Leaderboard
Italy
Italy
Worldwide
Rank
Level
1
Sergio Iommi
Abruzzo
95
2
Michele Celentano
Lazio
91
3
User
Piemonte
83
4
User
Lombardia
83
5
Davide Accordini
Veneto
82
6
User
Lazio
81
7
User
Lombardia
80
8
Alberto Piva
Lombardia
75
9
skandro the comeback
Lombardia
75
10
Alessandro Naia
Lombardia
73
11
Angelo Velardi
Sicilia
71
12
User
Lombardia
64
13
User
Lombardia
63
14
User
Lombardia
63
15
User
Lombardia
61
16
Florian W
Lombardia
60
17
User
Lombardia
60
18
User
Liguria
59
19
User
Lombardia
59
20
Leonardo andreoni
Emilia-Romagna
58
21
User
Sicilia
58
22
Tom
Lombardia
58
23
User
Lombardia
57
24
User
Lombardia
57
25
User
Lombardia
56
26
User
Lombardia
54
27
User
Lombardia
54
28
User
Veneto
54
29
Giovanni Angelotti
Lombardia
53
30
User
Lazio
53
31
User
Lombardia
52
32
User
Lombardia
52
33
Chimm K
Lombardia
52
34
L’ipotesi di Riemann
Campania
51
35
User
Lazio
51
36
User
Lombardia
51
37
User
Lombardia
51
38
User
Lombardia
51
39
User
Lombardia
51
40
User
Lombardia
50
41
User
Lombardia
50
42
User
Toscana
50
43
diana aparo
Lazio
50
44
User
Lombardia
50
45
User
Lombardia
49
46
User
Puglia
49
47
User
Lombardia
49
48
Edoardo tribulato
Lombardia
49
49
User
Lombardia
49
50
User
Lombardia
48
51
User
Campania
48
52
User
Lombardia
48
53
User
Lombardia
48
54
User
Lombardia
46
55
User
Lombardia
46
56
User
Lombardia
46
57
User
Lombardia
45
58
User
Lombardia
45
59
User
Lazio
45
60
User
Lombardia
45
61
User
Lombardia
45
62
User
Lombardia
44
63
User
Lombardia
44
64
User
Lombardia
44
65
User
Piemonte
44
66
Nathaniel Corley
Toscana
44
67
Antonio
Lazio
44
68
User
Lazio
43
69
User
Lombardia
43
70
User
Lombardia
43
71
zahraranjbar
Lazio
42
72
User
Veneto
42
73
User
Lombardia
41
74
User
Lombardia
41
75
User
Lombardia
41
76
Simone
Lombardia
41
77
Daniele La Martina
Pisa
41
78
Nacho S
Emilia-Romagna
40
79
User
Lombardia
40
80
User
Lombardia
40
81
User
Lombardia
40
82
User
Calabria
40
83
User
Lombardia
40
84
User
Lombardia
40
85
User
Lombardia
40
86
Riccardo Ronzani
Veneto
40
87
User
Lazio
39
88
User
Lombardia
39
89
User
Lombardia
39
90
User
Toscana
39
91
Bernhard Winter
Milano
39
92
User
Lombardia
38
93
Anto
Sardegna
38
94
User
Veneto
38
95
User
Lombardia
38
96
User
Lombardia
38
97
Ondiola
Sardegna
38
98
User
Lombardia
37
99
User
Lombardia
37
100
User
Lombardia
37
101
User
Lombardia
36
102
Matteo Magni
Lombardia
36
103
Roberto
Lombardia
36
104
User
Lombardia
36
105
User
Lombardia
36
106
User
Lombardia
35
107
User
Lombardia
35
108
Zhou Erika
Toscana
35
109
Marco
Campania
35
110
User
Lombardia
35
112
User
Lombardia
34
113
User
Lombardia
34
114
User
Lombardia
34
115
Steven Chen
Udine
34
116
Gabriele
Campania
33
117
User
Lombardia
33
118
User
Piemonte
33
119
User
Lombardia
33
120
User
Calabria
33
121
User
Veneto
33
122
User
Lombardia
32
123
Arturo Mariani
Toscana
32
124
User
Lombardia
32
125
User
Veneto
32
126
User
Lombardia
32
127
Brian Dervishi
Lombardia
32
129
Giovanni Passuello
Veneto
31
130
User
Piemonte
31
131
User
Piemonte
31
132
User
Piemonte
31
133
User
Lazio
31
134
User
Lombardia
