Leaderboard
Italy
Italy
Worldwide
Rank
Level
1
Sergio Iommi
Abruzzo
95
2
Michele Celentano
Lazio
91
3
User
Sicilia
88
4
User
Piemonte
83
5
User
Lombardia
83
6
Davide Accordini
Veneto
82
7
User
Lazio
81
8
User
Lombardia
80
10
skandro the comeback
Lombardia
75
11
BackToZeon
Trentino-Alto Adige
74
12
Alessandro Naia
Lombardia
73
13
User
Lombardia
67
14
Leonardo andreoni
Emilia-Romagna
65
15
User
Lazio
65
16
User
Lombardia
64
17
User
Lombardia
63
18
User
Lombardia
63
19
User
Lombardia
63
20
L’ipotesi di Riemann
Campania
62
21
User
Lombardia
61
22
Angelo Velardi
Sicilia
60
23
User
Veneto
60
24
Florian W
Lombardia
60
25
User
Lombardia
60
26
User
Liguria
59
27
User
Lombardia
59
28
Giuseppe Mazzara
Sicilia
58
29
User
Sicilia
58
30
Tom
Lombardia
58
31
Fabi89
Lazio
57
32
User
Lazio
57
33
User
Lombardia
57
34
User
Lombardia
57
35
Giuseppe Tringali
Sicilia
56
36
User
Lombardia
56
37
Jeremy Alvarez
Lombardia
55
38
User
Veneto
54
39
User
Lombardia
54
40
User
Lombardia
54
41
User
Veneto
54
42
Giovanni Angelotti
Lombardia
53
43
User
Lombardia
52
44
User
Lombardia
52
45
Chimm K
Lombardia
52
46
Wyatt Swanson
Campania
51
47
User
Lombardia
51
48
User
Lombardia
51
49
User
Lazio
51
50
User
Lombardia
51
51
User
Lombardia
51
52
User
Lombardia
51
53
User
Lombardia
51
54
Massimo Caliaro
Lombardia
50
55
User
Lombardia
50
56
User
Lombardia
50
57
User
Toscana
50
58
diana aparo
Lazio
50
59
User
Lombardia
50
60
User
Sardegna
49
61
Alessandro Caroli
Lombardia
49
62
User
Lombardia
49
63
User
Puglia
49
64
User
Lombardia
49
65
Edoardo tribulato
Lombardia
49
66
User
Lombardia
49
67
User
Lombardia
48
68
User
Campania
48
69
User
Lombardia
48
70
User
Lombardia
48
71
Cris Milano
Lombardia
47
72
User
Lazio
46
73
User
Piemonte
46
74
User
Lombardia
46
75
User
Lombardia
46
76
User
Lombardia
46
77
User
Lombardia
46
78
User
Lazio
45
79
User
Lombardia
45
80
User
Lazio
45
81
User
Lombardia
45
82
User
Lombardia
45
83
User
Lazio
45
84
User
Lombardia
45
85
User
Lombardia
45
86
User
Sicilia
44
87
User
Lombardia
44
88
User
Lombardia
44
89
User
Lombardia
44
90
User
Piemonte
44
91
Nathaniel Corley
Toscana
44
92
Antonio
Lazio
44
93
User
Lombardia
43
94
User
Lazio
43
95
User
Lombardia
43
96
User
Lombardia
43
97
zahraranjbar
Lazio
42
98
User
Veneto
42
99
User
Lombardia
41
100
User
Lombardia
41
101
User
Lombardia
41
102
Simone
Lombardia
41
103
Daniele La Martina
Pisa
41
104
User
Veneto
40
105
Nacho S
Emilia-Romagna
40
106
User
Lombardia
40
107
User
Lombardia
40
108
User
Lombardia
40
109
User
Calabria
40
110
User
Lombardia
40
111
User
Lombardia
40
112
User
Lombardia
40
113
Riccardo Ronzani
Veneto
40
114
Gaetano
Lazio
39
115
User
Lazio
39
116
User
Lombardia
39
117
User
Lombardia
39
118
User
Toscana
39
119
User
Lombardia
38
120
Anto
Sardegna
38
121
User
Veneto
38
122
User
Lombardia
38
123
User
Lombardia
38
124
Ondiola
Sardegna
38
125
User
Piemonte
37
126
User
Marche
37
127
User
Lazio
37
128
Fabrice Legrange
Toscana
37
129
User
Campania
37
130
User
Lombardia
37
131
User
Lombardia
37
132
User
Lombardia
37
133
User
Campania
36
134
User
Lombardia
36
135
Giovanni Passuello
Veneto
36
136
User
Lombardia
36
137
Matteo Magni
Lombardia
36
138
Roberto
Lombardia
36
139
User
Lombardia
36
140
User
Lombardia
36
141
Arturo Mariani
Puglia
35
142
User
Lombardia
35
143
User
Sicilia
35
144
User
Lombardia
35
145
User
Lombardia
35
146
Zhou Erika
Toscana
35
147
Marco
Campania
35
148
User
Lombardia
35
149
Nicolò P.
