Leaderboard
Michigan
Rank
Level
1
User
Adrian
97
2
User
Iron River
93
3
Anand Bahl
Ann Arbor
91
4
Hui Cai
Ann Arbor
84
5
Pi
Iron River
81
6
Ashwin Sreevatsa
Ann Arbor
74
7
User
Iron River
72
8
User
Hartford
71
9
User
Ann Arbor
71
10
User
Grass Lake Charter Township
70
11
User
Grass Lake Charter Township
70
12
User
Grass Lake Charter Township
69
13
User
Iron River
68
14
User
Grass Lake Charter Township
67
15
Jeu
Clinton Township
66
16
User
Iron River
64
17
User
Grass Lake Charter Township
59
18
User
Ann Arbor
59
19
Promotion Please
Iron River
56
20
User
Ann Arbor
55
21
User
Ann Arbor
55
22
User
Grass Lake Charter Township
53
23
User
East Lansing
53
24
User
Grass Lake Charter Township
52
25
User
Grass Lake Charter Township
52
26
User
Ann Arbor
52
27
User
Ann Arbor
52
28
User
Detroit
52
29
User
Ann Arbor
51
30
User
Parma
51
31
User
Grass Lake Charter Township
51
32
User
Grass Lake Charter Township
51
33
User
Ann Arbor
50
34
User
Ann Arbor
50
35
User
Ann Arbor
50
36
User
Ann Arbor
49
37
User
Clinton Township
48
38
User
Ann Arbor
48
39
User
Grass Lake Charter Township
48
40
User
Ann Arbor
48
41
michigan monster hunting squad2 monster
Saginaw
47
42
Jake Hart
Plymouth
46
43
User
Sterling Heights
45
44
User
Ann Arbor
44
45
User
Roscommon
44
46
User
Adrian
44
47
User
Ann Arbor
44
48
User
Iron River
43
49
User
Ann Arbor
43
50
Carter Pryor
Ann Arbor
42
51
User
Rochester Hills
42
52
Chris Q
Lyon Charter Township
40
53
Sára Losonczy
Novi
39
54
User
Grass Lake Charter Township
39
55
User
Grass Lake Charter Township
38
56
Ben Stensen
Ann Arbor
38
57
User
Farmington Hills
38
58
User
Ann Arbor
38
59
User
Ann Arbor
38
60
Maria Dawson
Sterling Heights
37
61
Michael Levesque
Saline
37
62
User
Shepherd
35
63
User
Ann Arbor
35
64
User
Ann Arbor
35
65
User
Niles
35
66
Vahn Brail
Grand Blanc Township
34
67
Vanni T
Detroit
34
68
User
Detroit
33
69
User
Ann Arbor
33
70
User
Grass Lake Charter Township
33
71
User
Ann Arbor
33
72
User
Ann Arbor
33
73
Continuous Peach
Ann Arbor
33
74
Alexa Williams-Lall
Ypsilanti Charter Township
32
75
User
Ann Arbor
32
76
User
Dearborn
32
77
User
Ann Arbor
31
78
User
Ann Arbor
31
79
User
Ann Arbor
31
80
User
Kalamazoo
30
81
User
Marquette
30
82
maes
Blackman
30
83
KAT
Oscoda Township
30
84
Alex Zureick
West Bloomfield Township
30
85
User
Madison Charter Township
29
86
User
Sterling Heights
29
87
Aliya Renee Khan
Northville
29
88
User
Georgetown Township
29
89
User
Ann Arbor
29
90
Saarah Alam
Plymouth
29
91
User
Holland
28
92
Claire
Orion Charter Township
28
93
Alan Tilley
Traverse City
28
94
User
Ann Arbor
28
95
User
Ann Arbor
28
96
User
Ann Arbor
28
97
User
Kentwood
27
98
User
Sterling Heights
27
99
User
West Bloomfield Township
27
100
User
Detroit
27
101
User
Dearborn
26
102
User
Petoskey
26
103