31
135
User
Lombardia
31
136
User
Lombardia
30
137
User
Lazio
30
138
User
Campania
30
139
User
Toscana
30
141
User
Campania
30
142
User
Lombardia
30
143
User
Lombardia
30
144
User
Lazio
30
145
User
Lazio
30
146
User
Lombardia
29
147
User
Campania
28
148
User
Lazio
28
149
User
Lombardia
28
150
Reyromy
Lombardia
28
151
Francesco Boraso
Veneto
28
152
User
Lazio
27
153
Alessandro Caroli
Lombardia
27
154
User
Lombardia
27
155
User
Campania
27
156
User
Lombardia
27
157
Daniele Nerstini
Piemonte
27
158
User
Lombardia
27
159
User
Piemonte
27
160
User
Lombardia
26
161
Anna Caturano
Lazio
26
162
Lorenzo
Campania
26
163
User
Lombardia
26
164
Veronica Pietrobon
Lombardia
26
165
Marco
Campania
26
166
User
Toscana
25
167
User
Campania
25
168
User
Campania
25
169
User
Campania
25
170
User
Lombardia
25
171
User
Lombardia
25
172
User
Lombardia
25
173
User
Liguria
25
174
Rafael Nistor
Bolzano
25
175
User
Lombardia
24
176
User
Veneto
24
178
Libreria Belvedere
Campania
24
179
User
Lombardia
24
180
User
Piemonte
24
181
User
Piemonte
24
182
User
Lombardia
24
183
User
Lombardia
23
184
User
Lombardia
23
186
User
Lombardia
23
187
User
Campania
23
188
User
Liguria
23
189
User
Piemonte
23
190
Greta Grimaldi
Lombardia
23
191
Giacomo Mazzotti
Lombardia
23
192
User
Lombardia
23
193
User
Lombardia
23
194
User
Lombardia
23
195
Ally Segreto
Nuoro
23
196
User
Marche
22
197
User
Sardegna
22
199
User
Lombardia
22
200
User
Lazio
22
201
User
Lombardia
22
202
User
Piemonte
22
204
User
Lombardia
21
205
Andrea
Lombardia
21
207
User
Lombardia
21
208
User
Campania
21
209
User
Lombardia
21
210
User
Campania
21
211
User
Campania
21
212
User
Lombardia
20
213
User
Campania
20
214
User
Lazio
20
215
User
Campania
20
216
User
Lombardia
20
217
User
Lombardia
20
218
User
Lazio
20
219
User
Toscana
20
220
User
Lombardia
20
221
User
Puglia
20
222
User
Campania
20
223
User
Lombardia
20
224
User
Lazio
20
225
User
Lazio
20
226
richard abrokwa
Emilia-Romagna
20
227
Pippo Calcoli
Toscana
20
228
Fabrice Legrange
Toscana
19
229
User
Veneto
19
230
User
Lombardia
19
231
User
Lombardia
19
232
User
Lazio
19
233
Marco
Lazio
19
234
User
Lombardia
19
235
User
Lazio
19
236
Stefano Morabito
Lazio
19
237
User
Campania
19
238
User
Calabria
18
239
User
Lombardia
18
240
Michał Gren
Lombardia
18
242
Pedro Pedrito
Lombardia
18
243
User
Lombardia
18
244
User
Lombardia
18
245
User
Lombardia
18
246
User
Campania
18
247
User
Campania
18
248
User
Toscana
17
249
User
Veneto
17
250
User
Lombardia
17
251
User
Liguria
17
252
User
Lazio
17
253
User
Campania
17
254
User
Lombardia
17
255
User
Lombardia
17
256
User
Campania
17
257
User
Campania
17
258
User
Lombardia
17
259
User
Lombardia
17
260
User
Campania
17
261
User
Piemonte
17
262
User
Veneto
17
263
Paolo Marini
Lombardia
16
264
User
Lombardia
16
265
User
Lombardia
16
266
User
Lazio
16
267
Luca Luciani
Lombardia
16
268
User
Veneto
16
269
User
Puglia
16
271
User
Sicilia
16
272
Jag
Siena
16
273
Ion Prodan
Lombardia
15
274
anna marca
Lombardia
15
275
User
Toscana
15
276
User
Lombardia
15
277