Puglia
34
150
User
Sardegna
34
151
User
Sardegna
34
152
User
Lombardia
34
153
User
Lombardia
34
154
User
Lombardia
34
155
Steven Chen
Udine
34
156
User
Lazio
33
157
Gabriele
Campania
33
158
User
Lombardia
33
159
User
Piemonte
33
160
User
Lombardia
33
161
User
Calabria
33
162
User
Veneto
33
163
User
Lazio
32
164
User
Lombardia
32
165
User
Lombardia
32
166
User
Veneto
32
167
User
Lombardia
32
168
Brian Dervishi
Lombardia
32
169
User
Sicilia
31
170
User
Piemonte
31
171
User
Piemonte
31
172
User
Piemonte
31
173
User
Lazio
31
174
User
Lombardia
31
175
User
Lombardia
31
176
User
Piemonte
30
177
civilizedbug
Lombardia
30
178
Pingable
Lazio
30
179
User
Lombardia
30
180
User
Lombardia
30
181
User
Lazio
30
182
User
Campania
30
183
User
Toscana
30
185
User
Campania
30
186
User
Lombardia
30
187
User
Lombardia
30
188
User
Lazio
30
189
User
Lazio
30
190
User
Sardegna
29
191
User
Marche
29
192
User
Lombardia
29
193
User
Campania
28
194
User
Campania
28
195
User
Lombardia
28
196
Reyromy
Lombardia
28
197
Francesco Boraso
Veneto
28
198
User
Campania
27
199
User
Lombardia
27
200
User
Campania
27
201
User
Lombardia
27
202
Daniele Nerstini
Piemonte
27
203
User
Lombardia
27
204
User
Piemonte
27
205
User
Lombardia
26
206
User
Lombardia
26
207
User
Lombardia
26
208
Anna Caturano
Lazio
26
209
Lorenzo
Campania
26
210
User
Lombardia
26
211
Veronica Pietrobon
Lombardia
26
212
Marco
Campania
26
213
User
Lombardia
25
214
User
Campania
25
215
User
Lombardia
25
216
User
Lombardia
25
217
David Thomas
Veneto
25
218
User
Lombardia
25
219
User
Lazio
25
220
User
Toscana
25
221
User
Campania
25
222
User
Campania
25
223
User
Campania
25
224
User
Lombardia
25
225
User
Lombardia
25
226
User
Lombardia
25
227
User
Liguria
25
228
Rafael Nistor
Bolzano
25
229
Alessio Parolari
Lombardia
24
230
User
Lombardia
24
231
User
Lombardia
24
232
User
Veneto
24
234
Libreria Belvedere
Campania
24
235
User
Lombardia
24
236
User
Piemonte
24
237
User
Piemonte
24
238
User
Lombardia
24
239
Tom Verlaine
Lazio
23
240
User
Lombardia
23
242
User
Lombardia
23
243
User
Campania
23
244
User
Liguria
23
245
User
Piemonte
23
246
Greta Grimaldi
Lombardia
23
247
Giacomo Mazzotti
Lombardia
23
248
User
Lombardia
23
249
User
Lombardia
23
250
User
Lombardia
23
251
Ally Segreto
Nuoro
23
252
User
Lazio
22
253
User
Marche
22
254
User
Sardegna
22
256
User
Lombardia
22
257
User
Lazio
22
258
User
Lombardia
22
259
User
Piemonte
22
261
User
Lombardia
21
262
Andrea
Lombardia
21
264
User
Lombardia
21
265
User
Campania
21
266
User
Lombardia
21
267
User
Campania
21
268
User
Campania
21
269
User
Lazio
20
270
User
Lombardia
20
271
User
Campania
20
272
User
Lazio
20
273
User
Campania
20
274
User
Lombardia
20
275
User
Lombardia
20
276
User
Lazio
20
277
User
Toscana
20
278
User
Lombardia
20
279
User
Puglia
20
280
User
Campania
20
281
User
Lombardia
20
282
User
Lazio
20
283
User
Lazio
20
284
richard abrokwa
Emilia-Romagna
20
285
Pippo Calcoli
Toscana
20
286
User
Puglia
