Leaderboard
Pennsylvania
Rank
Level
1
ManInJelly
Narberth
92
2
User
Pittsburgh
91
3
User
Philadelphia
86
4
User
Pittsburgh
86
5
User
Park Forest Village
81
6
James Hu
Philadelphia
81
7
Michelle Abigan
Philadelphia
78
8
User
Irwin
77
9
Josh Jung
Pittsburgh
77
10
User
Philadelphia
77
11
Krit
Harrisburg
66
12
User
Philadelphia
65
13
User
Philadelphia
63
14
User
Philadelphia
63
15
frances
Havertown
62
16
User
Philadelphia
62
17
User
Marlin
61
18
User
Leacock-Leola-Bareville
60
19
User
New Holland
60
20
1. Thou Shant Lewd Kaori
Bethlehem
59
21
User
Pittsburgh
59
22
L. J.
Pittsburgh
57
23
Zack Lee
Pittsburgh
55
24
User
Pittsburgh
55
25
User
New Holland
54
26
Latissimus65
Fountain Hill
53
27
User
Philadelphia
53
28
User
Philadelphia
53
29
User
Leacock-Leola-Bareville
51
30
pei254
State College
51
31
User
New Holland
51
32
User
Leacock-Leola-Bareville
50
33
User
Philadelphia
50
34
Saku
Philadelphia
50
35
User
Lewisburg
49
36
User
Pittsburgh
49
37
User
Philadelphia
48
38
User
Lititz
48
39
User
Birdsboro
47
40
User
Philadelphia
47
41
User
Philadelphia
46
42
User
Philadelphia
45
43
Sanjib
Chester Springs
45
44
User
Conshohocken
44
45
User
Pittsburgh
44
46
seth chizmar
Sandy Lake
43
47
User
Pittsburgh
43
48
User
Pittsburgh
43
49
Chris Doms
Bethlehem
42
50
User
Bethlehem
42
51
User
Burgettstown
41
52
User
State College
41
53
User
Philadelphia
41
54
User
New Holland
41
55
User
Newtown
40
56
User
Stockertown
40
57
User
Philadelphia
40
58
User
Mccandless
39
59
User
Doylestown
39
60
User
Philadelphia
39
61
User
Pittsburgh
38
62
User
Sandy Lake
38
63
User
Philadelphia
37
64
User
Gettysburg
37
65
User
Pittsburgh
36
66
User
Philadelphia
36
67
User
Philadelphia
35
68
User
Doylestown
35
69
User
Sewickley Hills
35
70
User
Philadelphia
35
71
User
Philadelphia
35
72
User
Philadelphia
34
73
User
Easton
33
74
User
State College
33
75
User
Philadelphia
33
76
User
Philadelphia
32
77
User
State College
31
78
User
Philadelphia
31
79
User
Philadelphia
31
80
User
Pittsburgh
31
81
User
Philadelphia
31
82
User
Haverford
30
83
User
Hummelstown
30
84
User
Pittsburgh
30
85
User
Swarthmore
30
86
User
Fawn Grove
30
87
User
New Holland
30
88
User
Bensalem
29
89
William Pratt
Broomall
29
90
User
Philadelphia
28
91
Bethann Dolbow
York
28
92
User
Philadelphia
28
93
User
Pittsburgh
28
94
John Thomas
Greensburg
28
95
User
Philadelphia
27
96
User
Burgettstown
27
97
User
Philadelphia
27
98
User
Philadelphia
27
99
User
Philadelphia
27
100
User
Philadelphia
27
101
User
Lancaster
27
102
User
Bloomsburg
26
103
User
Pittsburgh
26
104
User
Hanover
26
105
User
State College
26
106
User
Philadelphia
26
107
jinjin wang
Pittsburgh
26
108
User
State College
26
109
User
Pittsburgh
26
110
User
Allentown
26
111
Emily Horne
Lebanon
26
112
User
Leith-Hatfield
25
113
User
Acme
25
114
User
Swarthmore
25
115
User
Fawn Grove
