Leaderboard
Germany
Germany
Worldwide
Rank
Level
1
Bruno Jahn
Berlin
97
2
Jan K
Hessen
95
3
User
Hessen
95
6
Daniel
Bayern
86
7
User
Bayern
85
10
Dekaz Turuz
Hessen
84
11
User
Hamburg
84
12
moormaster
Baden-Württemberg
83
13
User
Berlin
83
14
User
Hessen
82
15
Seth Robles
Bayern
82
16
User
Bayern
82
17
ooer
Bremen
82
18
Armin Becker
Nordrhein-Westfalen
81
20
Tim-p
Hessen
80
21
Timmy
Niedersachsen
80
22
User
Bayern
80
23
S H
Hessen
78
24
User
Bayern
78
25
Tobias Nack
Brandenburg
78
27
Karna Potwar
Bayern
74
28
NJOI
Hamburg
73
29
James Krano
Bayern
73
30
AFGHANISTAN
Nordrhein-Westfalen
73
31
User
Bayern
73
32
Ohnohnsen
Nordrhein-Westfalen
71
34
User
Region
70
35
Elfy
Hamburg
69
36
User
Bayern
69
37
User
Bremen
69
38
User
Bayern
68
39
User
Sachsen
68
40
User
Berlin
66
41
User
Berlin
65
42
Erich
Hamburg
64
44
User
Hessen
63
45
Leifur Eriksson
Berlin
63
46
User
Bayern
63
47
User
Bremen
63
48
User
Hessen
62
49
User
Hessen
62
50
Jakob Bohnacker
Hessen
62
51
User
Bremen
61
52
User
Bremen
61
53
Ferdi W
Bayern
60
54
PreppingForGS
Bayern
60
55
Loek Oprinsen
Bayern
59
56
User
Hessen
59
57
User
Bayern
58
58
I am going to be top
Rheinland-Pfalz
58
59
User
Bremen
58
60
User
Bremen
58
61
User
Bremen
57
62
phipsae
Bayern
57
63
User
Bremen
57
64
User
Bremen
57
66
User
Berlin
56
67
Hussein Wael Al-Ani
Nordrhein-Westfalen
56
69
User
Bremen
56
70
User
Bremen
56
72
User
Bayern
55
73
User
Bayern
55
74
User
Bremen
55
76
Warm Tea
Berlin
54
77
User
Bremen
54
78
User
Bremen
54
79
User
Bremen
54
80
User
Hessen
53
82
User
Bayern
52
83
User
Bremen
52
84
pete pete
Niedersachsen
52
85
User
Bremen
52
86
Serhat
Hessen
52
88
User
Berlin
51
93
User
Hessen
50
94
dgdgdffrg
Sachsen
50
95
User
Bremen
50
97
Eric
Hessen
49
99
User
Bremen
49
100
Abhilasha Singh
Hamburg
49
101
User
Bremen
49
102
User
Hessen
49
103
Gergely Horváth
Bayern
48
104
CP
Hessen
48
105
Caroline Matthis
Berlin
48
106
User
Bremen
48
107
Ostroco maths team two
Baden-Württemberg
48
108
User
Bremen
48
109
Joschka Ziesche
Rheinland-Pfalz
48
110
User
Bremen
47
111
User
Berlin
47
116
User
Bremen
47
117
piidde
Bayern
46
118
Kristian Ericsson
Berlin
46
119
Maik Schneider
Berlin
46
120
Mátyás Mérőfi
Baden-Württemberg
46
122
User
Hessen
46
123
User
Bayern
46
126
User
Hessen
45
127
Jan
Bayern
45
128
User
Bayern
45
129
User
Bremen
45
130
User
Bremen
45
131
Yosh
Berlin
44
132
NiaiserieAllemande
Bayern
44
133
DamulaAtra
Bayern
44
134
Helios Chromatic
Baden-Württemberg
44
135
Login Online
Nordrhein-Westfalen
44
138
User
Hessen
44
141
User
Bremen
44
142
Michael Pencil
Schleswig-Holstein
44
143
Janosz Schäfer
Berlin
43
144
User
Bayern
43
146
NoC4
Sachsen
43
148
User
Hamburg
43
149
User
Hessen
43
150
Miguel Sanz
Baden-Württemberg
43
151
User
Bayern
43
152
Chris
Bayern
43
153
User
Bremen
43
154
Sören Krüger
Niedersachsen
43
155
User
Bremen
43
156
José Ricardo
Rheinland-Pfalz
42
157
User
Hessen
42
159
User
Bremen
42
160
David
Bayern
41
162
User
Bayern
41
164
Felix Frenzel
Nordrhein-Westfalen
41
165
User
Hessen
41
166
User
Hessen
41
167
User
Hessen
41
168
User
Berlin
41
169
User
Niedersachsen
41
170
