Leaderboard
Brazil
Brazil
Worldwide
Rank
Level
1
ackuhl2@gmail.com
Rio de Janeiro
96
2
Milan “youngscience” v.de Meeberg
Paraná
94
4
Marcos Cabral
Pernambuco
84
5
Milan
Paraná
82
7
Filipe de Castro
Minas Gerais
80
8
User
Paraná
78
9
User
Paraná
75
10
Gabriel
Paraíba
74
11
User
São Paulo
69
12
User
Paraná
67
16
User
São Paulo
60
17
bruno couto
Rio de Janeiro
58
18
User
Ceará
58
19
User
Amazonas
58
20
Fernando Henrique
Pará
56
21
User
São Paulo
54
22
José Freire
Sergipe
51
23
Tiago Teodoro
Paraná
51
24
Michelle
Rio de Janeiro
51
25
User
Pernambuco
50
26
User
São Paulo
50
27
User
Paraná
50
28
Lyfaenia
Rio Grande do Sul
49
29
victor gouveia
São Paulo
49
30
User
São Paulo
48
31
User
São Paulo
48
32
Roderiko
São Paulo
47
33
User
São Paulo
47
35
Vandré
Paraná
46
36
Lyfaenia
Rio Grande do Sul
45
37
xyz
São Paulo
45
38
Juliano Negri
São Paulo
44
39
Brian Moret
Rio de Janeiro
44
41
User
São Paulo
44
42
Thiago Gulmine
Distrito Federal
42
43
User
São Paulo
41
44
User
Pernambuco
40
45
Eduardo Freire
Distrito Federal
40
46
Fierce Deity
Rio de Janeiro
40
47
User
São Paulo
39
48
User
Ceará
39
50
User
Minas Gerais
39
51
User
Paraná
38
52
User
São Paulo
38
53
User
Paraná
38
55
lucas teles
Rio Grande do Sul
36
56
User
São Paulo
36
57
User
São Paulo
36
59
User
São Paulo
35
60
Loust 055
Amapá
35
61
User
Pará
35
62
Gabriel Küter
Santa Catarina
35
64
User
Paraná
35
65
User
Pernambuco
34
66
User
Paraná
34
67
User
São Paulo
34
68
Gustavo Henrik
São Paulo
33
69
User
Bahia
33
70
User
São Paulo
33
73
User
Pernambuco
32
74
User
São Paulo
32
75
Fabio Ocana Vieira
São Paulo
32
76
User
São Paulo
32
79
User
São Paulo
31
80
Rubia
São Paulo
30
81
User
Minas Gerais
30
83
User
São Paulo
30
84
Leandro Davila
Sergipe
30
85
User
Minas Gerais
29
86
User
Pernambuco
29
87
Taquion
Goiás
29
88
Matheus Rech
São Paulo
29
90
User
Minas Gerais
28
91
User
São Paulo
28
92
Caio Kaercher
São Paulo
28
93
User
São Paulo
28
94
User
São Paulo
28
95
User
Bahia
28
96
User
São Paulo
27
98
Jessica Messias
São Paulo
27
100
User
Pernambuco
26
101
MestreGnomo
São Paulo
26
102
User
São Paulo
26
103
Vinicius de Santana
São Paulo
26
104
User
Minas Gerais
26
105
User
São Paulo
26
108
User
Ceará
25
109
Luis Masuko
Amapá
25
110
User
Maranhão
25
111
Larissa Saori Seino
Paraná
25
112
Juliana Cunha
Rio de Janeiro
25
113
User
São Paulo
24
114
User
São Paulo
24
115
User
São Paulo
24
117
User
São Paulo
24
118
User
Goiás
23
119
Jonathan Costa
Minas Gerais
23
121
User
São Paulo
23
122
Renan W.
