Leaderboard
Portugal
Portugal
Worldwide
Rank
Level
1
Miguel Juncais
Lisboa
97
2
Mário Gomes
Porto
92
3
João Gomes
Lisboa
90
4
MartimG
Lisboa
84
5
vitor shadows
Lisboa
81
6
Agualva Cacem
Lisboa
80
7
User
Lisboa
74
8
vitor shadows
Lisboa
71
9
El Barto
Porto
70
10
Dinis Santos
Leiria
69
11
User
Lisboa
65
12
User
Lisboa
62
13
User
Lisboa
61
14
User
Aveiro
58
15
Gonçalo Ganhão
Lisboa
58
16
Vitor Hugo
Aveiro
57
17
User
Lisboa
56
18
User
Lisboa
56
19
sff asfd
Faro
56
20
Lucas
Lisboa
55
21
BichodoMato
Porto
54
22
Vinicius Fernandes
Faro
53
23
User
Lisboa
51
24
User
Lisboa
51
25
User
Lisboa
51
26
João Alves
Lisboa
50
27
User
Lisboa
50
28
User
Lisboa
50
29
Roberto Cortez
Porto
49
30
User
Lisboa
48
31
User
Lisboa
48
32
Miguel Vasconcelos
Lisboa
48
33
lawi
Aveiro
48
34
User
Lisboa
47
35
Gayle
Porto
45
36
Samuel Wejchert
Faro
44
37
TomTom
Setúbal
44
38
Tomás Santos
Lisboa
44
39
User
Lisboa
44
40
User
Lisboa
42
41
Ricardo Correia
Aveiro
41
43
Tiago
Setúbal
38
44
User
Faro
38
45
Inês Martins
Lisboa
37
46
Fernando Bacelar de Sousa
Lisboa
36
47
Pedro
Lisboa
35
48
User
Lisboa
35
49
User
Aveiro
35
50
User
Lisboa
34
51
Afonso CV
Castelo Branco
33
52
User
Lisboa
33
53
User
Faro
33
54
User
Porto
33
55
User
Viseu
32
56
Carolina Paes de Faria
Lisboa
31
57
João Tiago Bairras Gorgulho
Porto
31
58
User
Porto
31
59
b 8444
Lisboa
30
60
Phil
Évora
30
61
User
Lisboa
30
62
User
Santarém
30
63
User
Lisboa
29
64
User
Lisboa
29
65
User
Porto
29
66
cha erg
Setúbal
29
67
User
Faro
29
68
User
Porto
28
69
User
Porto
28
70
User
Aveiro
28
71
User
Lisboa
28
72
User
Porto
28
73
User
Porto
27
74
User
Lisboa
27
75
User
Porto
27
76
Tiago Morais
Porto
27
77
User
Lisboa
27
78
User
Santarém
27
79
User
Porto
26
80
User
Faro
26
81
User
Porto
26
82
User
Porto
26
83
User
Porto
25
84
User
Lisboa
25
85
User
Porto
25
86
User
Lisboa
25
87
User
Porto
25
88
User
Lisboa
24
89
User
Porto
24
90
User
Porto
24
91
User
Porto
23
92
User
Portalegre
23
93
User
Portalegre
23
94
User
Porto
23
95
User
Porto
23
96
User
Faro
23
97
User
Porto
23
98
User
Porto
22
99
User
Porto
22
100
User
Porto
22
101
SyNThw8Ve
Évora
22
102
User
Porto
22
103
User
Coimbra
22
104
User
Porto
22
105
User
Porto
21
106
User
Porto
21
107
User
Porto
21
108
User
Porto
21
109
User
Porto
21
110
User
Porto
21
111
User
Porto
21
112
User
Porto
21
113
User
Porto
21
114
User
Faro
21
115
User
Porto
21
116
Vilgeforte Águas-Santas
Lisboa
20
117
User
Porto
20
118
User
Porto
20
119
User
Porto
20
120
User
Porto
20
121
User
Porto
20
122
User
Porto
20
123
User
Porto
20
124
User
Porto
20
125
Solange Silva
Lisboa
20
126
User
Porto
20
127
User
Porto
19
128
User
Lisboa
19
129
User
Porto
19
130
Joaquim Silva
Lisboa
19
132
User
Porto
19
133
User
Lisboa
19
134
User
Lisboa
19
135
User
Porto
19
136
User
Lisboa
19
137
User
Lisboa
19
138
User
Setúbal
19
139
User
