Leaderboard
Germany
Germany
Worldwide
Rank
Level
1
imbahero NO one
Hessen
95
2
Woodrow Wilson
Schleswig-Holstein
95
3
User
Berlin
93
4
Dekaz Turuz
Nordrhein-Westfalen
93
5
Smith
Bayern
89
6
Marco Julian Hummel
Niedersachsen
87
7
imbahero
Hessen
87
8
Karna Potwar
Bayern
87
10
User
Hessen
77
11
Bruno Jahn
Berlin
75
12
Tim
Hessen
75
14
Christian K
Bayern
72
15
F CDN
Hamburg
71
16
phipsae
Berlin
71
17
GreenSmurf
Baden-Württemberg
70
18
User
Bayern
68
19
Adele Holzener
Bayern
67
20
NJOI
Hamburg
66
21
User
Bayern
65
22
Armin Becker
Nordrhein-Westfalen
64
23
Diana Sofia
Baden-Württemberg
63
24
Yoshy
Berlin
62
25
Peter Kretschmer
Rheinland-Pfalz
61
30
S H
Hessen
57
31
User
Bayern
57
32
User
Hessen
57
33
Ohnohnsen
Nordrhein-Westfalen
57
34
User
Region
57
35
User
Bayern
57
36
Tobias Nack
Brandenburg
56
38
User
Hessen
55
39
User
Hamburg
55
40
User
Berlin
54
42
James Krano
Bayern
54
43
AFGHANISTAN
Nordrhein-Westfalen
54
44
User
Hamburg
54
46
Seth Robles
Bayern
53
47
User
Bayern
53
48
User
Berlin
52
49
User
Berlin
52
50
Erich
Hamburg
52
51
User
Hamburg
51
53
Warm Tea
Berlin
51
54
User
Bayern
51
55
User
Bayern
51
56
Helios Chromatic
Rheinland-Pfalz
50
57
User
Bayern
50
58
User
Bayern
50
59
User
Hamburg
50
61
User
Hamburg
49
62
User
Hessen
49
64
Loek Oprinsen
Bayern
48
66
User
Hamburg
48
67
User
Berlin
47
68
User
Hamburg
47
69
User
Berlin
47
70
Stefan
Berlin
46
72
Saakeli
Berlin
45
74
User
Hessen
45
75
User
Bayern
45
76
User
Bremen
44
77
User
Saarland
44
78
Ferdi W
Bayern
44
80
User
Hessen
44
81
Peerawat_Fluke
Berlin
43
83
secondinferno93@gmail.com
Hessen
42
84
ah ha
Hessen
42
85
Conrad Prillwitz
Nordrhein-Westfalen
42
86
Jan K
Hessen
42
87
User
Hessen
41
89
PreppingForGS
Bayern
41
91
User
Hessen
40
92
Hussein Wael Al-Ani
Nordrhein-Westfalen
39
93
Caroline Matthis
Berlin
39
94
Gergely Horváth
Bayern
39
96
Simon Stähler
Nordrhein-Westfalen
39
97
Konrad Gritschneder
Saarland
38
98
User
Bremen
38
99
User
Hamburg
38
100
Daniel Beumers
Nordrhein-Westfalen
37
102
User
Bayern
37
103
I. Ang.
Berlin
37
104
User
Sachsen
37
105
Timmy
Niedersachsen
36
106
User
Bayern
36
107
piidde
Bayern
36
108
User
Bayern
36
109
Elfy
Hamburg
36
110
User
Bayern
36
111
CP
Hessen
35
112
User
Hessen
35
113
Yosh
Berlin
35
118
User
Sachsen
33
119
User
Hamburg
33
120
Janosz Schäfer
Berlin
33
121
User
Hessen
33
122
User
Bremen
33
123
User
Bayern
33
124
Kristian Ericsson
Berlin
33
125
User
Berlin
33
126
EL
Berlin
33
127
User
Hamburg
32
128
Fufufem Poelbeq
Berlin
32
131
User
Hamburg
32
132
Daniel
Bayern
32
133
User
Bayern
32
134
Michal Besler
Bayern
32
135
moormaster
Baden-Württemberg
32
136
User
Bayern
32
137
Jose
Bayern
32
138
Hans Baum
Hessen
32
139
hfdghdfgh
Hamburg
31
142
User
Hessen
31
144
NiaiserieAllemande
Bayern
31
145
debez
Hessen
30
147
User
Bayern
30
150
John Park
Hessen
30
151
User
Hessen
29
152
DamulaAtra
Bayern
29
154
User
Bayern
29
155
Bao Phan
Bayern
29
157
User
Brandenburg
29
158
User
Bayern
29
159
José Ricardo
Rheinland-Pfalz
29
160
Mala Mala
Rheinland-Pfalz
29
161
User
Hamburg
29
162
Maik Schneider
Berlin
29
163
User
Hessen
28
165
User
Bayern
28
166
