Leaderboard
Switzerland
Switzerland
Worldwide
Rank
Level
1
Bunk McGregor
Vaud
84
2
Gaetan Cretton
Vaud
82
3
Sandro
Sankt Gallen
77
4
Bruno
Zug
77
5
JovBla
Luzern
74
6
Trade Ticker
Aargau
72
7
Lucas Alava
Zürich
70
8
Joel
Zürich
70
9
tom mueller
Zürich
67
10
Seb
Vaud
64
11
User
Luzern
63
12
User
Zürich
60
13
Sean
Genève
60
14
User
Sankt Gallen
59
15
User
Genève
57
16
Jomini
Genève
55
17
User
Aargau
55
18
SneakyRobot
Basel-Landschaft
55
19
Lo Dro
Genève
55
20
User
Neuchâtel
53
21
User
Graubünden
52
22
User
Zürich
52
23
User
Zürich
51
24
Xingyu Yang
Zürich
51
25
User
Zürich
50
26
Torsten Hendrich
Zürich
48
27
Lamparsky
Zürich
48
28
User
Genève
47
29
User
Genève
46
30
EP
Genève
46
31
Pepe
Zürich
46
32
Hönzger
Aargau
45
33
Miklós Horváth
Zürich
45
34
User
Vaud
45
35
User
Neuchâtel
44
36
User
Zürich
44
37
Thebananana
Zürich
43
38
User
Zürich
42
39
User
Valais
42
40
User
Basel-Stadt
41
41
AA BBB
Vaud
41
42
Yanli Wang
Vaud
40
43
Gianmarco Fabbri
Zug
40
44
User
Zürich
40
45
User
Zürich
40
46
User
Zürich
40
47
tschu gui
Genève
40
48
User
Neuchâtel
40
49
Attila Mathe
Zürich
39
50
Matt
Genève
38
51
User
Aargau
35
52
Bernt Nicht
Sankt Gallen
35
53
User
Zürich
35
54
User
Vaud
32
55
User
Zürich
31
56
Phoomraphee Luenam
Zürich
30
57
SRA
Zürich
29
58
User
Zürich
29
59
User
Zürich
28
60
User
Zürich
27
61
Joshua Rohrer
Genève
27
62
sonewone
Genève
26
63
User
Zürich
25
64
User
Bern
25
65
User
Zug
25
66
Bramart
Zürich
25
67
User
Vaud
25
68
User
Sankt Gallen
24
69
User
Vaud
24
70
User
Neuchâtel
24
71
Boris Sangiorgio
Sankt Gallen
23
72
E_4B-2_J EFFAH
Genève
22
73
Louisa
Bern
22
74
SeaDiamond
Vaud
22
75
knuddelbaer
Zürich
22
76
User
Zürich
22
77
User
Zürich
22
78
User
Zürich
22
79
User
Vaud
21
80
User
Neuchâtel
21
81
User
Sankt Gallen
21
82
User
Zürich
21
83
Clau Soliva
Sankt Gallen
21
84
ChessKing
Genève
20
85
User
Zürich
20
86
Y A
Zürich
20
87
User
Zürich
20
88
User
Zürich
20
89
User
Zürich
20
90
Andy Bahou
Zürich
20
91
User
Genève
19
92
User
Zürich
19
93
User
Sankt Gallen
19
94
User
Zürich
19
95
User
Genève
19
96
User
Bern
19
97
User
Basel-Stadt
18
98
User
Zürich
18
99
User
Vaud
18
100
User
Neuchâtel
18
101
User
Zürich
18
102
User
Zürich
18
103
User
Zürich
17
104
User
Bern
17
105
User
Zürich
17
106
User
Ticino
17
107
User
Zürich
17
108
User
Zürich
17
109
User
Luzern
17
110
Mohamed Belguendouz
Zürich
16
111
User
Genève
16
112
User
Vaud
16
113
User
Zürich
16
114
User
Zürich
16
115
User
Zürich
16
116
User
Vaud
16
117
Mark (Yu-Cheng) Cheng
Vaud
16
118
User
Vaud
16
119
User
Aargau
16
120
User
Zürich
16
121
User
Vaud
16
122
User
Zürich
15
123
User
Graubünden
15
124
User
Zürich
15
125
User
Vaud
15
126
User
Zürich
15
127
User
Zürich
15
128
User
Zürich
15
129
User
Zürich
15
130
User
Valais
15