User
Lombardia
15
278
User
Lombardia
15
279
User
Lazio
15
280
User
Lombardia
15
281
User
Campania
15
282
User
Piemonte
14
283
User
Campania
14
284
User
Campania
14
285
Carmen
Lombardia
14
286
User
Campania
14
287
User
Puglia
14
289
User
Lombardia
14
290
User
Campania
14
291
Nic
Puglia
13
292
User
Lombardia
13
293
User
Lombardia
13
294
User
Marche
13
295
User
Lombardia
13
296
User
Lombardia
13
297
User
Lombardia
13
298
User
Lombardia
13
299
User
Campania
13
300
User
Piemonte
13
301
User
Campania
13
302
User
Lombardia
13
304
User
Veneto
13
305
User
Sicilia
13
306
Cris
Catania
13
307
Gabriele Zigurella
Emilia-Romagna
12
308
Ric
Calabria
12
309
User
Lazio
12
310
User
Lombardia
12
311
User
Lombardia
12
312
User
Lazio
12
313
Maya
Veneto
12
314
User
Campania
12
315
User
Veneto
12
316
User
Sicilia
12
317
User
Lombardia
12
318
User
Lombardia
12
319
User
Campania
12
320
User
Campania
12
321
User
Lombardia
12
322
User
Lombardia
12
323
User
Campania
12
324
User
Marche
11
325
User
Campania
11
328
User
Campania
11
329
User
Lazio
11
330
User
Marche
11
331
User
Lombardia
11
332
User
Piemonte
11
333
User
Veneto
11
334
User
Lazio
10
335
User
Lombardia
10
336
User
Lazio
10
337
User
Campania
10
338
User
Lombardia
10
340
User
Lombardia
10
342
User
Lombardia
10
343
User
Lombardia
10
344
User
Piemonte
10
345
User
Lombardia
10
346
User
Lombardia
10
347
Juan Diego Donneys
Lombardia
9
348
User
Piemonte
9
349
User
Lombardia
9
350
User
Lombardia
9
351
User
Lazio
9
352
User
Piemonte
9
353
User
Lombardia
9
354
User
Lombardia
9
355
User
Lazio
9
356
User
Campania
9
357
User
Lombardia
9
358
User
Campania
9
359
User
Veneto
9
361
User
Campania
9
362
User
Piemonte
9
363
User
Campania
9
364
User
Campania
9
365
User
Lombardia
8
366
User
Piemonte
8
367
User
Lombardia
8
368
User
Abruzzo
8
370
Luca Greco
Lombardia
8
371
User
Lombardia
8
372
User
Lombardia
8
373
User
Lazio
8
374
User
Campania
8
376
User
Lazio
8
377
User
Lombardia
8
378
User
Veneto
8
380
User
Lazio
8
381
User
Campania
8
382
User
Lombardia
8
384
User
Veneto
8
385
OswyChow
Brescia
8
386
User
Puglia
7
387
User
Campania
7
388
User
Lazio
7
389
User
Lombardia
7
390
User
Lombardia
7
391
User
Campania
7
392
User
Lombardia
7
393
User
Marche
7
394
User
Lazio
7
395
Matteo .M
Emilia-Romagna
7
396
User
Campania
7
397
User
Lombardia
7
398
User
Toscana
7
399
User
Lazio
7
400
User
Toscana
7
401
User
Campania
7
402
User
Lombardia
7
404
User
Puglia
7
405
User
Campania
7
406
User
Lombardia
7
407
User
Lombardia
6
408
User
Lazio
6
409
User
Lombardia
6
410
User
Lombardia
6
411
Simone Bombelli
Lombardia
6
412
User
Lombardia
6
413
User
Veneto
6
414
User
Lombardia
6
415
User
Lazio
6
416
User
Piemonte
6
417
User
Lombardia
6
418
User
Lombardia
6
419
User
Lazio
6
420
User
Lazio
6
421
User
Lombardia
6
422
User
Sicilia
6
423
User
Lazio
6
424
User
Lombardia
6
425
User
Lombardia
6
426
User
Val d\'Aoste
6
427
User
Campania
6
428
User
Lombardia
6
430
Jacopo Campedel
Lombardia
6
431
User
Puglia
5
432
User
Puglia
5
433
User
Abruzzo
5
434
User
Lombardia
5
435
User