User
Sachsen
41
174
User
Bayern
40
175
User
Hamburg
40
176
User
Bremen
40
177
User
Hessen
39
180
User
Berlin
38
181
User
Niedersachsen
38
182
Chris
Hessen
38
183
User
Bayern
38
184
User
Bayern
38
185
User
Bremen
38
187
Platen
Niedersachsen
37
188
Victor ARMINJON
Bayern
37
190
User
Bayern
37
193
User
Niedersachsen
37
194
User
Bayern
36
196
Charlie
Hessen
36
197
User
Berlin
36
198
Wolle
Niedersachsen
36
200
Max
Bayern
36
201
zormit
Hessen
36
203
User
Bremen
36
204
Harsha Kumar
Nordrhein-Westfalen
36
205
undefined
Hamburg
36
206
User
Berlin
36
207
Bliblu
Niedersachsen
35
208
Peter Hans
Baden-Württemberg
35
209
User
Hamburg
35
210
User
Niedersachsen
35
211
User
Hessen
35
213
User
Bayern
35
214
User
Niedersachsen
35
216
User
Bayern
35
217
User
Hamburg
35
219
User
Hessen
35
220
Ghostdog
Schleswig-Holstein
35
221
User
Berlin
34
222
Lidan nadiL
Nordrhein-Westfalen
34
224
User
Niedersachsen
34
225
User
Hessen
34
226
User
Bremen
34
227
User
Hessen
34
228
User
Hessen
34
229
User
Niedersachsen
34
230
User
Saarland
34
231
User
Berlin
34
232
User
Niedersachsen
34
233
Martin Shkreli
Bayern
33
234
f mustermann
Nordrhein-Westfalen
33
236
User
Hessen
33
237
User
Hessen
33
240
Dan D
Hessen
32
241
User
Bremen
32
242
User
Hessen
32
243
User
Hessen
32
244
User
Bayern
32
246
User
Hamburg
32
247
User
Bremen
32
248
User
Berlin
32
249
User
Hamburg
31
251
User
Niedersachsen
31
254
User
Berlin
31
255
User
Bayern
31
256
User
Berlin
31
257
User
Hamburg
31
259
User
Bayern
31
260
User
Hessen
31
262
Laurenz Schroer
Hessen
30
265
User
Hessen
30
267
User
Hessen
30
269
User
Sachsen
30
272
User
Bremen
30
273
User
Bayern
30
274
User
Berlin
30
275
User
Bayern
30
276
User
Berlin
30
277
User
Berlin
30
280
User
Hessen
29
282
User
Hessen
29
283
User
Hamburg
29
284
User
Bayern
29
285
User
Hessen
29
287
Andreas W.
Hessen
29
288
Karim
Rheinland-Pfalz
29
291
User
Hessen
29
294
User
Berlin
28
295
User
Berlin
28
297
User
Hessen
28
298
Ildiko Benke
Bayern
28
299
Hi
Berlin
28
301
mathdumb
Berlin
28
303
User
Berlin
28
304
Marcus Bannenberg
Nordrhein-Westfalen
28
305
User
Hessen
28
307
User
Hessen
28
308
User
Hessen
28
309
User
Hamburg
28
310
User
Bayern
28
311
User
Berlin
28
312
Bliblu
Niedersachsen
27
314
User
Berlin
27
315
User
Bayern
27
318
User
Hamburg
27
319
EL
Berlin
26
321
Simon Stähler
Nordrhein-Westfalen
26
322
User
Berlin
26
324
User
Hessen
26
325
User
Berlin
26
326
User
Bayern
26
328
User
Berlin
26
329
User
Hessen
26
330
User
Hessen
26
331
User
Berlin
25
333
User
Berlin
25
336
User
Berlin
25
337
moormaster
Baden-Württemberg
25
338
User
Bayern
25
340
User
Berlin
25
341
User
Berlin
25
342
User
Berlin
25
343
User
Bayern
25
344
Plus Plus
Nordrhein-Westfalen
25
345
Kubko
Hamburg
25
348
User
Hessen
24
350
James Whynot
Nordrhein-Westfalen
24
351
User
Sachsen
24
352
User
Hessen
24
353
User
Hamburg
24
354
Grahmbler
Baden-Württemberg
24
355
User
Niedersachsen
24
356
Irina Lebedeva
Hessen
24
358
User
Niedersachsen
24
359
User
Berlin
23
360
User
Hessen
23
361
Samuel Scandola
Bayern
23
362
User
Bremen
23
363
User
Bayern
23
365
User
Hamburg
23
366
User
Bayern
23
367