131
tty 1
Zürich
14
132
User
Bern
14
133
User
Genève
14
134
User
Zürich
14
135
Joona Soratie
Vaud
14
136
User
Zürich
14
137
User
Genève
14
138
User
Zürich
14
139
User
Zürich
13
140
User
Zürich
13
141
Soraya
Zürich
13
142
User
Genève
13
143
User
Vaud
13
144
User
Zürich
13
145
User
Vaud
13
146
User
Bern
13
147
User
Zürich
13
148
User
Aargau
13
149
Jeton
Zürich
13
150
User
Zürich
13
151
User
Valais
13
152
User
Thurgau
13
153
User
Vaud
13
154
User
Bern
13
155
Alix Dups
Freiburg
12
156
Lizzy Lisander
Zürich
12
157
User
Zürich
12
158
User
Valais
12
159
Andy
Zürich
12
160
User
Zürich
12
161
Harry Tuttle
Basel-Landschaft
12
162
jean-mickael nounahon
Zürich
12
163
User
Zürich
12
164
dcdavid84
Zürich
12
165
User
Genève
12
166
User
Genève
12
167
User
Bern
12
168
User
Zürich
12
169
User
Zürich
12
170
User
Vaud
12
171
User
Aargau
12
172
User
Zürich
12
173
User
Genève
12
174
User
Zürich
12
175
User
Obwalden
12
176
User
Basel-Stadt
12
177
User
Zürich
11
178
User
Zürich
11
179
User
Genève
11
180
User
Zürich
11
181
User
Genève
11
182
User
Genève
11
183
User
Sankt Gallen
11
184
User
Vaud
11
185
User
Genève
11
186
User
Zürich
11
187
User
Genève
11
188
User
Vaud
11
189
User
Vaud
11
190
User
Zürich
11
191
User
Zürich
10
192
User
Zürich
10
193
Lou
Vaud
10
194
User
Neuchâtel
10
195
User
Zürich
10
196
User
Basel-Stadt
10
197
User
Vaud
10
198
Mal sar
Zürich
10
199
User
Genève
10
200
User
Vaud
10
201
User
Zürich
10
202
User
Zürich
10
203
User
Zürich
10
204
User
Bern
10
205
TheMaster
Luzern
9
206
User
Zug
9
207
Alex M
Zürich
9
208
User
Basel-Stadt
9
209
User
Aargau
9
210
User
Vaud
9
211
User
Vaud
9
212
Priscilla
Sankt Gallen
9
213
User
Aargau
9
215
User
Genève
9
216
Michal Grabowski
Vaud
9
217
User
Vaud
9
218
User
Valais
9
219
User
Zürich
9
220
Charlotte Sertic
Zürich
9
221
User
Genève
9
222
User
Ticino
8
223
User
Genève
8
224
User
Genève
8
225
User
Vaud
8
226
User
Valais
8
227
User
Sankt Gallen
8
228
User
Vaud
8
229
User
Genève
8
230
User
Vaud
8
231
User
Sankt Gallen
8
232
andy
Zürich
8
233
User
Aargau
8
234
User
Valais
8
235
Tomas Carbo Medem
Vaud
8
236
Kopanda
Genève
8
237
User
Sankt Gallen
8
238
User
Genève
8
239
User
Zürich
8
240
User
Zürich
8
241
User
Zürich
8
242
User
Zürich
8
243
Honzger
Zürich
8
244
User
Vaud
8
245
Raphaël Cardinaux
Vaud
8
246
User
Zürich
8
247
User
Vaud
8
248
User
Zürich
8
249
User
Ticino
8
250
User
Vaud
7
251
User
Genève
7
252
User
Zürich
7
253
User
Vaud
7
254
User
Basel-Stadt
7
255
User
Vaud
7
256
User
Zürich
7
257
User
Genève
7
258
User
Zürich
7
259
User
Vaud
7
260
User
Zürich
7
261
User
Vaud
7
262
Mònica Morera Ruocco
Basel-Stadt
7
263
User
Vaud
7
264
Night Thief
Zürich
7
265
User
Genève
7
266
User
Zürich
7
267
User
Vaud
7
268
User
Zürich
7
269
User
Zürich
7
270
