249
User
Philadelphia
10
250
Tyler Jordan
Ligonier
10
251
User
Philadelphia
10
252
User
Philadelphia
10
253
User
Jersey Shore
10
254
Brent Xiao
Pittsburgh
10
255
User
Philadelphia
10
256
User
Seneca
10
257
User
Mercersburg
10
258
Daniel Clark
Elkins Park
10
259
Yuan Huang
Philadelphia
10
260
User
Hanover
10
261
User
State College
10
262
User
Franklin Park
10
263
User
Swarthmore
10
264
User
Philadelphia
10
265
User
Warminster
10
266
User
Philadelphia
10
267
User
State College
10
268
User
Philadelphia
10
269
User
Middletown
10
270
User
Mcalisterville
10
271
User
Lake Meade
10
272
User
Fawn Grove
10
273
User
Philadelphia
10
274
User
Pittsburgh
10
275
User
Fountain Hill
10
276
User
Swarthmore
10
277
User
Chambersburg
10
278
User
Pottstown
9
279
User
Philadelphia
9
280
User
Philadelphia
9
281
User
Haverford
9
282
User
Boyertown
9
283
User
Chambersburg
9
284
User
Elkins Park
9
285
User
Philadelphia
9
286
User
Fawn Grove
9
287
jinjin wang
Pittsburgh
9
288
User
City
9
289
User
East Stroudsburg
9
290
User
Philadelphia
9
291
User
Emlenton
9
292
User
Shippensburg
9
293
User
Franklin Park
9
294
User
Philadelphia
9
295
Grey McAlaine
Wayne
9
296
User
Pittsburgh
9
297
User
State College
9
298
User
Fawn Grove
9
299
User
Whitehall
9
300
User
Philadelphia
9
301
User
Philadelphia
9
302
User
Doylestown
9
303
User
Seneca
9
304
User
Stroudsburg
9
305
Nah Bruh
Greensburg
9
306
User
Mars
8
307
User
Philadelphia
8
308
User
Bloomsburg
8
309
User
State College
8
310
User
Philadelphia
8
311
User
Philadelphia
8
312
Kumar
Philadelphia
8
313
User
Bethlehem
8
314
User
Red Lion
8
315
User
Mercersburg
8
316
User
Exton
8
317
User
Plymouth Meeting
8
318
User
Horsham
8
319
User
New Salem Borough
8
320
Luis Rupp
York
8
321
User
Wilson
8
322
User
Doylestown
8
323
User
Wexford
8
324
User
Pittsburgh
8
325
User
Carlisle
8
326
User
Philadelphia
8
327
H AM
Pittsburgh
8
328
User
Park Forest Village
8
329
User
Philadelphia
8
330
User
Philadelphia
8
331
User
Economy
8
332
Dean Boyer
Lebanon
8
333
User
Pittsburgh
8
334
User
California
8
335
User
Bethlehem
8
336
User
Philadelphia
8
337
User
Philadelphia
8
338
User
Philadelphia
8
339
User
East Stroudsburg
8
340
User
Philadelphia
8
341
User
Lancaster
8
342
User
Pittsburgh
8
343
User
Lancaster
8
344
User
Haverford
8
345
User
Pittsburgh
8
346
User
Waynesboro
8
347
User
Philadelphia
8
348
User
Bethlehem
7
349
User
Springfield
7
350
User
Lancaster
7
351
Zachary Carter
York
7
352
User
Philadelphia
7
353
User
Philadelphia
7
354
User
Harrisburg
7
355
User
Philadelphia
7
356
User
Chadds Ford
7
357
Suyash Sinkar
Franklin Park
7
358
User
Pittsburgh
7
359
User
Topton
7
360
User
Philadelphia
7
361
User
Quakertown
7
362
User
City
7
363
User
City
7
364
User
Philadelphia
7
365
User
State College
7
366
User
Bethel Park
7
367
User
New Castle
7
368
User
Meadville
7
369
User
Philadelphia
7
370
User
Mars
7
371
User
Philadelphia
7
372
User
Philadelphia
7
373
User
Philadelphia
7
374
User
East Norriton
7
375
User
Pittsburgh
7
376
User
Meadville
7
377
User
Valley Green
7
378
Brian Herrmann
Selinsgrove
7
379
User
Emmaus
7
380
User
Mount Lebanon
7
381
User
Pittsburgh
7
382
Nancy
Hamburg
7
383
User
Philadelphia
7
384
User
Houserville
7
385
User
Bala Cynwyd
7
386
User
Swarthmore
7
387
User
Philadelphia
7
388
User
Bradford
7
389
Tom Zhang
Villanova
7
390
User
Pittsburgh
7
391
User
State College
6
392
User
Pittsburgh
6
393
User
East Stroudsburg
6
394
User
Bala Cynwyd
6
395
Mia Will
Ligonier
6
396
User
East Stroudsburg
6
397
User
State College
6
398
User
Philadelphia
6
399
Andrew Moore
Ligonier
6
400
User
Franklin
6
401
Saku
Philadelphia
6
402
User
Pittsburgh
6
403
User
Newtown Grant
6
404
User
Philadelphia
6
405
User
Breinigsville
6
406
User
Philadelphia
6
407
User
Northampton
6
408
Brian Neuner
Ligonier
6
409
User
Philadelphia
6
410
User
Philadelphia