Lombardia
5
436
User
Sardegna
5
438
User
Campania
5
439
User
Lombardia
5
440
User
Lombardia
5
441
User
Lombardia
5
442
User
Sicilia
5
444
User
Lombardia
5
445
User
Campania
5
446
User
Campania
5
447
User
Lombardia
5
448
User
Campania
5
449
Danilo Carrassi
Bari
5
450
User
Veneto
4
451
User
Toscana
4
452
User
Campania
4
453
User
Campania
4
454
User
Lombardia
4
455
User
Campania
4
456
User
Lombardia
4
457
FM
Lombardia
4
458
User
Lazio
4
459
User
Lombardia
4
460
User
Lombardia
4
461
User
Lazio
4
462
User
Sardegna
4
463
User
Lombardia
4
465
User
Lazio
4
466
User
Campania
4
467
User
Campania
4
469
User
Veneto
4
470
User
Lombardia
4
471
Matt Lejocondi
Firenze
4
472
User
Lazio
3
473
User
Lombardia
3
474
User
Lombardia
3
475
User
Campania
3
476
User
Lazio
3
479
User
Lombardia
3
480
User
Lombardia
3
481
User
Sicilia
3
482
User
Lombardia
3
483
User
Lazio
3
484
User
Lombardia
3
485
User
Lombardia
3
486
Cristian Scarpulla
Lombardia
3
487
User
Liguria
3
488
User
Lombardia
3
489
User
Lazio
3
490
User
Piemonte
3
491
User
Lombardia
3
492
User
Campania
3
493
User
Veneto
3
494
User
Lombardia
3
495
User
Sicilia
3
496
User
Lombardia
3
497
User
Lombardia
3
498
User
Toscana
3
499
User
Veneto
3
500
User
Campania
3
501
User
Lazio
3
502
User
Piemonte
3
503
User
Lombardia
3
505
User
Venezia
3
507
User
Lombardia
2
508
User
Lombardia
2
509
User
Lombardia
2
510
User
Lombardia
2
511
User
Lombardia
2
512
User
Veneto
2
513
User
Campania
2
515
User
Veneto
2
516
User
Lombardia
2
517
User
Lombardia
2
518
User
Lombardia
2
519
User
Lazio
2
520
User
Liguria
2
521
User
Campania
2
522
User
Sardegna
2
523
User
Lazio
2
524
User
Campania
2
526
User
Lombardia
2
527
User
Veneto
2
528
User
Lazio
2
529
User
Veneto
2
530
User
Lombardia
2
531
User
Lombardia
2
532
User
Lombardia
2
533
User
Piemonte
2
534
User
Lombardia
2
535
User
Lombardia
2
536
User
Lombardia
2
537
User
Lombardia
2
538
User
Lombardia
2
539
User
Lombardia
2
540
User
Lombardia
2
541
User
Campania
2
542
User
Lazio
2
543
Sean Nguyen
Sicilia
2
544
User
Marche
2
545
User
Veneto
2
546
User
Veneto
2
547
User
Lombardia
2
549
User
Lombardia
2
550
User
Sicilia
2
551
User
Lombardia
2
552
User
Campania
2
553
User
Campania
2
554
User
Lazio
2
555
User
Sicilia
2
556
User
Toscana
2
557
User
Campania
2
558
User
Lombardia
2
559
User
Lazio
2
560
User
Lombardia
1
561
User
Lazio
1
562
User
Toscana
1
563
User
Lombardia
1
564
User
Campania
1
565
User
Lazio
1
566
User
Campania
1
567
User
Liguria
1
568
User
Lombardia
1
569
User
Puglia
1
570
User
Lombardia
1
571
User
Lombardia
1
572
User
Piemonte
1
573
User
Lombardia
1
574
User
Toscana
1
575
User
Abruzzo
1
576
User
Lazio
1
577
Ondiola Due
Sardegna
1
578
Marco Melis
Sardegna
1
579
Cris
Catania
1

Free

Free basic access

Login Optional

Progress saved automatically

Students

Works great in classrooms

20,000,000
Response times
650,000
Unique questions
100
Difficulty levels

Cross-Platform

Use any operating system with Chrome, Safari, or Firefox

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.


Privacy Policy