User
Hessen
23
368
User
Thüringen
23
369
User
Berlin
23
370
User
Berlin
23
371
User
Berlin
23
372
User
Hessen
23
373
User
Bayern
23
374
User
Hessen
22
375
User
Sachsen
22
376
User
Berlin
22
379
User
Bayern
22
381
User
Berlin
22
383
User
Berlin
22
384
User
Bayern
22
385
User
Berlin
22
388
User
Niedersachsen
22
390
User
Berlin
22
391
Giorgos Gk
Hamburg
21
392
User
Hessen
21
393
User
Hamburg
21
394
User
Bayern
21
395
User
Niedersachsen
21
396
User
Niedersachsen
21
397
Anne-Sophie Korotov
Berlin
21
400
User
Brandenburg
21
403
User
Hessen
21
404
User
Thüringen
21
406
User
Berlin
21
408
User
Berlin
21
409
User
Berlin
21
410
User
Bremen
21
412
overwatchKatze
Hessen
21
415
sven hero kaufmann
Bayern
20
418
User
Berlin
20
421
User
Hamburg
20
422
János Géczi
Baden-Württemberg
20
423
User
Hamburg
20
427
User
Berlin
20
428
User
Bremen
20
429
User
Bayern
20
431
road text
Rheinland-Pfalz
20
432
Justyna Kurzela
Hessen
20
433
User
Niedersachsen
20
434
Jihan Gervasoni
Hamburg
20
435
User
Bayern
20
436
User
Bremen
20
437
User
Berlin
20
438
User
Hamburg
20
441
User
Berlin
19
443
User
Bayern
19
444
User
Hessen
19
445
User
Hamburg
19
446
User
Berlin
19
447
User
Bayern
19
450
User
Berlin
19
452
User
Niedersachsen
19
453
User
Bayern
19
455
User
Berlin
19
456
Manfred Klöppel
Hamburg
18
457
User
Hessen
18
462
User
Bayern
18
464
User
Hamburg
18
465
User
Bayern
18
466
User
Hessen
18
469
User
Berlin
18
471
Stella Matthiessen Krausenecker
Bayern
18
473
User
Berlin
18
474
User
Bayern
18
475
User
Sachsen
18
476
Mahesh Mahajan
Hamburg
17
477
User
Bayern
17
479
User
Bayern
17
480
User
Bayern
17
483
User
Bayern
17
484
User
Niedersachsen
17
488
User
Hessen
17
491
User
Hamburg
17
492
User
Bayern
17
493
User
Sachsen
17
495
User
Bayern
16
496
User
Hessen
16
497
User
Bayern
16
498
User
Bayern
16
500
User
Berlin
16
501
cherrycoke
Berlin
16
502
Caio Martins
Baden-Württemberg
16
503
User
Bayern
16
504
User
Berlin
16
505
User
Bayern
16
506
User
Bayern
16
511
User
Hessen
16
512
User
Bayern
16
513
FARES
Hessen
16
514
User
Hessen
16
515
User
Bremen
16
517
User
Region
15
518
User
Berlin
15
519
User
Berlin
15
520
User
Bayern
15
523
User
Berlin
15
524
User
Hessen
15
525
Im Doing This For Yuna
Hessen
15
526
User
Region
15
527
User
Bayern
15
528
User
Berlin
15
529
User
Niedersachsen
15
531
User
Bayern
15
536
User
Hessen
15
539
User
Sachsen
15
542
Abhilasha Mohania
Bayern
14
543
User
Saarland
14
544
User
Region
14
548
User
Hessen
14
549
User
Bayern
14
550
User
Bayern
14
551
Nigel Armino
Nordrhein-Westfalen
14
552
User
Berlin
14
553
User
Bayern
14
557
User
Berlin
14
558
User
Berlin
14
559
User
Bayern
14
562
User
Bayern
13
563
G
Bayern
13
564
User
Berlin
13
565
User
Niedersachsen
13
566
User
Bayern
13
567
User
Bayern
13
568
User
Hamburg
13
569
Noah Afkir
Nordrhein-Westfalen
13
570
User
Niedersachsen
13
571
User
Sachsen
13
575
User
Hessen
13
576
User
Berlin
13
578
User
Niedersachsen
13
579
User
Hessen
13
580
User
Hessen
13
582
User
Hessen
13
583
User
Bremen
13
584
User
Bayern
13
586
User
Berlin
13
587
User
Hamburg
13
588
User
Hessen
13
589
User
Niedersachsen
13
592