19
287
André Ferretti
Piemonte
19
288
User
Lombardia
19
289
User
Veneto
19
290
User
Lombardia
19
291
User
Lombardia
19
292
User
Lazio
19
293
Marco
Lazio
19
294
User
Lombardia
19
295
User
Lazio
19
296
Stefano Morabito
Lazio
19
297
User
Campania
19
298
User
Campania
18
299
User
Abruzzo
18
300
User
Veneto
18
301
User
Calabria
18
302
User
Lombardia
18
303
Michał Gren
Lombardia
18
305
Pedro Pedrito
Lombardia
18
306
User
Lombardia
18
307
User
Lombardia
18
308
User
Lombardia
18
309
User
Campania
18
310
User
Campania
18
311
User
Puglia
17
312
User
Sicilia
17
313
User
Campania
17
314
User
Lombardia
17
315
User
Toscana
17
316
User
Veneto
17
317
User
Lombardia
17
318
User
Liguria
17
319
User
Lazio
17
320
User
Campania
17
321
User
Lombardia
17
322
User
Lombardia
17
323
User
Campania
17
324
User
Campania
17
325
User
Lombardia
17
326
User
Lombardia
17
327
User
Campania
17
328
User
Piemonte
17
329
User
Veneto
17
330
User
Lombardia
16
331
User
Sardegna
16
332
Emeraldvv Vertex
Sardegna
16
333
User
Lombardia
16
334
User
Veneto
16
335
User
Campania
16
336
User
Lazio
16
337
User
Lombardia
16
338
User
Lombardia
16
339
Paolo Marini
Lombardia
16
340
User
Lombardia
16
341
User
Lombardia
16
342
User
Lazio
16
343
Luca Luciani
Lombardia
16
344
User
Veneto
16
345
User
Puglia
16
347
User
Sicilia
16
348
Jag
Siena
16
349
User
Lazio
15
350
User
Toscana
15
351
User
Lombardia
15
352
anna marca
Lombardia
15
353
User
Toscana
15
354
User
Lombardia
15
355
User
Lombardia
15
356
User
Lombardia
15
357
User
Lazio
15
358
User
Lombardia
15
359
User
Campania
15
360
User
Puglia
14
361
Luciano Grermorio
Lombardia
14
362
Adriano Vlad
Piemonte
14
363
User
Piemonte
14
364
User
Campania
14
365
User
Campania
14
366
User
Campania
14
367
User
Puglia
14
369
User
Lombardia
14
370
User
Campania
14
371
User
Lombardia
13
372
User
Puglia
13
373
User
Basilicata
13
374
User
Puglia
13
375
User
Puglia
13
376
User
Puglia
13
377
User
Puglia
13
378
User
Puglia
13
379
User
Campania
13
380
User
Sicilia
13
381
User
Lazio
13
382
User
Lombardia
13
383
User
Lombardia
13
384
User
Marche
13
385
User
Lombardia
13
386
User
Lombardia
13
387
User
Lombardia
13
388
User
Lombardia
13
389
User
Campania
13
390
User
Piemonte
13
391
User
Campania
13
392
User
Lombardia
13
394
User
Veneto
13
395
User
Sicilia
13
396
Cris
Catania
13
397
User
Lazio
12
398
User
Puglia
12
399
User
Puglia
12
400
User
Puglia
12
401
GryZzlw
Lombardia
12
402
User
Lombardia
12
403
User
Lombardia
12
404
User
Lombardia
12
405
User
Lombardia
12
406
Gabriele Zigurella
Emilia-Romagna
12
407
Ric
Calabria
12
408
User
Lazio
12
409
User
Lombardia
12
410
User
Lombardia
12
411
User
Lazio
12
412
Maya
Veneto
12
413
User
Campania
12
414
User
Veneto
12
415
User
Sicilia
12
416
User
Lombardia
12
417
User
Lombardia
12
418
User
Campania
12
419
User
Campania
12
420
User
Lombardia
12
421
User
Lombardia
12
422
User
Campania
12
423
User
Lombardia
11
424
User
Puglia
11
425
User
Puglia
11
426
User
Puglia
11
427
