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

Do you want to be fast at mental math? Many people do, but the options for doing the necessary exercises are simply too cumbersome for all but the most dedicated of trainees. In physical fitness, many people are interested in training their bodies but allocating the time, energy, and money for it is a significant obstacle. Likewise, lugging around books and whatnot for math practice is a threshold that just doesn't meet the standards of modern life.

Training yourself to be skilled at mental math needs to be quick and convenient. mathtrainer.org is a web app that works in your browser rather than a program you have to download and install on your computer or phone. This allows users to try and use the app without having to install new software. As a web app, updates are also much simpler. There is no need to download endless updates—the website will always be the most current version.

You can access a web app from any device connected to the internet and equipped with a web browser, including smartphones, tablets, and desktop computers. Moreover, you are free to use whichever browser you prefer, including Google Chrome, Safari, Firefox, and others. Google Chrome is the recommended browser for the best maths training since it tends to lead the pack in supporting the latest web technologies.

Math Trainer is designed to offer a similar experience regardless of what you’re using to access it, whether it be Android, iOS, Windows, or another operating system. Though an on-screen touch keyboard will appear on mobile devices, you may prefer to use the app on a desktop with a keyboard. Hopefully the advantages of a web app for convenient mathematics training are apparent.

Another part of making the app easy to use is eliminating the need for signing up and logging in. Users can get started with their math training as soon as they click the start button on this page. After progressing to higher levels in the app, your progress is automatically saved so long as your return to the site through the same browser.


What's a Mental Math Tip?

A mental math tip is a sequence of steps that can be taken to solve a math problem in your head. Click the arrow below to see an example for the following problem:

÷
984
3
328

A tip like this one is available for every problem in Math Trainer, so there's always help if you get stuck. With enough 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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