User
Bayern
13
593
User
Berlin
12
594
User
Berlin
12
597
User
Niedersachsen
12
598
User
Berlin
12
601
User
Bayern
12
602
User
Bayern
12
603
User
Bayern
12
604
User
Bayern
12
607
User
Bayern
12
608
User
Berlin
12
610
User
Berlin
12
611
User
Bayern
12
612
User
Bayern
12
613
User
Hessen
12
615
User
Bayern
12
616
User
Berlin
12
620
User
Berlin
12
624
User
Bayern
11
625
User
Sachsen
11
626
User
Hessen
11
628
Alena Soukupova
Niedersachsen
11
630
Gnedla Abozriba
Berlin
11
632
User
Bayern
11
633
Eugen Klein
Baden-Württemberg
11
634
User
Bayern
11
638
User
Bayern
11
639
User
Berlin
11
641
User
Bayern
11
644
User
Bayern
11
645
User
Berlin
11
647
Tim Wind
Bayern
11
650
User
Hessen
11
652
User
Berlin
11
653
User
Bayern
11
655
User
Berlin
11
657
User
Berlin
11
658
User
Bayern
10
660
User
Hessen
10
661
User
Region
10
662
Hawk 0802
Nordrhein-Westfalen
10
663
User
Bayern
10
664
User
Berlin
10
666
User
Bayern
10
668
Flemca
Hamburg
10
670
User
Hessen
10
674
User
Bayern
10
678
User
Bayern
10
680
User
Hessen
10
681
User
Hessen
10
683
User
Hessen
10
684
User
Niedersachsen
10
685
User
Berlin
10
687
User
Hamburg
10
689
User
Berlin
10
690
Hallucigenia
Hessen
10
692
User
Niedersachsen
10
694
User
Hamburg
10
696
User
Berlin
10
697
User
Bayern
10
701
User
Bayern
9
702
User
Bremen
9
703
User
Berlin
9
705
Omar Afkir
Nordrhein-Westfalen
9
707
User
Berlin
9
710
User
Hessen
9
714
User
Bayern
9
716
User
Hessen
9
718
User
Bayern
9
719
User
Bayern
9
720
whyme
Hessen
9
721
User
Bayern
9
723
User
Berlin
9
724
User
Berlin
9
728
User
Thüringen
9
730
User
Bayern
9
731
User
Sachsen
8
733
User
Berlin
8
734
User
Hamburg
8
736
User
Hessen
8
737
User
Hessen
8
739
User
Hessen
8
742
User
Hessen
8
743
User
Berlin
8
746
User
Hessen
8
747
User
Hessen
8
749
User
Berlin
8
751
User
Bayern
8
754
User
Hessen
8
757
User
Region
7
762
Arjun Sreekumar
Baden-Württemberg
7
769
Jamal
Berlin
7
771
User
Hessen
7
772
User
Sachsen
7
773
User
Hessen
7
775
User
Hamburg
7
776
chris-stud@gmx.de
Schleswig-Holstein
7
777
User
Berlin
7
782
User
Sachsen
7
784
User
Hessen
7
785
User
Berlin
7
788
SaraS
Berlin
7
789
User
Hessen
7
790
User
Berlin
7
792
User
Berlin
7
793
User
Bayern
7
794
User
Bremen
7
795
User
Hessen
7
796
User
Bremen
7
798
User
Hessen
7
802
User
Bayern
6
803
User
Hessen
6
808
User
Bayern
6
809
User
Hessen
6
813
User
Hessen
6
814
User
Bayern
6
815
User
Berlin
6
816
User
Bayern
6
818
User
Bayern
6
819
User
Sachsen
6
821
User
Bayern
6
822
User
Hamburg
6
824
User
Bayern
6
825
User
Bayern
6
826
User
Bayern
6
827
User
Bayern
6
829
User
Hamburg
6
831
User
Hessen
6
832
User
Bayern
6
833
User
Saarland
6
834
Solange
Bayern
6
835
User
Bayern
6
838
User
Bayern
6
839
User
Bayern
6
840
User
Bayern
6
841
User
Bayern
6
842
User
Hessen
6
845
User
Hessen
6
846
User
Bremen
6
847
User
Bayern
6
848
User
Hessen
6
850
User
Bremen
6
856
User
Bayern
5
857
User
Bayern
5
860
User
Hessen
5
863
User
Region
5
864
User
Berlin
5
866
User
Bayern
5
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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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