São Paulo
23
124
User
Minas Gerais
23
125
User
São Paulo
23
127
Vinicius Fonseca
Rio de Janeiro
22
128
User
São Paulo
22
129
User
São Paulo
22
130
User
Paraná
22
131
User
Minas Gerais
22
132
User
Minas Gerais
22
134
User
São Paulo
21
135
User
Minas Gerais
21
136
User
Goiás
21
137
Eder de Faria
Minas Gerais
21
138
User
São Paulo
21
139
User
Sergipe
21
140
User
Paraná
20
141
User
Paraná
20
142
memes
São Paulo
20
144
User
Minas Gerais
20
145
Paulo Ormonde
São Paulo
20
147
User
São Paulo
20
148
Igor Edington
São Paulo
20
149
User
São Paulo
19
150
Leonardo
Paraná
19
151
Paulo Henrique dos Santos
Rio Grande do Sul
19
152
Bruno
Rio de Janeiro
19
153
User
Paraíba
19
154
User
São Paulo
19
156
User
São Paulo
19
157
geovana p
São Paulo
18
158
User
São Paulo
18
159
User
São Paulo
18
160
Giovana Flores
São Paulo
18
161
User
São Paulo
18
162
math math
Pernambuco
18
164
User
Goiás
17
165
User
São Paulo
17
166
Vinicius
Minas Gerais
17
168
joana wanderley
Paraná
17
169
User
Paraná
17
170
User
Minas Gerais
17
171
User
Minas Gerais
17
173
Lucas Valença
Santa Catarina
16
174
User
Sergipe
16
175
User
São Paulo
16
177
User
São Paulo
16
178
User
Bahia
16
180
Talyta dos Santos Capitan Dias
São Paulo
15
181
Valter M.K.
Maranhão
15
182
User
Pernambuco
15
183
User
São Paulo
15
184
User
Pernambuco
15
185
Best Botter
São Paulo
15
186
User
São Paulo
15
187
User
São Paulo
15
188
User
Maranhão
15
189
User
São Paulo
15
191
arielnc
São Paulo
14
192
User
Amazonas
14
193
User
São Paulo
14
194
User
São Paulo
14
195
User
Minas Gerais
14
196
User
São Paulo
14
197
Joan Manuel Del Valle
Ceará
14
198
User
São Paulo
13
199
User
São Paulo
13
200
Veronica Cantarelli
São Paulo
13
201
User
São Paulo
13
204
User
Bahia
13
205
User
São Paulo
13
207
User
Paraíba
13
210
User
São Paulo
13
211
User
São Paulo
12
212
Rhuan Paulo Lopes Barros
Rio Grande do Sul
12
215
Alex C
São Paulo
12
216
User
São Paulo
12
217
User
Paraná
12
218
Tadeu Moraes
Rio de Janeiro
12
219
User
São Paulo
12
221
Vitor
São Paulo
12
223
User
São Paulo
12
224
User
Paraná
12
225
User
Paraná
12
226
User
São Paulo
12
227
User
Paraná
12
228
User
São Paulo
11
230
User
Tocantins
11
231
User
São Paulo
11
232
User
São Paulo
11
233
Felipe Albarelli
São Paulo
11
234
User
São Paulo
11
235
User
Paraná
11
238
User
Bahia
11
240
User
São Paulo
11
241
User
Paraná
11
242
User
Amazonas
11
243
User
Paraná
11
244
User
São Paulo
11
245
Riquelmy Oliveira
Bahia
11
246
User
São Paulo
11
247
User
Minas Gerais
11
249
User
Minas Gerais
10
250
Cleto
São Paulo
10
251
User
São Paulo
10
252
User
Paraná
10
253
Flyx Hoy
Paraíba
10
255
User
São Paulo
10
256
User
São Paulo
10
257
User
São Paulo
10
258
User
São Paulo
10
259
User
Minas Gerais
10
260
User
Minas Gerais
10
261
User
São Paulo
10
262
User
São Paulo
10
263
User
São Paulo
10
264
Darcy Borg
Region
10
265
User
São Paulo
10
267
Anna
São Paulo
9
268
User
São Paulo
9
269
User
São Paulo
9
271
Gustavo Mazzarollo
Rio Grande do Sul
9
272
User
Pernambuco
9
273
User
São Paulo
9
274
User
São Paulo
9
275
User
Piauí
9
276
Mareval Soares Pereira
São Paulo
9
277
pato
Minas Gerais
9
279
User
São Paulo
9
281
User
Sergipe
9
285
Heitor
São Paulo
9
287
User
Minas Gerais
9
289
User
São Paulo
9
290
User
São Paulo
9
291
User
São Paulo
9
293
User
Paraná
8
294
User
São Paulo
8
296
User
Minas Gerais
8
297
User
São Paulo
8
298
bruno pessoa
Minas Gerais
8
299
User
Paraná
8
300
User
Ceará
8
303
filipe 2357
Paraíba
8
304
User
São Paulo
8
305
User
São Paulo
8
306
User
São Paulo
8
307
User
São Paulo
8
309
User
São Paulo
8
311
User
Mato Grosso
8
312
User
Pernambuco
8
313
Camelzebra
São Paulo
8
314
User
São Paulo
8
316
User
São Paulo
8
317
User
São Paulo
8
318
User
São Paulo
8
319
User
São Paulo
8
320
User
São Paulo
8
321
User
São Paulo
8
322
User
São Paulo
8
323
User
São Paulo
8
324
User
São Paulo
8
325
João Paulo Y.
