Leaderboard
Michigan
Rank
Level
1
User
Ann Arbor
91
2
User
Coldwater
86
3
KRZ
Garfield Township
82
4
Qiushi Yu
Ann Arbor
81
5
Ronit Tiwary
Novi
71
6
User
Grass Lake Charter Township
66
7
User
Iron River
65
8
User
Ann Arbor
64
9
User
Dearborn
63
10
User
Novi
60
11
Pi
Iron River
57
12
User
Ann Arbor
53
13
User
Iron River
53
14
User
Grass Lake Charter Township
52
15
Jane
Eaton Rapids
51
16
User
Grass Lake Charter Township
51
17
User
Grass Lake Charter Township
49
18
User
Ann Arbor
49
19
User
Iron River
48
20
Jeu
Clinton Township
46
21
User
Grass Lake Charter Township
46
22
User
Iron River
45
23
User
White Lake Charter Township
43
24
Vahn Brail
Grand Blanc Township
41
25
User
Ann Arbor
41
26
Promotion Please
Iron River
40
27
Ashwin Sreevatsa
Ann Arbor
40
28
Ben Hayward
Kalamazoo
40
29
User
Ann Arbor
39
30
User
Adrian
37
31
Anthony Kaminski
Dearborn
37
32
User
Royal Oak
36
33
User
Clinton Township
35
34
User
Grass Lake Charter Township
34
35
michigan monster hunting squad2 monster
Saginaw
34
36
User
Sterling Heights
34
37
User
Grass Lake Charter Township
33
38
User
Ann Arbor
33
39
Alexa Williams-Lall
Ypsilanti Charter Township
32
40
User
Ann Arbor
32
41
User
Grass Lake Charter Township
31
42
User
Iron River
31
43
User
Grass Lake Charter Township
31
44
User
Ann Arbor
31
45
User
Ann Arbor
31
46
User
Iron River
30
47
User
Ann Arbor
30
48
User
Ann Arbor
30
49
User
Ann Arbor
30
50
User
Detroit
30
51
User
Ann Arbor
30
52
User
Roscommon
30
53
User
Grass Lake Charter Township
28
54
Chris Q
Lyon Charter Township
27
55
User
Ann Arbor
27
56
Hui Cai
Ann Arbor
26
57
User
Ann Arbor
25
58
User
Ann Arbor
24
59
User
Shepherd
24
60
User
Ann Arbor
24
61
User
Ann Arbor
24
62
User
Detroit
23
63
User
Ann Arbor
23
64
User
Detroit
23
65
User
Parma
23
66
User
Ann Arbor
23
67
User
Grass Lake Charter Township
22
68
User
Grass Lake Charter Township
22
69
User
Ann Arbor
22
70
User
Commerce Charter Township
22
71
User
Grass Lake Charter Township
21
72
User
Ann Arbor
21
73
User
Kalamazoo
21
74
User
Ann Arbor
21
75
User
Grass Lake Charter Township
21
76
User
Grass Lake Charter Township
21
77
Ben Stensen
Ann Arbor
21
78
User
Auburn Hills
21
79
Maria Dawson
Sterling Heights
21
80
User
Kentwood
21
81
Vanni T
Detroit
20
82
User
Farmington Hills
20
83
User
Madison Charter Township
20
84
User
Eaton Rapids
19
85
Carter Pryor
Ann Arbor
19
86
User
Davison
19
87
User
Holland
19
88
User
Detroit
19
89
Claire
Orion Charter Township
19
90
Sára Losonczy
Novi
19
91
User
Saint Clair Shores
19
92
User
Warren
19
93
Julio Peralta Terrazas
Rochester Hills
19
94
User
Romulus
18
95
User
Detroit
18
96
User
Rochester Hills
18
97
User
Dearborn
18
98
User
Ann Arbor
18
99
User
Adrian
18
100
Nicholas Boose
Redford Charter Township
17
101
User
City
16
102
User
Ann Arbor
16
103
User