Lisboa
19
140
User
Porto
19
141
User
Porto
18
142
User
Porto
18
143
User
Porto
18
144
User
Lisboa
18
145
User
Porto
18
146
User
Lisboa
18
147
User
Porto
18
148
User
Porto
18
149
User
Porto
18
150
User
Porto
18
151
User
Lisboa
18
152
Shahrukh khan
Lisboa
18
153
User
Porto
18
154
User
Porto
18
155
User
Porto
18
156
User
Porto
18
157
User
Porto
18
158
User
Porto
18
159
User
Porto
18
160
User
Porto
17
161
User
Porto
17
162
User
Lisboa
17
163
User
Lisboa
17
164
User
Bragança
17
165
User
Porto
17
166
User
Porto
17
167
User
Lisboa
17
168
User
Porto
17
169
User
Lisboa
17
170
User
Porto
17
171
User
Lisboa
17
172
User
Porto
17
173
Afonso Miranda
Porto
17
174
User
Porto
17
175
User
Porto
17
176
User
Porto
17
177
User
Porto
17
178
User
Porto
17
179
User
Porto
17
180
User
Porto
17
181
User
Porto
17
182
User
Porto
17
183
User
Porto
17
184
User
Porto
16
185
User
Setúbal
16
186
User
Porto
16
187
User
Porto
16
188
User
Porto
16
189
User
Porto
16
190
User
Porto
16
191
User
Porto
16
192
User
Aveiro
16
193
User
Porto
16
194
NM
Lisboa
16
195
User
Lisboa
16
196
User
Porto
16
197
User
Porto
16
198
User
Lisboa
16
199
User
Porto
16
200
User
Porto
16
201
User
Porto
16
202
User
Porto
16
203
User
Setúbal
15
204
User
Faro
15
205
User
Lisboa
15
206
User
Porto
15
207
User
Porto
15
208
Pedro Caeiro
Lisboa
15
209
User
Lisboa
15
210
User
Porto
15
211
User
Porto
15
212
User
Porto
15
213
User
Porto
15
214
User
Braga
15
215
User
Porto
15
216
Hes pt
Braga
15
217
User
Lisboa
15
218
User
Porto
15
219
User
Lisboa
15
220
User
Lisboa
15
221
User
Lisboa
15
222
User
Porto
15
223
User
Lisboa
15
224
User
Porto
15
225
User
Porto
15
226
User
Porto
15
227
User
Porto
15
228
User
Porto
15
229
User
Setúbal
15
230
User
Lisboa
15
231
Paulo Pesqueira
Lisboa
15
232
User
Lisboa
15
233
User
Porto
14
234
User
Porto
14
235
User
Porto
14
236
User
Porto
14
237
User
Porto
14
238
User
Lisboa
14
239
User
Porto
14
240
User
Porto
14
241
User
Vila Real
14
242
User
Porto
14
243
joao.miguel
Setúbal
14
244
User
Porto
14
245
User
Vila Real
14
246
User
Lisboa
14
247
User
Porto
14
248
User
Lisboa
14
249
User
Lisboa
14
250
User
Porto
14
251
User
Lisboa
14
252
User
Porto
14
253
User
Lisboa
14
254
User
Lisboa
14
255
User
Lisboa
14
256
User
Faro
14
257
User
Porto
14
258
User
Coimbra
14
259
User
Lisboa
14
260
User
Porto
14
261
User
Lisboa
14
262
User
Lisboa
14
263
User
Porto
14
264
User
Porto
14
265
User
Coimbra
14
266
User
Porto
14
267
User
Lisboa
14
268
User
Lisboa
14
269
User
Porto
14
270
User
Porto
14
271
User
Lisboa
14
272
User
Porto
14
273
User
Lisboa
14
274
Inês Machado
Porto
14
275
User
Porto
13
276
User
Porto
13
277
adr
Lisboa
13
278
User
Lisboa
13
279
User
Lisboa
13
280
User
Porto
13
281
User
Porto
13
282
User
Porto
13
283
User
Porto
13
284
User
Lisboa
13
285
User
Porto
13
286
User
Santarém
13
287
User