6
411
User
Phoenixville
6
412
User
York
6
413
User
Philadelphia
6
414
no
Industry
6
415
User
Sinking Spring
6
416
User
Philadelphia
6
417
User
Bala Cynwyd
6
418
User
Waynesboro
6
419
User
Fayetteville
6
420
User
Harleysville
6
421
Keondrick Lucas
Ligonier
6
422
User
Upper Darby
6
423
User
Fullerton
6
424
Maxwell Kriberney
Ligonier
6
425
Luke Barkley
Ligonier
6
426
0 HZ
Philadelphia
6
427
User
King Of Prussia
6
428
User
East Stroudsburg
6
429
User
Philadelphia
6
430
User
Philadelphia
6
431
User
Bradford
6
432
User
Pittsburgh
6
433
User
Pittsburgh
6
434
User
Pittsburgh
6
435
User
Philadelphia
6
436
User
Shippensburg
6
437
User
Pittsburgh
6
438
User
Swarthmore
6
439
User
Ligonier
6
440
Brody Krieger
Ligonier
6
441
Tylan Wilkins
Ligonier
6
442
User
Philadelphia
6
443
User
Philadelphia
6
444
User
Philadelphia
6
445
User
Mount Joy
6
446
User
Gilbertsville
6
447
User
Quakertown
6
448
User
City
6
449
User
Paxtonia
6
450
User
Philadelphia
5
451
User
East Stroudsburg
5
452
User
Philadelphia
5
453
User
Phoenixville
5
454
User
Philadelphia
5
455
User
Levittown
5
456
User
Philadelphia
5
457
User
Petrolia
5
458
User
Philadelphia
5
459
User
City
5
460
User
Scranton
5
461
User
Fawn Grove
5
462
User
Chester
5
463
User
Philadelphia
5
464
User
Bala Cynwyd
5
465
User
Lancaster
5
466
User
Mechanicsburg
5
467
User
York
5
468
User
Pittsburgh
5
469
User
Philadelphia
5
470
User
Bryn Athyn
5
471
User
Yardley
5
472
User
Fort Washington
5
473
anish nangare
State College
5
474
User
State College
5
475
User
Fullerton
5
476
User
Erie
5
477
User
Economy
5
478
User
Oil City
5
479
User
Upper Darby
5
480
User
Mcalisterville
5
481
User
Philadelphia
5
482
User
Williamsport
5
483
Drake Roan
Williamsport
5
484
User
Greensburg
5
485
Aja L Woolfolk
Philadelphia
5
486
User
Fairview
5
487
User
Waynesboro
5
488
User
Philadelphia
5
489
User
Progress
5
490
User
Upper Darby
5
491
User
Lancaster
5
492
Mitchell Findley
Ligonier
5
493
User
Dillsburg
5
494
User
Wellsboro
5
495
User
Fawn Grove
5
496
User
Lansdale
5
497
User
Bethlehem
5
498
User
Scranton
5
499
User
Philadelphia
5
500
User
Pittsburgh
5
501
User
Bethel Park
5
502
User
Bethlehem
5
503
User
Philadelphia
5
504
User
Philadelphia
5
505
Ben Dawn-Cross
Allentown
5
506
User
Skippack
5
507
Logan Woodyard
Ligonier
5
508
Kia Deemer
Ligonier
5
509
User
Allentown
5
510
User
Ephrata
5
511
User
East Stroudsburg
5
512
Bella Rabaiotti
York
5
513
User
Trappe
5
514
User
City
5
515
User
City
5
516
User
Bensalem
4
517
User
Philadelphia
4
518
User
Philadelphia
4
519
User
State College
4
520
User
Philadelphia
4
521
User
Philadelphia
4
522
User
Philadelphia
4
523
User
Allentown
4
524
User
State College
4
525
User
Moon
4
526
User
King Of Prussia
4
527
User
West Point
4
528
User
Philadelphia
4
529
User
Philadelphia
4
530
User
Philadelphia
4
531
User
Doylestown
4
532
Joshy Hunter
Allenport
4
533
User
Kingston
4
534
User
State College
4
535
User
Philadelphia
4
536
User
Philadelphia
4
537
User
State College
4
538
User
State College
4
539
User
East Stroudsburg
4
540
User
Doylestown
4
541
Jonathan Shawley
Ligonier
4
542
User
Pittsburgh
4
543
User
Philadelphia
4
544
User
Meadville
4
545
User
Gladwyne
4
546
User
Philadelphia
4
547
User
Philadelphia
4
548
User
Palmer Heights
4
549
User
Linglestown
4
550
User
Philadelphia
4
551
Harish G
Pittsburgh
4
552
User
Aston
4
553
Chet Dillaman
Ligonier
4
554
User
York
4
555
User
York
4
556
User
Pittsburgh
4
557
Dominick Thiel
Ligonier
4
558
User
Pittsburgh
4
559
User
Altoona
4
560
User
Fountain Hill
4
561
User
Wynnewood
4
562
User
Franklin Park
4
563
User
Philadelphia
4
564
User
Philadelphia
4
565
User
New Salem Borough
4
566
Elora Miller
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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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