Grass Lake Charter Township
16
104
User
Ann Arbor
16
105
User
Ann Arbor
16
106
User
Sterling Heights
16
107
User
Sterling Heights
15
108
User
Ann Arbor
15
109
Michael Levesque
Saline
15
110
User
East Lansing
15
111
User
Midland
15
112
User
Marquette
15
113
User
Taylor
15
114
User
Ann Arbor
15
115
User
Ann Arbor
15
116
User
Grass Lake Charter Township
14
117
Cora McHugh
Ann Arbor
14
118
Aliya Renee Khan
Northville
14
119
User
Farmington Hills
14
120
User
Houghton
14
121
User
Troy
14
122
User
Ann Arbor
14
123
User
Ann Arbor
14
124
User
Georgetown Township
14
125
Alan Tilley
Traverse City
13
126
User
Ann Arbor
13
127
User
Detroit
13
128
User
Fair Plain
13
129
User
West Bloomfield Township
13
130
User
Albion
13
131
User
Ann Arbor
13
132
User
Ann Arbor
13
133
User
Ann Arbor
12
134
User
Ann Arbor
12
135
User
Houghton
12
136
User
Iron River
12
137
User
Wyoming
12
138
User
Beechwood
12
139
User
Benton Harbor
12
140
maes
Blackman
12
141
User
Ann Arbor
11
142
User
Bangor
11
143
User
Madison Heights
11
144
User
Petoskey
11
145
User
Niles
11
146
User
Dearborn
11
147
User
Ann Arbor
11
148
KAT
Oscoda Township
11
149
User
Ann Arbor
11
150
User
Beechwood
11
151
User
Rochester Hills
11
152
User
Clinton Township
11
153
User
Carleton
10
154
User
Dearborn
10
155
User
Ann Arbor
10
156
User
Ann Arbor
10
157
User
Plymouth Charter Township
10
158
User
Mount Pleasant
10
159
Evan Leathorn
Port Huron
10
160
User
Ann Arbor
10
161
User
Ann Arbor
10
162
Continuous Peach
Ann Arbor
10
163
User
Texas Charter Township
10
164
László Losonczy
Novi
10
165
User
Ann Arbor
10
166
User
Alpena
10
167
Shelagh Fehrenbach
Traverse City
10
168
User
Novi
10
169
User
Capac
10
170
User
Ann Arbor
10
171
Erin Glumb
Pinckney
9
172
User
Lansing
9
173
User
Ann Arbor
9
174
User
Sterling Heights
9
175
User
East Lansing
9
176
User
Capac
9
177
User
St. Joseph
9
178
User
Dearborn
9
179
User
Midland
9
180
Advik Dogra
Troy
9
181
User
Grand Rapids
9
182
User
Midland
9
183
User
Ann Arbor
9
184
User
Ann Arbor
9
185
User
West Bloomfield Township
9
186
User
Ann Arbor
9
187
User
Milan
9
188
Seminar
Homer
8
189
User
Clinton Township
8
190
User
Ann Arbor
8
191
User
Beechwood
8
192
User
Ann Arbor
8
193
User
West Bloomfield Township
8
194
User
Lapeer
8
195
Ryan Meister
Marquette
8
196
User
Charlotte
8
197
Jennifer Kusch
Saline
8
198
User
Charter Township Of Clinton
8
199
User
Detroit
8
200
User
Rochester Hills
8
201
User
East Lansing
8
202
User
Capac
8
203
User
Ann Arbor
8
204
Albert Cai
Ann Arbor
8
205
User
Oak Park
8
206
Anshu ankita raj
Ann Arbor
8
207
David Postiff
York Charter Township
8
208
User
Rochester Hills
8
209
Nick Reymann
Detroit
8
210
User
Beechwood
8
211
Milo Pappas
Ann Arbor
8
212
User
Detroit
7
213
User
Detroit
7
214
User
Ferndale
7
215
KrisViana
Kentwood
7
216
User