User
Zürich
7
271
User
Luzern
7
272
User
Zürich
7
273
User
Bern
7
274
User
Valais
6
275
User
Genève
6
276
User
Vaud
6
277
User
Basel-Stadt
6
278
User
Zürich
6
279
User
Solothurn
6
280
Mingjie Ma
Vaud
6
281
User
Genève
6
282
User
Genève
6
283
Owen Gibbs
Genève
6
284
User
Vaud
6
285
User
Basel-Stadt
6
286
User
Zug
6
287
User
Vaud
6
288
User
Genève
6
289
User
Genève
6
290
User
Genève
6
291
User
Zürich
6
292
User
Genève
6
293
virgil
Zürich
6
294
User
Genève
6
295
User
Vaud
6
296
User
Valais
6
297
User
Genève
5
298
User
Zürich
5
299
User
Genève
5
300
User
Zürich
5
301
User
Zürich
5
302
User
Vaud
5
303
User
Zürich
5
304
User
Freiburg
5
305
User
Aargau
5
306
User
Thurgau
5
307
User
Zug
5
308
User
Aargau
5
309
User
Vaud
5
310
User
Vaud
5
311
User
Genève
5
312
User
Vaud
5
313
User
Vaud
5
314
User
Zürich
5
315
User
Genève
5
316
User
Zürich
5
317
User
Genève
5
318
User
Zürich
5
319
User
Sankt Gallen
5
320
User
Vaud
5
321
User
Basel-Stadt
5
322
User
Zürich
5
323
User
Genève
5
324
User
Vaud
5
325
User
Basel-Stadt
5
326
User
Vaud
5
327
Jackson Jackson
Vaud
5
328
User
Zürich
5
329
User
Zürich
5
330
User
Zürich
5
331
User
Zürich
4
332
User
Luzern
4
333
User
Freiburg
4
334
Matt
Vaud
4
335
User
Freiburg
4
336
User
Genève
4
337
User
Sankt Gallen
4
338
User
Zürich
4
339
User
Genève
4
340
User
Zürich
4
341
User
Vaud
4
342
User
Zürich
4
343
User
Vaud
4
344
User
Vaud
4
345
User
Vaud
4
346
User
Zürich
4
347
Andrey Tolkachev
Zürich
4
348
User
Vaud
4
349
User
Zürich
4
350
User
Vaud
4
351
User
Neuchâtel
4
352
User
Vaud
4
353
User
Zürich
4
354
User
Vaud
4
355
User
Vaud
4
356
User
Genève
3
357
User
Zürich
3
358
User
Zürich
3
359
User
Sankt Gallen
3
360
User
Vaud
3
361
User
Zürich
3
362
User
Bern
3
363
User
Sankt Gallen
3
364
User
Zürich
3
365
User
Genève
3
366
User
Genève
3
367
IMA
Bern
3
368
User
Zürich
3
369
User
Zürich
3
370
User
Bern
3
371
User
Vaud
3
372
User
Luzern
3
373
User
Sankt Gallen
3
374
User
Zürich
3
375
User
Zürich
3
376
User
Aargau
3
377
Daniel Bucher
Luzern
3
378
User
Zürich
3
379
User
Vaud
3
380
User
Zürich
3
381
User
Genève
3
382
User
Zürich
3
383
User
Zürich
3
384
User
Vaud
3
385
User
Vaud
3
386
User
Zürich
3
387
User
Zürich
3
388
User
Zürich
3
389
User
Sankt Gallen
3
390
User
Zürich
3
391
User
Zug
3
392
User
Aargau
2
393
User
Vaud
2
394
User
Sankt Gallen
2
395
User
Zürich
2
396
User
Vaud
2
397
User
Zürich
2
398
User
Vaud
2
399
User
Genève
2
400
User
Vaud
2
401
User
Zürich
2
402
User
Zürich
2
403
User
Vaud
2
404
User
Vaud
2
405
Natalie Tommey
Vaud
2
406
User
Zug
2
407
User
Genève
2
408
MinnieMouse
Zürich
2
409
Radinka Siddiqi
Genève
2
411
User
Zürich
2
412
User
Zürich
2
413
Camille
Zürich
2
414
User
Vaud
2
415
User
Sankt Gallen
2
416
User
Bern
2
417
User
Solothurn
2
418
User