User
Bayern
28
168
User
Berlin
27
169
User
Bayern
27
170
User
Hessen
27
171
Leifur Eriksson
Berlin
27
172
I am going to be top
Nordrhein-Westfalen
27
174
User
Hessen
27
175
Peyo Peev
Hessen
27
176
Mátyás Mérőfi
Baden-Württemberg
27
177
User
Berlin
26
178
User
Hamburg
26
179
sfsdafa
Hessen
26
180
Martin Shkreli
Bayern
26
181
Lidan nadiL
Nordrhein-Westfalen
26
182
User
Niedersachsen
26
183
User
Hessen
25
184
User
Hamburg
25
187
User
Bayern
25
188
User
Niedersachsen
25
189
Ericson
Berlin
25
191
User
Hamburg
24
194
User
Hamburg
24
197
User
Hessen
24
198
Charlie
Hessen
24
199
Chris Inman
Niedersachsen
24
202
ooer
Bremen
24
204
User
Hessen
24
205
Bliblu
Niedersachsen
24
206
User
Niedersachsen
23
207
User
Hessen
23
209
User
Bayern
23
210
User
Hamburg
23
211
User
Bayern
23
212
User
Berlin
23
214
User
Hessen
23
216
SuLcOREf
Nordrhein-Westfalen
23
217
User
Hessen
23
218
Jakob Bohnacker
Hessen
23
219
User
Bayern
23
221
User
Bayern
22
222
User
Berlin
22
224
NoC4
Sachsen
22
225
User
Bayern
22
226
Platen
Niedersachsen
22
227
Felix Frenzel
Nordrhein-Westfalen
22
228
Ostroco maths team two
Baden-Württemberg
22
229
Login Online
Nordrhein-Westfalen
22
230
f mustermann
Nordrhein-Westfalen
22
232
Victor ARMINJON
Bayern
21
233
User
Berlin
21
236
User
Bremen
21
237
User
Berlin
21
238
User
Bayern
21
239
User
Bremen
21
242
User
Bremen
20
245
User
Bayern
20
247
User
Hessen
20
250
User
Berlin
20
251
Ivan Tulumovic
Baden-Württemberg
20
252
Peter Hans
Baden-Württemberg
20
254
User
Berlin
20
256
User
Hamburg
20
257
User
Berlin
20
258
User
Niedersachsen
20
259
User
Berlin
20
261
User
Bremen
19
264
User
Hessen
19
268
User
Hessen
19
269
User
Hessen
19
271
User
Hamburg
19
272
User
Bremen
19
273
User
Hamburg
19
274
User
Bayern
18
275
Jan
Bayern
18
276
User
Hessen
18
277
User
Niedersachsen
18
278
User
Hessen
18
279
User
Hamburg
18
280
Bliblu
Niedersachsen
18
281
Chris
Hessen
18
282
User
Hessen
18
283
User
Bremen
18
284
User
Bremen
18
285
User
Hessen
18
286
User
Bremen
18
287
User
Niedersachsen
18
288
User
Hessen
18
290
Wolle
Niedersachsen
18
292
User
Bayern
18
293
User
Bayern
18
297
Christina Noskevich
Berlin
17
298
User
Bayern
17
299
User
Bayern
17
300
User
Hessen
17
301
User
Hessen
17
302
User
Bremen
17
303
User
Hessen
17
304
User
Bayern
17
306
User
Bayern
17
307
User
Bremen
17
308
User
Hessen
17
309
User
Bayern
17
310
User
Bremen
17
311
User
Bayern
17
312
User
Bremen
17
313
User
Bayern
17
314
User
Hamburg
17
315
User
Hessen
17
316
Miguel Sanz
Baden-Württemberg
17
318
User
Sachsen
17
319
selda aran
Berlin
17
320
User
Bremen
17
321
Giorgos Gk
Hamburg
16
322
User
Bayern
16
323
User
Berlin
16
325
User
Hessen
16
326
User
Bremen
16
329
Ildiko Benke
Bayern
16
330
Marteyn
Berlin
16
331
User
Niedersachsen
16
333
User
Bremen
16
334
User
Bayern
16
335
Joschka Ziesche
Rheinland-Pfalz
16
336
User
Sachsen
16
337
mass
Hamburg
16
338
Max
Bayern
16
340
User
Hessen
16
341
User
Bayern
16
342
User
Bremen
16
343
User
Hamburg
16
345
User
Bayern
16
346
User
Bayern
16
347
User
Hessen
16
348
User
Bremen
16
350
User
Sachsen
15
351
User
Hessen
15
352
User
Hessen
15
354
User