Shelagh Fehrenbach
Traverse City
26
104
User
Capac
26
105
Albert Cai
Ann Arbor
26
106
Ananya
Saginaw
26
107
User
Ann Arbor
25
108
User
Capac
25
109
User
Ann Arbor
25
110
User
Charlotte
25
111
KrisViana
Kentwood
25
112
User
Grass Lake Charter Township
24
113
User
Ann Arbor
24
114
Anshu ankita raj
Ann Arbor
24
115
User
Ann Arbor
23
116
User
Ann Arbor
23
117
User
Ann Arbor
23
118
User
Beechwood
23
119
David Postiff
York Charter Township
23
120
User
Detroit
23
121
User
Ann Arbor
23
122
User
Detroit
23
123
User
Houghton
22
124
User
Fair Plain
22
125
User
Rochester Hills
22
126
User
Ann Arbor
22
127
Erin Glumb
Pinckney
22
128
User
Charlotte
22
129
User
Ann Arbor
22
130
User
Midland
21
131
User
Farmington Hills
21
132
User
Ann Arbor
21
133
User
Beechwood
21
134
User
Ann Arbor
21
135
User
Ann Arbor
21
136
Seminar
Homer
21
137
Jennifer Kusch
Saline
21
138
User
Capac
21
139
User
Beechwood
21
140
User
Dearborn
21
141
User
Ann Arbor
21
142
lbaroudi
Ann Arbor
21
143
User
Canton
21
144
User
Ann Arbor
21
145
User
Detroit
20
146
László Losonczy
Novi
20
147
Nick Reymann
Detroit
20
148
User
Detroit
20
149
User
Lyon Charter Township
20
150
User
Ann Arbor
19
151
User
West Bloomfield Township
19
152
User
Ann Arbor
19
153
User
Lansing
19
154
User
Ferndale
19
155
User
Capac
19
156
User
Midland
19
157
User
Midland
19
158
User
Charlotte
19
159
User
Troy
19
160
Cora McHugh
Ann Arbor
18
161
User
Alpena
18
162
User
Charter Township Of Clinton
18
163
User
South Lyon
18
164
User
Capac
18
165
User
Midland
18
166
User
Midland
18
167
Max
Ann Arbor
18
168
User
Plymouth
18
169
User
Ann Arbor
18
170
User
Wyoming
17
171
User
Oak Park
17
172
User
Ann Arbor
17
173
User
Ann Arbor
17
174
ISAIAH POTTER
Charlotte
17
175
User
Ann Arbor
17
176
User
Ann Arbor
17
177
Ryan Meister
Marquette
16
178
User
Dearborn
16
179
User
Ann Arbor
16
180
User
Rochester Hills
16
181
User
Beechwood
16
182
User
Ann Arbor
16
183
User
Ann Arbor
16
184
User
Ann Arbor
16
185
User
Ann Arbor
16
186
User
Carleton
15
187
User
Mount Pleasant
15
188
User
Novi
15
189
User
Ann Arbor
15
190
User
Detroit
15
191
User
Madison Heights
14
192
User
Milan
14
193
User
Meridian Charter Township
14
194
Deena Shah
Canton
14
195
User
Detroit
14
196
User
Midland
14
197
User
Ann Arbor
14
198
User
Ann Arbor
14
199
User
Ann Arbor
14
200
User
Charter Township Of Clinton
14
201
User
Ann Arbor
14
202
User
Ann Arbor
14
203
User
Charlotte
14
204
Patricia Collard
Delhi Charter Township
14
205
User
White Pigeon
13
206
User
Dearborn
13
207
User
Plymouth Charter Township
13
208
User
Midland
13
209
User
Ann Arbor
13
210
Advik Dogra
Troy
13
211
User
Ann Arbor
13
212
User
Caro
13
213
User
Grand Rapids
13
214
User
Ann Arbor
13
215
User
Midland
13
216
User
Charlotte
13
217
User
Romulus
13
218
User
Southfield
13
219
User