Aargau
2
419
User
Zürich
2
420
User
Zug
2
421
User
Zürich
2
422
User
Vaud
2
423
User
Zürich
2
424
User
Zürich
2
425
User
Vaud
2
426
User
Vaud
2
427
User
Bern
2
428
User
Zürich
2
429
User
Ticino
2
430
User
Zürich
2
431
User
Aargau
2
432
User
Zürich
2
433
User
Zürich
2
434
User
Vaud
2
435
User
Vaud
2
436
User
Zürich
2
437
User
Vaud
2
438
User
Zürich
2
439
User
Vaud
2
440
User
Zürich
2
441
User
Zürich
2
442
User
Genève
2
443
User
Vaud
2
444
User
Zürich
2
445
User
Vaud
2
446
User
Sankt Gallen
2
447
User
Zürich
2
448
User
Luzern
2
449
User
Vaud
2
450
User
Genève
2
451
User
Bern
2
452
Georg Felber
Thurgau
2
453
User
Zürich
2
454
User
Zürich
1
455
User
Zürich
1
456
User
Zürich
1
457
User
Genève
1
458
User
Vaud
1
459
AI
Genève
1
460
User
Zürich
1
461
User
Sankt Gallen
1
462
User
Freiburg
1
463
User
Zürich
1
464
User
Vaud
1
465
User
Zürich
1
466
User
Zürich
1
467
User
Luzern
1
468
User
Valais
1
469
John Byde
Vaud
1
470
User
Basel-Stadt
1
471
HappySquid
Zug
1
472
User
Zürich
1
473
User
Zürich
1
474
User
Valais
1
475
User
Aargau
1
476
User
Vaud
1
477
User
Vaud
1
478
User
Sankt Gallen
1
479
User
Bern
1
480
User
Thurgau
1
481
User
Vaud
1
482
User
Ticino
1
483
User
Aargau
1
484
User
Zürich
1
485
User
Zürich
1
486
User
Zürich
1
487
User
Vaud
1
488
User
Vaud
1
489
User
Genève
1
490
User
Genève
1
491
User
Ticino
1
492
User
Zürich
1
493
User
Solothurn
1
494
User
Genève
1
495
User
Vaud
1
496
User
Vaud
1
497
User
Obwalden
1
498
User
Basel-Stadt
1
499
User
Zürich
1
500
User
Vaud
1
501
User
Zürich
1
502
User
Vaud
1
503
User
Zürich
1
504
Chey
Zürich
1
505
User
Zürich
1
506
coco
Zürich
1
507
User
Zürich
1
508
User
Valais
1
509
User
Sankt Gallen
1
510
User
Vaud
1
511
User
Vaud
1
512
User
Genève
1
513
User
Genève
1
514
User
Zürich
1
515
User
Genève
1
516
User
Vaud
1
517
User
Vaud
1
518
User
Zürich
1
519
User
Zürich
1
520
User
Freiburg
1
521
User
Vaud
1
522
User
Zürich
1
523
User
Zürich
1
524
User
Genève
1
525
User
Bern
1
526
User
Zürich
1
527
User
Zürich
1
528
User
Zürich
1
529
User
Zürich
1
530
User
Vaud
1
531
User
Valais
1
532
User
Zürich
1
533
User
Zürich
1
534
User
Zug
1
535
User
Thurgau
1
536
User
Sankt Gallen
1
537
User
Vaud
1
538
User
Vaud
1
539
User
Zürich
1
540
User
Genève
1
541
User
Genève
1
542
User
Genève
1
543
User
Zürich
1
544
User
Freiburg
1
545
User
Zürich
1
546
User
Zürich
1
547
User
Vaud
1
548
User
Genève
1
549
User
Zürich
1
550
User
Zürich
1
551
User
Zürich
1
552
User
Genève
1
553
User
Vaud
1
554
User
Schwyz
1
555
User
Aargau
1
556
User
Vaud
1
557
User
Zürich
1
558
User
Zürich
1
559
User
Valais
1
560
User
Luzern
1

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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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