Paraná
8
326
User
Mato Grosso
7
327
User
São Paulo
7
329
User
São Paulo
7
330
Jonatas Rodrigues
Distrito Federal
7
331
User
São Paulo
7
332
User
São Paulo
7
333
User
São Paulo
7
334
User
São Paulo
7
335
User
Minas Gerais
7
336
User
São Paulo
7
337
User
Pará
7
338
Wlasmir Cavalcanti
Rio de Janeiro
7
340
User
São Paulo
7
341
User
Maranhão
7
342
Gabriel Tiberio
São Paulo
7
343
User
Bahia
7
344
User
São Paulo
7
345
User
São Paulo
7
346
User
Maranhão
7
347
User
Bahia
7
348
User
São Paulo
7
349
User
Minas Gerais
7
350
Mario
São Paulo
6
351
User
São Paulo
6
352
User
Paraná
6
353
User
São Paulo
6
355
User
São Paulo
6
356
User
São Paulo
6
357
User
São Paulo
6
358
User
São Paulo
6
359
User
Goiás
6
360
User
Paraná
6
362
User
Paraíba
6
363
User
São Paulo
6
364
User
São Paulo
6
365
User
São Paulo
6
366
User
Ceará
6
368
User
São Paulo
6
369
User
São Paulo
6
371
User
São Paulo
6
372
User
São Paulo
6
373
User
São Paulo
6
374
User
Sergipe
6
376
User
Paraná
5
377
User
São Paulo
5
378
User
São Paulo
5
379
User
São Paulo
5
380
User
São Paulo
5
382
User
São Paulo
5
384
User
São Paulo
5
385
User
Alagoas
5
386
Marcio Todeschini
São Paulo
5
387
Alisson Santos
Paraná
5
388
User
São Paulo
5
389
User
São Paulo
5
390
User
Piauí
5
391
User
São Paulo
5
392
User
São Paulo
5
393
User
São Paulo
5
394
User
São Paulo
5
396
User
Paraíba
5
397
User
Tocantins
5
398
User
Paraná
5
399
User
São Paulo
5
401
User
Minas Gerais
5
402
User
Minas Gerais
5
403
User
São Paulo
5
404
User
São Paulo
5
405
User
São Paulo
5
406
User
São Paulo
5
408
User
São Paulo
4
409
User
São Paulo
4
410
User
São Paulo
4
411
User
São Paulo
4
412
User
São Paulo
4
413
User
São Paulo
4
414
User
Sergipe
4
415
User
Minas Gerais
4
416
User
Minas Gerais
4
417
User
Goiás
4
418
User
São Paulo
4
422
User
Pernambuco
4
423
User
Bahia
4
424
User
São Paulo
4
425
User
Minas Gerais
4
426
User
São Paulo
4
427
User
Paraná
4
428
User
Minas Gerais
4
429
User
Minas Gerais
4
430
User
São Paulo
4
431
User
São Paulo
4
432
Pedro Cabral
Ceará
4
434
User
São Paulo
4
435
User
São Paulo
4
437
User
São Paulo
4
438
User
Minas Gerais
4
439
User
São Paulo
4
440
User
São Paulo
4
441
User
Bahia
4
442
Patricia de Souza Machado
São Paulo
4
443
User
Paraná
4
444
User
Maranhão
4
445
User
São Paulo
4
446
User
São Paulo
3
447
Julia Campiolo Porto
São Paulo
3
448
User
Pará
3
449
User
São Paulo
3
450
User
São Paulo
3
451
User
Tocantins
3
452
User
São Paulo
3
454
User
São Paulo
3
455
User
São Paulo
3
456
User
São Paulo
3
457
User
Paraíba
3
460
User
São Paulo
3
461
User
Ceará
3
462
User
Tocantins
3
466
Vandre
Paraná
3
467
User
São Paulo
3
470
User
São Paulo
3
471
User
Bahia
3
472
User
Bahia
3
473
User
São Paulo