25
116
User
Philadelphia
25
117
User
Philadelphia
25
118
User
Philadelphia
25
119
User
York
25
120
User
Philadelphia
25
121
User
Aston
24
122
H AM
Pittsburgh
24
123
User
State College
24
124
Amy
Wexford
24
125
User
Feasterville-Trevose
23
126
User
Havertown
23
127
User
Shippensburg
23
128
User
City
23
129
Tyler Jordan
Ligonier
23
130
User
Bloomsburg
23
131
User
Philadelphia
23
132
User
Philadelphia
23
133
User
Philadelphia
23
134
User
Pittsburgh
23
135
User
Harrisburg
22
136
User
Philadelphia
22
137
User
East Stroudsburg
22
138
User
State College
22
139
Luis Rupp
York
22
140
User
Wilson
22
141
User
California
22
142
User
Pittsburgh
22
143
User
Pittsburgh
22
144
User
Philadelphia
22
145
User
Houserville
22
146
User
Philadelphia
22
147
User
Philadelphia
22
148
User
Coatesville
22
149
Katie E
Easton
22
150
User
Exton
22
151
Mike Molaro
Penn Hills
21
152
User
Philadelphia
21
153
User
New Kensington
21
154
User
Jersey Shore
21
155
Daniel Clark
Elkins Park
21
156
User
Middletown
21
157
User
Pottstown
21
158
User
Philadelphia
21
159
User
Exton
21
160
User
Pittsburgh
21
161
User
Philadelphia
21
162
User
Breinigsville
21
163
User
York
21
164
User
Pittsburgh
21
165
User
Philadelphia
21
166
User
Chester Springs
21
167
User
Easton
20
168
Dana Yi
Philadelphia
20
169
Anna Hayes
Wilson
20
170
User
Pottsgrove
20
171
User
Philadelphia
20
172
User
Pittsburgh
20
173
User
Philadelphia
20
174
User
Jersey Shore
20
175
User
City
20
176
Nah Bruh
Greensburg
20
177
User
Horsham
20
178
User
Lancaster
20
179
Zachary Carter
York
20
180
User
Topton
20
181
Nancy
Hamburg
20
182
User
Pittsburgh
20
183
User
Philadelphia
20
184
User
Upper Darby
20
185
User
Malvern
20
186
User
Upper Darby
19
187
User
York
19
188
User
State College
19
189
Lily DeAngelis
New Hope
19
190
User
Elkins Park
19
191
User
Philadelphia
19
192
User
Meadville
19
193
User
Meadville
19
194
User
Philadelphia
19
195
User
Ebensburg
18
196
User
Bethlehem
18
197
User
Fountain Hill
18
198
User
Boyertown
18
199
User
East Stroudsburg
18
200
User
Philadelphia
18
201
User
Mars
18
202
User
Pittsburgh
18
203
Brent Xiao
Pittsburgh
17
204
User
Warminster
17
205
User
Bradford
17
206
User
York
17
207
User
Philadelphia
17
208
User
Swarthmore
17
209
User
Philadelphia
17
210
Aja L Woolfolk
Philadelphia
17
211
User
Philadelphia
17
212
User
Fawn Grove
16
213
User
New Salem Borough
16
214
User
Philadelphia
16
215
User
Chadds Ford
16
216
User
Bethel Park
16
217
User
Emmaus
16
218
User
Swarthmore
16
219
User
Sinking Spring
16
220
User
Bala Cynwyd
16
221
User
Fullerton
16
222
User
King Of Prussia
16
223
User
Philadelphia
16
224
anish nangare
State College
16
225
User
Mcalisterville
16
226
User
Philadelphia
16
227
User
Philadelphia
15
228
User
Lake Meade
15
229
User
Chambersburg
15
230
User
Haverford
15
231
Andrew Moore
Ligonier
15
232
Brian Neuner