Detroit
7
217
User
Warren
7
218
User
Ann Arbor
7
219
User
East Lansing
7
220
User
Redford Charter Township
7
221
User
Charlotte
7
222
User
Capac
7
223
User
Midland
7
224
User
Ann Arbor
7
225
Andy
Clinton Township
7
226
User
Detroit
7
227
User
Dearborn
7
228
User
Ann Arbor
7
229
Deja
Detroit
7
230
User
Ann Arbor
7
231
User
Harrison Charter Township
7
232
User
Ann Arbor
7
233
Deena Shah
Canton
7
234
User
South Lyon
7
235
User
Big Rapids Township
7
236
User
Midland
7
237
User
Capac
7
238
User
Midland
7
239
User
Ann Arbor
7
240
User
Westland
7
241
User
Shelby Charter Township
7
242
User
Detroit
6
243
User
Rockford
6
244
User
Rochester Hills
6
245
User
Ann Arbor
6
246
User
Detroit
6
247
User
Midland
6
248
User
Ann Arbor
6
249
User
Allendale Charter Township
6
250
User
Houghton
6
251
User
Sterling Heights
6
252
User
Detroit
6
253
lbaroudi
Ann Arbor
6
254
Max
Ann Arbor
6
255
User
Charlotte
6
256
User
Canton
6
257
User
Kentwood
6
258
User
Lyon Charter Township
6
259
User
Detroit
6
260
User
City
6
261
foggy
Independence Charter Township
6
262
User
Troy
6
263
ISAIAH POTTER
Charlotte
6
264
User
Grand Rapids
6
265
User
Ann Arbor
6
266
User
Sterling Heights
6
267
User
Beechwood
6
268
User
Ann Arbor
6
269
User
Farmington Hills
6
270
User
Macomb
6
271
User
Ann Arbor
6
272
User
Ann Arbor
6
273
User
Novi
6
274
User
Caro
6
275
User
Westland
6
276
User
Detroit
6
277
User
Georgetown Township
5
278
User
Detroit
5
279
User
Grand Rapids
5
280
sk
Vestaburg
5
281
User
Grand Rapids
5
282
User
Midland
5
283
User
Sterling Heights
5
284
User
Southfield
5
285
User
Detroit
5
286
Bobby b
Ann Arbor
5
287
User
Kalamazoo
5
288
User
Brighton
5
289
User
East Lansing
5
290
User
Ann Arbor
5
291
User
Ann Arbor
5
292
User
Coldwater
5
293
User
Brighton
5
294
User
Dearborn Heights
5
295
User
Lansing
5
296
User
Ann Arbor
5
297
User
Ann Arbor
5
298
User
Beechwood
5
299
User
Ann Arbor
5
300
User
Midland
5
301
User
Marquette
5
302
User
Ann Arbor
5
303
User
Ann Arbor
5
304
User
Clinton Township
5
305
User
Portage
5
306
User
Ann Arbor
5
307
User
Grand Rapids
5
308
User
Detroit
5
309
User
Detroit
4
310
User
South Lyon
4
311
User
Marquette
4
312
Patricia Collard
Delhi Charter Township
4
313
User
Ann Arbor
4
314
User
Detroit
4
315
User
Grass Lake Charter Township
4
316
User
Rochester Hills
4
317
User
Ann Arbor
4
318
Moe
Detroit
4
319
User
Detroit
4
320
User
Garfield Township
4
321
Retard
Commerce Charter Township
4
322
User
Ann Arbor
4
323
User
Rochester Hills
4
324
User
Charlotte
4
325
User
Capac
4
326
User
Midland
4
327
User
Ann Arbor
4
328
User
Detroit
4
329
User
Jackson
4
330
User
Coldwater
4
331
User
New Baltimore
4
332
User
Ann Arbor
4
333
User
Romulus
4
334
User
Troy
4
335
User
Ann Arbor
4
336
User
Dearborn
4
337
User
Southfield
4
338
User
Kalamazoo
4
339
User