User
Marche
11
428
User
Campania
11
431
User
Campania
11
432
User
Lazio
11
433
User
Marche
11
434
User
Lombardia
11
435
User
Piemonte
11
436
User
Veneto
11
437
User
Lombardia
10
438
User
Sardegna
10
439
User
Puglia
10
440
User
Puglia
10
441
User
Lombardia
10
442
User
Lombardia
10
443
User
Marche
10
444
User
Lazio
10
445
User
Campania
10
446
User
Lazio
10
447
User
Lombardia
10
448
User
Lazio
10
449
User
Campania
10
450
User
Lombardia
10
452
User
Lombardia
10
454
User
Lombardia
10
455
User
Lombardia
10
456
User
Piemonte
10
457
User
Lombardia
10
458
User
Lombardia
10
459
User
Sicilia
9
460
User
Sicilia
9
461
User
Campania
9
462
User
Liguria
9
463
Eduard Sokoli
Toscana
9
464
User
Lombardia
9
465
User
Piemonte
9
466
Juan Diego Donneys
Lombardia
9
467
User
Piemonte
9
468
User
Lombardia
9
469
User
Lombardia
9
470
User
Lazio
9
471
User
Piemonte
9
472
User
Lombardia
9
473
User
Lombardia
9
474
User
Lazio
9
475
User
Campania
9
476
User
Lombardia
9
477
User
Campania
9
478
User
Veneto
9
480
User
Campania
9
481
User
Piemonte
9
482
User
Campania
9
483
User
Campania
9
484
User
Lazio
8
485
User
Campania
8
486
User
Campania
8
487
User
Lombardia
8
488
User
Campania
8
489
User
Lombardia
8
490
User
Piemonte
8
491
User
Lombardia
8
493
Luca Greco
Lombardia
8
494
User
Lombardia
8
495
User
Lazio
8
496
User
Campania
8
498
User
Lazio
8
499
User
Lombardia
8
500
User
Veneto
8
502
User
Lazio
8
503
User
Campania
8
504
User
Lombardia
8
506
User
Veneto
8
507
OswyChow
Brescia
8
508
User
Lazio
7
509
User
Lazio
7
510
User
Puglia
7
511
User
Piemonte
7
512
User
Lombardia
7
513
User
Puglia
7
514
User
Lombardia
7
515
Carmen
Campania
7
516
User
Veneto
7
517
User
Lombardia
7
518
User
Puglia
7
519
User
Campania
7
520
User
Lazio
7
521
User
Lombardia
7
522
User
Lombardia
7
523
User
Campania
7
524
User
Lombardia
7
525
User
Marche
7
526
User
Lazio
7
527
Matteo .M
Emilia-Romagna
7
528
User
Campania
7
529
User
Lombardia
7
530
User
Toscana
7
531
User
Lazio
7
532
User
Toscana
7
533
User
Campania
7
534
User
Lombardia
7
536
User
Puglia
7
537
User
Campania
7
538
User
Lombardia
7
539
User
Lombardia
6
540
User
Campania
6
541
User
Lazio
6
542
User
Puglia
6
543
User
Lombardia
6
544
User
Lazio
6
545
User
Lombardia
6
546
User
Lombardia
6
547
Simone Bombelli
Lombardia
6
548
User
Lombardia
6
549
User
Veneto
6
550
User
Lombardia
6
551
User
Lazio
6
552
User
Piemonte
6
553
User
Lombardia
6
554
User
Lombardia
6
555
User
Lazio
6
556
User
Lazio
6
557
User
Lombardia
6
558
User
Sicilia
6
559
User
Lazio
6
560
User
Lombardia
6
561
User
Lombardia
6
562
User
Val d\'Aoste
6
563
User
Campania
6
564
User
Lombardia
6
566
Jacopo Campedel
Lombardia
6
567
User
Lombardia
5
568
User
Lombardia
5
569
User
Veneto
5
570
fiorile
Emilia-Romagna
5
571
User
Campania
5
572
User
Lazio
5
573
User
Veneto
5
574
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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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