Ligonier
15
233
User
Bradford
15
234
User
Philadelphia
15
235
User
Philadelphia
15
236
User
Bethlehem
15
237
User
Bethel Park
15
238
User
Philadelphia
15
239
User
Philadelphia
15
240
User
Philadelphia
15
241
User
Doylestown
15
242
User
State College
15
243
User
Willow Grove
15
244
User
State College
15
245
User
Cheswick
15
246
Joshua Baktay
Pittsburgh
14
247
User
Emlenton
14
248
User
Shippensburg
14
249
User
Fawn Grove
14
250
User
Mars
14
251
User
Lancaster
14
252
Keondrick Lucas
Ligonier
14
253
User
Aston
14
254
User
Lionville
14
255
User
Philadelphia
14
256
User
Fawn Grove
14
257
User
Philadelphia
14
258
User
Erie
14
259
User
Chambersburg
13
260
User
Bethlehem
13
261
User
Plymouth Meeting
13
262
User
East Stroudsburg
13
263
User
State College
13
264
User
Philadelphia
13
265
User
Trappe
13
266
User
Philadelphia
13
267
User
State College
13
268
User
Moon
13
269
User
Philadelphia
13
270
Kumar
Philadelphia
12
271
User
Bethlehem
12
272
User
State College
12
273
User
Philadelphia
12
274
User
State College
12
275
Mia Will
Ligonier
12
276
User
Philadelphia
12
277
User
Philadelphia
12
278
0 HZ
Philadelphia
12
279
User
Philadelphia
12
280
User
Quakertown
12
281
User
Philadelphia
12
282
User
Philadelphia
12
283
User
Fairview
12
284
User
Upper Darby
12
285
User
Bensalem
12
286
Brandon Lutz
Philadelphia
12
287
User
Emmaus
12
288
User
Philadelphia
12
289
Dean Boyer
Lebanon
11
290
User
Haverford
11
291
User
Quakertown
11
292
User
Bala Cynwyd
11
293
User
Pittsburgh
11
294
User
East Stroudsburg
11
295
User
East Stroudsburg
11
296
User
Fayetteville
11
297
Maxwell Kriberney
Ligonier
11
298
Luke Barkley
Ligonier
11
299
Brody Krieger
Ligonier
11
300
Tylan Wilkins
Ligonier
11
301
User
Ligonier
11
302
User
East Stroudsburg
11
303
User
York
11
304
User
Fullerton
11
305
User
Greensburg
11
306
User
Pittsburgh
11
307
User
Philadelphia
11
308
User
Philadelphia
11
309
User
Philadelphia
11
310
User
Philadelphia
11
311
User
State College
11
312
User
Hellertown
11
313
User
Pittsburgh
11
314
User
Philadelphia
10
315
User
Franklin
10
316
User
Newtown Grant
10
317
no
Industry
10
318
User
Pittsburgh
10
319
User
Philadelphia
10
320
User
Philadelphia
10
321
User
Petrolia
10
322
User
Pittsburgh
10
323
User
State College
10
324
User
Pittsburgh
10
325
User
Philadelphia
10
326
User
Philadelphia
10
327
User
Pittsburgh
10
328
User
Philadelphia
10
329
User
Philadelphia
10
330
User
Pittsburgh
10
331
User
Somerset
10
332
User
Harleysville
9
333
User
East Stroudsburg
9
334
User
Philadelphia
9
335
User
Philadelphia
9
336
User
Mount Joy
9
337
User
Philadelphia
9
338
User
State College
9
339
User
Lancaster
9
340
Mitchell Findley
Ligonier
9
341
User
Wellsboro
9
342
User
Bethlehem
9
343
Logan Woodyard
Ligonier
9
344
Kia Deemer
Ligonier
9
345
User
Allentown
9
346
User
Kingston
9
347
User
Linglestown
9