Plymouth
13
220
Evan Leathorn
Port Huron
12
221
User
East Lansing
12
222
User
Midland
12
223
User
East Lansing
12
224
Deja
Detroit
12
225
User
Harrison Charter Township
12
226
User
Rochester Hills
12
227
User
Ann Arbor
12
228
User
Ann Arbor
12
229
User
Ann Arbor
12
230
User
Novi
12
231
User
Dearborn Heights
12
232
User
Ann Arbor
12
233
User
Troy
12
234
User
Grand Rapids
11
235
User
Detroit
11
236
User
Rochester Hills
11
237
User
East Lansing
11
238
Milo Pappas
Ann Arbor
11
239
User
Detroit
11
240
User
Warren
11
241
User
Allendale Charter Township
11
242
Bobby b
Ann Arbor
11
243
User
Beechwood
11
244
User
Southfield
11
245
User
Sterling Heights
11
246
User
Genesee Township
11
247
User
Clinton Township
10
248
User
Ann Arbor
10
249
User
Westland
10
250
User
Georgetown Township
10
251
User
Coldwater
10
252
User
Ann Arbor
10
253
User
Portage
10
254
User
Grass Lake Charter Township
10
255
User
Ann Arbor
10
256
User
Saline
10
257
User
Royal Oak
10
258
User
Big Rapids Township
9
259
User
Ann Arbor
9
260
User
Houghton
9
261
User
Midland
9
262
User
Grand Rapids
9
263
User
South Lyon
9
264
User
Ann Arbor
9
265
User
Rochester Hills
9
266
User
Capac
9
267
User
Midland
9
268
User
Ann Arbor
9
269
User
Charlotte
9
270
User
Ann Arbor
9
271
User
Farmington Hills
9
272
User
Brighton
9
273
User
Ann Arbor
8
274
User
Shelby Charter Township
8
275
User
Rockford
8
276
User
Detroit
8
277
User
Detroit
8
278
User
City
8
279
User
Farmington Hills
8
280
User
Macomb
8
281
User
Detroit
8
282
sk
Vestaburg
8
283
User
Kalamazoo
8
284
User
East Lansing
8
285
User
Lansing
8
286
User
Marquette
8
287
User
Marquette
8
288
User
Coldwater
8
289
Mercedes
Midland
8
290
User
Midland
8
291
User
Midland
8
292
User
Ann Arbor
8
293
User
Midland
8
294
User
Charlotte
8
295
User
Ann Arbor
8
296
User
Grand Rapids
8
297
foggy
Independence Charter Township
7
298
User
Westland
7
299
User
Detroit
7
300
User
Sterling Heights
7
301
User
Southfield
7
302
User
Detroit
7
303
User
Brighton
7
304
User
Clinton Township
7
305
Retard
Commerce Charter Township
7
306
User
Detroit
7
307
User
Jackson
7
308
User
Farmington Hills
7
309
User
Holland
7
310
User
Ann Arbor
7
311
User
Beechwood
7
312
User
Capac
7
313
User
Midland
7
314
User
Detroit
7
315
User
Ann Arbor
7
316
User
West Bloomfield Township
7
317
User
Ann Arbor
7
318
User
Charlotte
7
319
User
Charlotte
7
320
Mara Rada
Troy
7
321
User
Detroit
7
322
User
Plymouth
7
323
User
Mason
7
324
User
Troy
7
325
User
Brighton
6
326
User
Ann Arbor
6
327
User
Detroit
6
328
User
Detroit
6
329
User
Garfield Township
6
330
User
New Baltimore
6
331
User
Troy
6
332
User
Kalamazoo
6
333
User
Dearborn
6
334
User
Ann Arbor
6
335
User
Grass Lake Charter Township
6
336
User
Capac
6
337
User
Capac
6
338
User
Capac
6
339
User
Capac
6
340
User