Porto
13
288
User
Porto
13
289
User
Lisboa
13
290
User
Porto
13
291
User
Porto
13
292
User
Porto
13
293
User
Porto
13
294
User
Porto
13
295
User
Porto
13
296
User
Porto
13
297
User
Aveiro
13
298
User
Lisboa
13
299
User
Coimbra
13
300
User
Porto
13
301
User
Porto
12
302
User
Santarém
12
303
User
Porto
12
304
User
Porto
12
305
User
Porto
12
306
User
Porto
12
307
User
Porto
12
308
User
Porto
12
309
User
Porto
12
310
User
Porto
12
311
Matthew Brews
Faro
12
312
edmond turquieh
Santarém
12
313
User
Porto
12
314
User
Porto
12
315
User
Porto
12
316
Derek Brennan
Lisboa
12
317
User
Lisboa
12
318
User
Coimbra
12
319
User
Porto
12
320
User
Porto
12
321
User
Lisboa
12
322
User
Porto
12
323
User
Porto
12
324
User
Porto
12
325
User
Porto
12
326
User
Porto
12
327
User
Porto
12
328
User
Lisboa
12
329
User
Porto
12
330
User
Porto
12
331
Tiago Cabo
Porto
12
332
User
Porto
11
333
User
Lisboa
11
334
User
Lisboa
11
335
User
Porto
11
336
User
Porto
11
337
User
Lisboa
11
338
User
Porto
11
339
User
Lisboa
11
340
User
Porto
11
341
User
Porto
11
342
User
Faro
11
343
User
Lisboa
11
344
User
Lisboa
11
345
User
Porto
11
346
User
Porto
11
347
User
Lisboa
11
348
User
Lisboa
11
349
User
Porto
11
350
User
Porto
11
351
User
Porto
11
352
User
Porto
11
353
User
Lisboa
11
354
User
Porto
11
355
User
Lisboa
11
356
Iara Serbaneci
Faro
11
357
User
Porto
11
358
User
Porto
11
359
User
Porto
11
360
User
Porto
11
361
User
Porto
11
362
Francisco Alves
Porto
11
363
User
Porto
10
364
User
Lisboa
10
365
User
Porto
10
366
GPB
Setúbal
10
367
User
Porto
10
368
User
Porto
10
369
User
Porto
10
370
User
Lisboa
10
371
User
Lisboa
10
372
User
Porto
10
373
User
Porto
10
374
mumbasa
Lisboa
10
375
User
Porto
10
376
User
Lisboa
10
377
User
Lisboa
10
378
User
Santarém
10
379
User
Lisboa
10
380
User
Porto
10
381
User
Leiria
10
382
User
Porto
10
383
User
Lisboa
10
384
User
Porto
10
385
User
Porto
10
386
User
Porto
10
387
User
Porto
10
388
User
Porto
10
389
User
Porto
10
390
User
Porto
10
391
User
Porto
10
392
User
Porto
10
393
User
Porto
10
394
User
Porto
9
395
User
Porto
9
396
User
Porto
9
397
User
Porto
9
398
User
Porto
9
399
User
Porto
9
400
User
Porto
9
401
User
Porto
9
402
User
Porto
9
403
User
Porto
9
404
Catarina Guerreiro
Lisboa
9
405
User
Porto
9
406
User
Lisboa
9
407
Simao Sarmento
Lisboa
9
408
User
Porto
9
409
User
Porto
9
410
User
Porto
9
411
User
Lisboa
9
412
Andreia Nogueira
Porto
9
413
User
Porto
9
414
User
Porto
9
415
Mariana Pitôrra
Lisboa
9
416
henrique proenca
Lisboa
9
417
User
Porto
9
418
User
Lisboa
9
419
User
Lisboa
9
420
User
Porto
9
421
User
Lisboa
9
422
User
Porto
9
423
User
Lisboa
8
424
Teresa Sa
Lisboa
8
425
User
Porto
8
426
User
Porto
8
427
P. M.
Lisboa
8
428
User
Lisboa
8
429
User
Lisboa
8
430
User
Porto
8
431
User
Porto
8
432
User
Porto
8
433
User
Porto
8
434
User
Porto
8