Charlotte
4
340
Mercedes
Midland
4
341
User
Farmington Hills
4
342
User
Midland
4
343
User
Midland
4
344
User
Novi
4
345
User
Kalamazoo
4
346
User
Farmington Hills
4
347
User
Ann Arbor
4
348
User
Ann Arbor
4
349
User
Holland
4
350
User
Warren
4
351
User
Midland
4
352
User
Dearborn
4
353
User
Detroit
4
354
User
Ann Arbor
3
355
User
Beechwood
3
356
User
Farmington Hills
3
357
User
Trenton
3
358
User
Ann Arbor
3
359
User
Edmore
3
360
User
York Charter Township
3
361
User
Ann Arbor
3
362
User
Livonia
3
363
User
Detroit
3
364
User
Charlotte
3
365
User
Midland
3
366
User
Marquette
3
367
User
Lyon Charter Township
3
368
User
Grand Blanc Township
3
369
User
Capac
3
370
User
Midland
3
371
User
Rochester Hills
3
372
User
Detroit
3
373
User
Allendale Charter Township
3
374
User
Detroit
3
375
User
Ann Arbor
3
376
User
Detroit
3
377
User
Royal Oak
3
378
User
Ann Arbor
3
379
User
Grass Lake Charter Township
3
380
User
West Bloomfield Township
3
381
User
Ann Arbor
3
382
User
Walker
3
383
yoyo
Brighton
3
384
User
Ann Arbor
3
385
User
Charlotte
3
386
User
Charlotte
3
387
User
Kalamazoo
3
388
User
Troy
3
389
User
Capac
3
390
User
Dearborn
3
391
User
Shelby Charter Township
3
392
Mara Rada
Troy
3
393
User
Detroit
3
394
User
Capac
3
395
User
Capac
3
396
User
Capac
3
397
User
Capac
3
398
User
Midland
3
399
User
Forest Hills
3
400
User
Midland
3
401
User
Midland
3
402
User
Midland
3
403
User
East Lansing
3
404
Jayani
Detroit
3
405
User
Detroit
3
406
User
Midland
3
407
User
Detroit
3
408
User
Livonia
3
409
User
Harrison
3
410
User
Detroit
3
411
User
Royal Oak
3
412
User
Troy
3
413
User
Warren
3
414
User
Saint Joseph
3
415
User
Auburn Hills
2
416
User
Charlotte
2
417
User
Grass Lake Charter Township
2
418
User
Charlotte
2
419
User
Comstock Township
2
420
User
Ann Arbor
2
421
User
Ann Arbor
2
422
User
Midland
2
423
User
Rochester Hills
2
424
User
Saginaw
2
425
User
Clay Township
2
426
User
Taylor
2
427
User
Marquette
2
428
User
Genesee Charter Township
2
429
User
Capac
2
430
User
Midland
2
431
User
Ann Arbor
2
432
Savvy Chae
Detroit
2
433
Scott Halasy
Ferndale
2
434
User
Sterling Heights
2
435
User
Ypsilanti
2
436
User
Forest Hills
2
437
User
Detroit
2
438
User
City
2
439
User
Ann Arbor
2
440
User
Charlotte
2
441
User
Charlotte
2
442
User
Charlotte
2
443
User
Farmington Hills
2
444
User
Ann Arbor
2
445
User
Midland
2
446
User
Clinton Township
2
447
User
Grand Rapids
2
448
User
Ann Arbor
2
449
Nicholas Porter
Ann Arbor
2
450
User
Detroit
2
451
User
Ann Arbor
2
452
User
Grand Rapids
2
453
User
Shelby Charter Township
2
454
User
Edmore
2
455
User
Dansville
2
456
User
Grand Rapids
2
457
User
Meridian Charter Township
2
458
User
Rochester Hills
2
459
User
Ann Arbor
2
460
User
Grand Rapids
2
461
Ryan
Commerce Charter Township
2
462
User
Rockford
2
463