Capac
6
341
User
Midland
6
342
User
Midland
6
343
User
Midland
6
344
User
Midland
6
345
User
Midland
6
346
User
Troy
6
347
User
Charlotte
6
348
User
Charlotte
6
349
User
Ann Arbor
6
350
User
Midland
6
351
User
Clay Township
6
352
User
Kent County
6
353
User
Ann Arbor
5
354
User
Kalamazoo
5
355
User
Warren
5
356
User
Novi
5
357
User
Edmore
5
358
User
York Charter Township
5
359
User
Ann Arbor
5
360
User
Detroit
5
361
User
Marquette
5
362
User
Detroit
5
363
User
Detroit
5
364
User
Ann Arbor
5
365
User
Ann Arbor
5
366
User
Shelby Charter Township
5
367
User
East Lansing
5
368
User
Warren
5
369
User
Saint Joseph
5
370
User
Capac
5
371
User
Midland
5
372
User
Forest Hills
5
373
User
Charlotte
5
374
User
Charlotte
5
375
User
Charlotte
5
376
Patricia Collard
Charlotte
5
377
User
Farmington Hills
5
378
User
Ann Arbor
5
379
User
Detroit
5
380
Crispin Contreras
East Lansing
5
381
Merit Amadi
Dearborn
5
382
User
Midland
4
383
User
Grand Blanc Township
4
384
User
Rochester Hills
4
385
User
Royal Oak
4
386
User
Walker
4
387
User
Kalamazoo
4
388
User
Troy
4
389
User
Forest Hills
4
390
User
Detroit
4
391
User
Detroit
4
392
User
Livonia
4
393
User
Harrison
4
394
User
Detroit
4
395
User
Royal Oak
4
396
User
Auburn Hills
4
397
User
Grass Lake Charter Township
4
398
User
Ann Arbor
4
399
User
Rochester Hills
4
400
User
Saginaw
4
401
User
Taylor
4
402
User
Marquette
4
403
User
Ann Arbor
4
404
Scott Halasy
Ferndale
4
405
User
Ypsilanti
4
406
User
Grand Rapids
4
407
User
Ann Arbor
4
408
User
Grand Rapids
4
409
Ryan
Commerce Charter Township
4
410
User
Ann Arbor
4
411
User
Holland
4
412
User
Capac
4
413
User
Midland
4
414
User
Pontiac
4
415
User
Texas Charter Township
4
416
User
Beechwood
4
417
User
Beechwood
4
418
User
Beechwood
4
419
User
Midland
4
420
User
Beechwood
4
421
User
Beechwood
4
422
User
Midland
4
423
User
Dansville
4
424
User
Midland
4
425
User
Charlotte
4
426
User
Charlotte
4
427
User
Detroit
4
428
User
Ann Arbor
4
429
User
Rochester
4
430
Dawn
Port Huron
4
431
User
Auburn Hills
4
432
User
Comstock Township
3
433
User
Genesee Charter Township
3
434
User
Sterling Heights
3
435
User
Detroit
3
436
User
City
3
437
User
Clinton Township
3
438
User
Ann Arbor
3
439
User
Grand Rapids
3
440
User
Shelby Charter Township
3
441
User
Edmore
3
442
User
Dansville
3
443
User
Rochester Hills
3
444
User
Grand Rapids
3
445
User
Rockford
3
446
User
Southfield
3
447
User
Detroit
3
448
User
Richland
3
449
User
Georgetown Township
3
450
User
Ann Arbor
3
451
Jackson
Ann Arbor
3
452
User
Ann Arbor
3
453
User
Roseville
3
454
User
Ann Arbor
3
455
User
Ann Arbor
3
456
User
Ann Arbor
3
457
User
Garfield Township
3
458
User
Ann Arbor
3
459
User
Petoskey
3
460
User
Beechwood
3
461
User
Ann Arbor
3
462
User
Coldwater
3
463
User
Coldwater
3
464
User