Bayern
15
355
pete pete
Niedersachsen
15
357
zormit
Hessen
15
358
User
Hessen
15
360
Abhilasha Mohania
Berlin
15
364
User
Berlin
15
365
User
Bayern
15
367
User
Hamburg
15
368
User
Berlin
15
369
User
Hessen
15
370
sven hero kaufmann
Bayern
15
371
User
Berlin
15
372
User
Bayern
15
373
mohsen Sali
Region
14
375
User
Bayern
14
376
User
Hamburg
14
377
User
Niedersachsen
14
379
User
Berlin
14
380
User
Berlin
14
381
User
Niedersachsen
14
382
User
Niedersachsen
14
384
User
Berlin
14
385
User
Bremen
14
386
Manfred Klöppel
Hamburg
14
387
User
Berlin
14
389
User
Bremen
14
390
User
Hessen
14
391
User
Hessen
14
392
Chris
Bayern
13
394
User
Berlin
13
395
User
Berlin
13
397
User
Bremen
13
398
User
Hessen
13
399
User
Bayern
13
400
User
Brandenburg
13
401
User
Berlin
13
402
User
Bremen
13
404
User
Hessen
13
405
User
Hamburg
13
406
Ali Kocak
Bayern
13
407
User
Bayern
13
408
User
Hessen
13
409
User
Berlin
13
410
User
Bayern
13
412
User
Hessen
13
414
User
Berlin
13
416
User
Bremen
13
417
User
Berlin
13
418
User
Niedersachsen
13
419
Sören Krüger
Niedersachsen
13
420
Andreas W.
Hessen
13
422
User
Niedersachsen
13
423
User
Berlin
13
424
User
Bremen
13
427
mathdumb
Berlin
13
428
User
Bayern
13
429
Anne-Sophie Korotov
Berlin
13
433
User
Bayern
12
436
User
Hamburg
12
437
User
Bremen
12
440
Karim
Rheinland-Pfalz
12
441
James Whynot
Nordrhein-Westfalen
12
442
User
Hamburg
12
445
User
Region
12
447
User
Sachsen
12
449
User
Bremen
12
450
User
Berlin
12
452
User
Berlin
12
453
User
Hessen
12
454
Ahlem أمازيغية Story Time
Bayern
12
455
User
Sachsen
12
458
User
Bayern
12
459
User
Bayern
12
460
User
Bayern
12
462
User
Berlin
12
463
User
Hessen
12
466
User
Hessen
11
467
User
Hessen
11
468
User
Bayern
11
469
User
Berlin
11
470
User
Hessen
11
472
User
Bremen
11
474
User
Hessen
11
476
Sebastian Kuhnow
Nordrhein-Westfalen
11
477
Samuel Scandola
Bayern
11
479
User
Hessen
11
480
User
Hamburg
11
483
User
Bremen
11
484
User
Berlin
11
485
User
Bayern
11
486
User
Bayern
11
487
dgdgdffrg
Sachsen
11
488
User
Hamburg
11
489
User
Hamburg
11
490
User
Hessen
11
491
User
Berlin
11
493
User
Hessen
11
494
User
Bayern
11
498
János Géczi
Baden-Württemberg
11
499
User
Hessen
11
501
Marcus Bannenberg
Nordrhein-Westfalen
11
503
User
Bremen
11
504
User
Niedersachsen
11
506
User
Bayern
11
508
User
Bayern
11
509
User
Hessen
11
512
Abdulkarim Saheed
Bremen
11
513
User
Saarland
11
515
User
Hessen
11
516
User
Hessen
11
518
User
Berlin
10
519
User
Berlin
10
520
John
Berlin
10
521
User
Bayern
10
522
User
Bayern
10
523
User
Hessen
10
524
Grahmbler
Baden-Württemberg
10
525
User
Hamburg
10
526
User
Bayern
10
527
User
Bayern
10
528
User
Sachsen
10
530
User
Berlin
10
531
User
Bayern
10
537
User
Bayern
10
539
User
Berlin
10
540
User
Bayern
10
541
User
Berlin
10
542
User
Region
10
543
User
Hamburg
10
545
G
Bayern
10
547
User
Niedersachsen
10
548
User
Berlin
10
549
User
Berlin
10
550
User
Bayern
10
551
User
Berlin
10
552
User
Hamburg
10
553
User
Bayern
10
554
User
Brandenburg
10
557
User
Berlin
10
558
User
Bayern
10
560
User
Niedersachsen
10
561
User
Hessen
10
562
User
Hessen
10
563
User