3
474
User
Pernambuco
3
475
User
São Paulo
3
476
User
São Paulo
3
477
User
Minas Gerais
3
479
User
Paraná
3
480
User
Bahia
3
481
User
Paraná
3
483
User
São Paulo
3
484
User
São Paulo
3
485
User
São Paulo
3
486
User
São Paulo
3
487
User
São Paulo
3
488
User
São Paulo
3
489
User
São Paulo
3
490
User
São Paulo
3
491
User
São Paulo
3
492
User
São Paulo
3
494
User
Minas Gerais
3
496
User
Paraná
3
497
User
São Paulo
3
498
User
Minas Gerais
3
499
User
Paraná
3
500
User
Paraná
3
501
User
São Paulo
2
502
User
São Paulo
2
503
User
São Paulo
2
504
User
Pernambuco
2
505
User
São Paulo
2
507
User
São Paulo
2
508
User
São Paulo
2
509
Luana Castro
Minas Gerais
2
511
User
São Paulo
2
512
User
São Paulo
2
513
User
São Paulo
2
514
User
São Paulo
2
515
User
Ceará
2
516
User
São Paulo
2
517
Camila Vecchi
São Paulo
2
518
User
Paraíba
2
519
User
Sergipe
2
520
User
São Paulo
2
521
Ruan Silva
Ceará
2
522
User
São Paulo
2
523
User
Paraná
2
524
User
São Paulo
2
525
User
Amazonas
2
527
User
São Paulo
2
529
User
Goiás
2
530
User
São Paulo
2
531
User
Amazonas
2
535
User
Sergipe
2
536
User
Sergipe
2
537
User
Paraná
2
538
Eduardo Marchetti
Santa Catarina
2
539
User
Bahia
2
540
User
São Paulo
2
543
User
São Paulo
2
544
User
São Paulo
2
545
User
Pará
2
546
User
São Paulo
2
547
User
São Paulo
2
548
User
São Paulo
2
549
rAPHAEL
Goiás
2
550
User
Minas Gerais
2
551
User
São Paulo
2
552
User
São Paulo
2
553
User
São Paulo
2
554
User
São Paulo
2
555
User
São Paulo
2
556
User
São Paulo
2
559
User
São Paulo
2
560
User
Paraná
2
561
User
Paraná
2
563
User
São Paulo
2
564
User
Paraná
2
565
User
São Paulo
2
567
User
Minas Gerais
2
569
User
São Paulo
2
572
User
Minas Gerais
2
573
User
São Paulo
2
574
User
São Paulo
2
575
User
São Paulo
1
576
User
Paraná
1
577
User
São Paulo
1
578
User
São Paulo
1
581
Vinicius Willer Alencar Haas
Distrito Federal
1
582
User
Pará
1
584
User
Paraná
1
586
User
Pará
1
587
User
Sergipe
1
588
User
Paraíba
1
589
User
Pernambuco
1
590
User
Paraná
1
591
User
Bahia
1
592
Lucio Neto
Goiás
1
593
User
São Paulo
1
594
User
São Paulo
1
595
User
Ceará
1
596
User
Bahia
1
598
User
São Paulo
1
600
User
São Paulo
1
601
User
Paraná
1
602
User
Minas Gerais
1
604
User
Pernambuco
1
605
User
São Paulo
1
606
User
São Paulo
1
607
User
Minas Gerais
1
608
User
Goiás
1
609
User
São Paulo
1
610
User
São Paulo
1
611
User
Paraná
1
612
User
Mato Grosso
1
613
User
Paraíba
1
614
João Dini
São Paulo
1
615
Luiz Farias
Distrito Federal
1
616
CAMILA BRETTAS
São Paulo
1
617
Investor br
São Paulo
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 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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