Charter Township Of Clinton
3
465
User
Dexter
3
466
User
Birch Run
3
467
User
Ann Arbor
3
468
User
Capac
3
469
User
Capac
3
470
User
Midland
3
471
User
Beechwood
3
472
User
Forest Hills
3
473
User
Livonia
3
474
User
Midland
3
475
User
Midland
3
476
User
Midland
3
477
User
Battle Creek
3
478
User
Ann Arbor
3
479
User
Ann Arbor
3
480
User
Midland
3
481
User
Midland
3
482
User
Midland
3
483
User
Ann Arbor
3
484
User
Ann Arbor
3
485
jadin
Charlotte
3
486
User
Charlotte
3
487
User
Charlotte
3
488
Kaylen Gangi
Charlotte
3
489
User
Detroit
3
490
User
Grand Rapids
3
491
User
Detroit
3
492
User
Ypsilanti
3
493
User
Dearborn Heights
3
494
User
Auburn Hills
2
495
User
Flint
2
496
User
Grand Blanc Township
2
497
User
Detroit
2
498
Sean Powers
Detroit
2
499
User
Detroit
2
500
User
Detroit
2
501
User
Detroit
2
502
User
Ann Arbor
2
503
User
Clinton Township
2
504
User
Capac
2
505
User
Capac
2
506
User
Sterling Heights
2
507
User
Detroit
2
508
Olivia Bowron
Ann Arbor
2
509
User
Township Of Hamburg
2
510
User
Ann Arbor
2
511
User
Traverse City
2
512
User
Jackson
2
513
User
Marquette
2
514
User
Dearborn
2
515
User
Taylor
2
516
User
Jackson
2
517
User
Rochester Hills
2
518
Valerie Chri
Ann Arbor
2
519
User
Union Charter Township
2
520
User
Grand Blanc Township
2
521
User
Ann Arbor
2
522
User
Kentwood
2
523
User
Sterling Heights
2
524
User
Parma
2
525
User
Coldwater
2
526
User
Coldwater
2
527
User
Coldwater
2
528
User
Georgetown Township
2
529
User
Ann Arbor
2
530
User
Coldwater
2
531
User
Coldwater
2
532
User
Coldwater
2
533
User
Coldwater
2
534
User
Coldwater
2
535
User
Livonia
2
536
User
Capac
2
537
User
Capac
2
538
User
Capac
2
539
User
Ann Arbor
2
540
User
Midland
2
541
User
Ann Arbor
2
542
User
Ann Arbor
2
543
User
Ann Arbor
2
544
User
Midland
2
545
User
Midland
2
546
Captain Moroni
Saint Johns
2
547
User
Detroit
2
548
User
Midland
2
549
User
Charlotte
2
550
User
Charlotte
2
551
User
Charlotte
2
552
User
Charlotte
2
553
User
Charlotte
2
554
User
Charlotte
2
555
User
Charlotte
2
556
User
Ann Arbor
2
557
User
Independence Charter Township
2
558
User
Farmington Hills
2
559
User
Ann Arbor
2
560
User
Ann Arbor
2
561
User
Ann Arbor
2
562
User
Novi
2
563
User
Plymouth
2
564
User
Zeeland
2
565
Lee
Detroit
2
566
User
East Lansing
2
567
User
Grand Blanc
2
568
User
Grand Rapids
2
569
User
Ann Arbor
2
570
User
City
1
571
User
Grand Rapids
1
572
User
Trenton
1
573
User
Midland
1
574
Michael Ramlow
Troy
1
575
User
Vienna Township
1
576
User
Bloomfield Township
1
577
User
Grand Haven
1
578
User
Detroit
1
579
User
Grand Blanc Township
1
580
User
Richland
1
581
User
Kalamazoo
1
582
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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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