435
User
Porto
8
436
User
Porto
8
437
User
Lisboa
8
438
User
Lisboa
8
439
User
Lisboa
8
440
User
Porto
8
441
Afonso Elvas
Porto
8
442
User
Porto
8
443
User
Lisboa
8
444
User
Porto
8
445
User
Lisboa
8
446
User
Porto
8
447
Kimi
Porto
8
448
User
Porto
8
449
User
Porto
8
450
User
Lisboa
8
451
User
Porto
8
452
User
Porto
8
453
User
Lisboa
8
454
Vanessa Muteia
Lisboa
8
455
User
Porto
8
456
User
Porto
8
457
User
Porto
7
458
User
Porto
7
459
User
Porto
7
460
User
Lisboa
7
461
User
Porto
7
462
User
Porto
7
463
User
Lisboa
7
464
User
Porto
7
465
User
Porto
7
466
User
Lisboa
7
467
User
Porto
7
468
User
Porto
7
469
User
Faro
7
470
User
Lisboa
7
471
User
Setúbal
7
472
User
Porto
7
473
User
Porto
7
474
User
Porto
7
475
User
Lisboa
7
476
User
Porto
7
477
User
Lisboa
7
478
User
Lisboa
7
479
User
Porto
7
480
User
Lisboa
7
481
User
Coimbra
7
482
User
Porto
7
483
User
Porto
7
484
User
Porto
7
485
User
Porto
7
486
User
Porto
7
487
User
Lisboa
7
488
User
Porto
7
489
User
Porto
7
490
Francis M
Bragança
7
491
User
Porto
7
492
User
Porto
7
493
User
Setúbal
7
494
User
Porto
7
495
User
Braga
7
496
User
Lisboa
7
497
User
Porto
7
498
User
Porto
7
499
User
Porto
7
500
User
Porto
7
501
User
Porto
7
502
User
Lisboa
7
503
User
Porto
7
504
User
Lisboa
6
505
User
Porto
6
506
User
Porto
6
507
User
Porto
6
508
User
Vila Real
6
509
User
Porto
6
510
User
Lisboa
6
511
User
Porto
6
512
User
Lisboa
6
513
User
Porto
6
514
User
Porto
6
515
User
Coimbra
6
516
User
Porto
6
517
XGariex
Lisboa
6
518
User
Porto
6
519
User
Porto
6
520
User
Porto
6
521
User
Porto
6
522
Rodrigo Maia da Silva
Região Autónoma dos Açores
6
523
User
Porto
6
524
User
Lisboa
6
525
User
Porto
6
526
User
Porto
6
527
User
Porto
6
528
User
Lisboa
6
529
User
Porto
6
530
User
Porto
6
531
User
Porto
6
532
User
Lisboa
6
533
User
Porto
6
534
User
Bragança
6
535
User
Lisboa
6
536
User
Porto
6
537
User
Lisboa
6
538
User
Porto
6
539
User
Porto
6
540
User
Lisboa
6
541
User
Lisboa
6
542
User
Porto
6
543
User
Braga
6
544
User
Porto
6
545
User
Lisboa
6
546
Matilde Pitôrra
Lisboa
6
547
User
Porto
6
548
User
Porto
6
549
User
Porto
6
550
User
Porto
6
551
User
Lisboa
6
552
User
Setúbal
6
553
User
Porto
6
554
User
Porto
6
555
User
Porto
5
556
User
Coimbra
5
557
User
Porto
5
558
User
Porto
5
559
User
Porto
5
560
User
Porto
5
561
User
Porto
5
562
User
Porto
5
563
User
Porto
5
564
User
Lisboa
5
565
User
Lisboa
5
566
Norin Stevensky
Porto
5
567
User
Porto
5
568
User
Porto
5
569
User
Porto
5
570
EzYRa
Faro
5
571
User
Porto
5
572
User
Porto
5
573
User
Lisboa
5
574
User
Lisboa
5
575
User
Porto
5
576
User
Porto
5
577
User
Lisboa
5
578
User
Porto
5
579
Calvin Duarte
Lisboa
5
580
User
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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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