User
Southfield
2
464
User
Ann Arbor
2
465
User
Ypsilanti Charter Township
2
466
User
Grand Rapids
2
467
User
Detroit
2
468
User
Warren
2
469
User
Ann Arbor
2
470
User
Richland
2
471
Patricia Collard
Charlotte
2
472
User
Holland
2
473
User
Georgetown Township
2
474
User
Auburn Hills
2
475
User
Ann Arbor
2
476
User
Flint
2
477
User
Capac
2
478
User
Ann Arbor
2
479
User
Midland
2
480
Jackson
Ann Arbor
2
481
User
Pontiac
1
482
User
Grand Blanc Township
1
483
Crispin Contreras
East Lansing
1
484
User
Beechwood
1
485
User
Beechwood
1
486
User
Midland
1
487
User
Beechwood
1
488
User
Beechwood
1
489
User
Beechwood
1
490
User
Roseville
1
491
User
Detroit
1
492
User
Ann Arbor
1
493
User
Midland
1
494
Sean Powers
Detroit
1
495
User
Ann Arbor
1
496
User
Dansville
1
497
User
Ann Arbor
1
498
User
Garfield Township
1
499
User
Midland
1
500
User
Detroit
1
501
User
Detroit
1
502
User
Detroit
1
503
User
Ann Arbor
1
504
User
Charlotte
1
505
User
Charlotte
1
506
User
Petoskey
1
507
User
Ann Arbor
1
508
User
Beechwood
1
509
User
Clinton Township
1
510
User
Ann Arbor
1
511
User
Coldwater
1
512
User
Detroit
1
513
User
Coldwater
1
514
User
Capac
1
515
User
Capac
1
516
User
Macomb
1
517
User
Muskegon Charter Township
1
518
User
Sterling Heights
1
519
User
Detroit
1
520
User
Ann Arbor
1
521
User
Livonia
1
522
Olivia Bowron
Ann Arbor
1
523
User
Charter Township Of Clinton
1
524
User
Township Of Hamburg
1
525
User
Ann Arbor
1
526
User
Traverse City
1
527
User
Dexter
1
528
User
East Lansing
1
529
User
Marquette
1
530
User
Jackson
1
531
User
Birch Run
1
532
User
Ann Arbor
1
533
User
Dearborn
1
534
User
Capac
1
535
User
Taylor
1
536
User
Jackson
1
537
User
Capac
1
538
User
Midland
1
539
User
Rochester Hills
1
540
User
Beechwood
1
541
User
Forest Hills
1
542
User
Detroit
1
543
User
Harrison Charter Township
1
544
User
Livonia
1
545
User
Midland
1
546
User
Midland
1
547
User
Midland
1
548
Valerie Chri
Ann Arbor
1
549
User
Battle Creek
1
550
User
Union Charter Township
1
551
User
Grand Blanc Township
1
552
User
Ann Arbor
1
553
User
Ann Arbor
1
554
User
Midland
1
555
User
Midland
1
556
User
Birmingham
1
557
User
Warren
1
558
User
Midland
1
559
User
Ann Arbor
1
560
User
Ann Arbor
1
561
User
Ann Arbor
1
562
jadin
Charlotte
1
563
User
Charlotte
1
564
User
Charlotte
1
565
Kaylen Gangi
Charlotte
1
566
User
Kentwood
1
567
User
Sterling Heights
1
568
User
Detroit
1
569
User
Detroit
1
570
User
Parma
1
571
User
Grand Rapids
1
572
User
Kalamazoo
1
573
User
Coldwater
1
574
User
Coldwater
1
575
User
Coldwater
1
576
User
Georgetown Township
1
577
User
Ann Arbor
1
578
User
Coldwater
1
579
User
Coldwater
1
580
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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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