348
User
Pittsburgh
9
349
User
State College
9
350
User
Dunmore
9
351
User
Doylestown
9
352
User
Somerset
9
353
User
Philadelphia
9
354
User
Levittown
9
355
User
Hazleton
9
356
User
Pittsburgh
8
357
User
Shippensburg
8
358
User
Fawn Grove
8
359
User
Philadelphia
8
360
User
Scranton
8
361
User
Fort Washington
8
362
User
Dillsburg
8
363
Bella Rabaiotti
York
8
364
Jonathan Shawley
Ligonier
8
365
Harish G
Pittsburgh
8
366
User
York
8
367
User
York
8
368
User
Wynnewood
8
369
User
Doylestown
8
370
User
Philadelphia
8
371
User
Glenside
8
372
User
Hershey
8
373
User
State College
8
374
User
Philadelphia
8
375
User
Philadelphia
8
376
User
Philadelphia
8
377
User
Erie
8
378
User
Philadelphia
8
379
User
York
8
380
User
Philadelphia
8
381
User
Philadelphia
8
382
Sohum Kshirsagar
Philadelphia
7
383
User
Waynesboro
7
384
User
Fawn Grove
7
385
Ben Dawn-Cross
Allentown
7
386
User
Skippack
7
387
User
Allentown
7
388
User
East Stroudsburg
7
389
User
Philadelphia
7
390
User
Philadelphia
7
391
User
East Stroudsburg
7
392
User
Meadville
7
393
User
Palmer Heights
7
394
Chet Dillaman
Ligonier
7
395
Dominick Thiel
Ligonier
7
396
User
Philadelphia
7
397
Elora Miller
Ligonier
7
398
User
Pittsburgh
7
399
User
Pittsburgh
7
400
User
Pittsburgh
7
401
User
Pittsburgh
7
402
User
Philadelphia
7
403
User
Philadelphia
7
404
User
Philadelphia
7
405
Austin M Patchett
Philadelphia
7
406
User
Philadelphia
7
407
User
Philadelphia
7
408
User
Philadelphia
7
409
User
State College
7
410
User
Philadelphia
7
411
User
West Point
6
412
Joshy Hunter
Allenport
6
413
User
State College
6
414
User
Doylestown
6
415
User
Pittsburgh
6
416
User
State College
6
417
User
Aston
6
418
User
Fountain Hill
6
419
User
New Salem Borough
6
420
User
Philadelphia
6
421
User
Philadelphia
6
422
User
Oil City
6
423
Ethan Will
Ligonier
6
424
User
Phoenixville
6
425
User
Pittsburgh
6
426
User
Philadelphia
6
427
User
Wynnewood
6
428
User
Philadelphia
6
429
User
Fort Indiantown Gap
6
430
User
Pittsburgh
6
431
User
Pittsburgh
6
432
User
Philadelphia
6
433
User
North Versailles
6
434
User
Philadelphia
6
435
User
Philadelphia
6
436
User
Philadelphia
6
437
Papa Nii Vanderpuye
Lancaster
6
438
User
Philadelphia
6
439
User
Philadelphia
6
440
User
Philadelphia
5
441
User
Lake Ariel
5
442
User
Mansfield
5
443
User
Pittsburgh
5
444
User
Scalp Level
5
445
james maszle
Pittsburgh
5
446
User
Coatesville
5
447
User
Wexford
5
448
User
Philadelphia
5
449
User
East Stroudsburg
5
450
User
East Stroudsburg
5
451
User
East Stroudsburg
5
452
User
Philadelphia
5
453
User
Park Forest Village
5
454
User
Spring House
5
455
User
Ligonier
5
456
User
Ligonier
5
457
User
West Norriton
5
458
Ace Vavrek
Ligonier
5
459
User
Philadelphia
5
460
User
Philadelphia
5
461
User
Parkville
5
462
User
Breinigsville
5
463
User
Philadelphia
5
464
User
Philadelphia
5
465