Bayern
10
564
User
Hessen
10
566
User
Bremen
10
568
User
Bayern
9
570
Im Doing This For Yuna
Hessen
9
571
User
Berlin
9
573
User
Berlin
9
574
User
Bayern
9
576
User
Hessen
9
578
User
Bayern
9
579
User
Hamburg
9
580
moormaster
Baden-Württemberg
9
581
User
Bayern
9
582
User
Bayern
9
583
User
Bayern
9
585
User
Hamburg
9
586
User
Hessen
9
587
User
Berlin
9
588
cherrycoke
Berlin
9
589
Caio Martins
Baden-Württemberg
9
590
User
Berlin
9
592
User
Berlin
9
593
User
Bremen
9
594
User
Bayern
9
595
User
Bayern
9
598
User
Saarland
9
599
User
Berlin
9
600
User
Region
9
602
User
Berlin
9
603
User
Region
9
605
Christoph Büttner
Berlin
9
609
User
Bayern
9
612
User
Bayern
9
614
Alena Soukupova
Niedersachsen
9
617
User
Hessen
9
619
User
Bayern
9
620
User
Berlin
9
621
User
Berlin
9
622
User
Bayern
9
623
User
Berlin
9
625
User
Berlin
8
626
User
Bayern
8
629
User
Bayern
8
630
Gnedla Abozriba
Berlin
8
631
User
Hessen
8
632
User
Hessen
8
633
User
Hessen
8
634
User
Berlin
8
637
User
Bayern
8
639
User
Berlin
8
641
User
Hessen
8
642
User
Hessen
8
643
User
Berlin
8
644
Noah Afkir
Nordrhein-Westfalen
8
645
User
Bayern
8
646
User
Bayern
8
647
User
Hessen
8
648
User
Thüringen
8
649
User
Bayern
8
650
User
Bayern
8
652
User
Berlin
8
654
User
Hessen
8
655
User
Bayern
8
656
User
Hamburg
8
657
User
Berlin
8
659
User
Bayern
8
660
User
Region
8
661
Hawk 0802
Nordrhein-Westfalen
8
663
User
Bayern
8
665
User
Berlin
8
666
User
Berlin
8
669
User
Bayern
8
670
road text
Rheinland-Pfalz
8
673
User
Bremen
8
675
User
Sachsen
8
679
User
Bayern
8
684
User
Berlin
8
689
User
Bayern
8
690
User
Thüringen
8
691
User
Bayern
8
692
User
Berlin
7
693
User
Bayern
7
694
User
Berlin
7
695
Eugen Klein
Baden-Württemberg
7
696
User
Berlin
7
697
Irina Lebedeva
Hessen
7
700
Justyna Kurzela
Hessen
7
702
User
Bayern
7
703
User
Bayern
7
705
User
Bayern
7
707
User
Berlin
7
710
User
Berlin
7
716
User
Bayern
7
718
User
Bremen
7
720
User
Bayern
7
721
User
Hessen
7
722
Jean Carlo Machado
Baden-Württemberg
7
723
User
Berlin
7
725
User
Berlin
7
727
User
Bayern
7
729
User
Hessen
7
730
Flemca
Hamburg
7
734
User
Bayern
7
736
User
Berlin
7
738
Nigel Armino
Nordrhein-Westfalen
7
739
User
Berlin
7
740
User
Berlin
7
741
User
Thüringen
7
745
User
Berlin
7
746
User
Hessen
7
747
User
Bayern
7
750
User
Bayern
7
751
User
Thüringen
7
752
User
Hessen
7
753
User
Bayern
7
754
User
Hamburg
7
755
User
Hessen
7
756
User
Berlin
7
759
User
Sachsen
7
762
Jihan Gervasoni
Hamburg
6
764
User
Bayern
6
768
User
Berlin
6
769
User
Hessen
6
770
User
Hamburg
6
771
User
Berlin
6
772
User
Hessen
6
773
User
Hessen
6
774
User
Hessen
6
775
User
Berlin
6
777
Omar Afkir
Nordrhein-Westfalen
6
778
User
Berlin
6
780
User
Bremen
6
787
User
Hessen
6
788
User
Bayern
6
790
User
Hessen
6
791
User
Region
6
792
User
Bayern
6
794
User
Bayern
6
795
User
Bremen
6
798
User
Berlin
6
799
User
Hessen
6
802
User
Berlin
6
805
User
Hessen
6
806
User
Bayern
6
808
User
Bayern
6
810
User
Bayern
6
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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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