User
Mccandless
5
466
User
Sinking Spring
5
467
User
Lititz
5
468
User
Philadelphia
5
469
User
Moon
5
470
User
Philadelphia
5
471
User
Fawn Grove
5
472
User
Philadelphia
5
473
User
Philadelphia
5
474
User
Philadelphia
5
475
User
Philadelphia
5
476
User
Philadelphia
5
477
User
Philadelphia
5
478
User
Pittsburgh
5
479
User
Glenolden
5
480
User
Philadelphia
5
481
User
Philadelphia
5
482
User
Pittsburgh
5
483
User
Philadelphia
5
484
User
Philadelphia
5
485
User
Pittsburgh
4
486
User
Fawn Grove
4
487
User
East Stroudsburg
4
488
User
Philadelphia
4
489
User
Philadelphia
4
490
User
Pittsburgh
4
491
Dalton Reitterer
Fawn Grove
4
492
User
Norristown
4
493
User
East Stroudsburg
4
494
User
East Stroudsburg
4
495
User
East Stroudsburg
4
496
User
Philadelphia
4
497
User
Pittsburgh
4
498
User
Bala Cynwyd
4
499
User
Bala Cynwyd
4
500
User
Philadelphia
4
501
User
Ligonier
4
502
Wyatt Piper
Ligonier
4
503
Tevin Brittingham
Ligonier
4
504
User
Ligonier
4
505
User
Pittsburgh
4
506
User
Aston
4
507
User
State College
4
508
User
Emmaus
4
509
User
Wyomissing
4
510
User
Spring Grove
4
511
User
Fawn Grove
4
512
User
Hatfield
4
513
User
Philadelphia
4
514
User
Ardmore
4
515
User
Pittsburgh
4
516
User
Aston
4
517
User
Berwyn
4
518
User
Easton
4
519
User
Philadelphia
4
520
User
Carbondale
4
521
User
Collegeville
4
522
User
Philadelphia
4
523
User
State College
4
524
User
Philadelphia
4
525
User
Harrisburg
4
526
User
Chester Springs
4
527
User
Philadelphia
4
528
User
Wayne
4
529
User
Chester Springs
4
530
User
Allentown
3
531
User
Fleetwood
3
532
User
East Stroudsburg
3
533
User
Philadelphia
3
534
User
Philadelphia
3
535
User
Devault
3
536
User
Park Forest Village
3
537
Lyric Parkins
East Stroudsburg
3
538
User
East Stroudsburg
3
539
User
Elkins Park
3
540
User
New Oxford
3
541
User
Philadelphia
3
542
User
Philadelphia
3
543
Jacob Roadman
Ligonier
3
544
Rehmatullah Cheema
Philadelphia
3
545
William Sugden
Ligonier
3
546
Kelsie Graham
Ligonier
3
547
Audrey Fruehauf
Ligonier
3
548
Hayden Hart
Ligonier
3
549
User
Philadelphia
3
550
User
Bryn Mawr
3
551
User
Philadelphia
3
552
User
Wyncote
3
553
User
Gettysburg
3
554
User
Red Lion
3
555
User
Bryn Athyn
3
556
User
Plum
3
557
User
Carnegie
3
558
User
Philadelphia
3
559
User
Pittsburgh
3
560
User
State College
3
561
User
Macungie
3
562
User
Pittsburgh
3
563
User
Philadelphia
3
564
User
Exton
3
565
User
Willow Grove
3
566
User
Philadelphia
3
567
User
Philadelphia
3
568
User
Philadelphia
3
569
User
Lancaster
3
570
User
Philadelphia
3
571
User
Sharon Hill
3
572
User
Johnstown
3
573
User
Philadelphia
3
574
User
Wayne
3
575
User
Philadelphia
3
576
User
Pittsburgh
3
577
User
Pittsburgh
3
578
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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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