Leaderboard
Italy
Italy
Worldwide
Rank
Level
1
Giulio
Lazio
98
2
Diego Caligiore
Sicilia
97
3
Giovanni Casalegno
Lombardia
89
4
User
Lombardia
76
5
Daniele Transerici
Lazio
75
6
Jordi
Lombardia
71
7
Evrard
Toscana
69
8
Fabrizio Lilliu
Sardegna
67
9
User
Sicilia
67
10
Anna Di Lorenzo
Campania
65
11
Giuliano
Lazio
64
12
Francesco Perniciaro
Sicilia
64
13
Giovanni Angelotti
Lombardia
62
14
Fabi89
Lazio
58
15
Babi
Lombardia
58
16
L’ipotesi di Riemann
Campania
58
17
Gianluca Ghigliano
Piemonte
57
18
Enrico Persico
Sicilia
57
20
User
Lombardia
53
21
Leonardo andreoni
Emilia-Romagna
51
22
User
Lazio
51
23
User
Marche
51
24
User
Lombardia
49
25
dvdnoveotto
Lombardia
49
26
User
Lombardia
49
27
Martino
Emilia-Romagna
49
28
User
Lombardia
48
29
BackToZeon
Trentino-Alto Adige
47
30
User
Piemonte
46
31
Laura Garavoglia
Piemonte
46
32
Angelo Velardi
Sicilia
46
33
Alessandro Caroli
Veneto
46
34
Fregnolento
Toscana
46
35
User
Lombardia
46
36
Giuseppe Mazzara
Sicilia
45
37
User
Lombardia
45
38
skandro the comeback
Lombardia
44
39
User
Lazio
43
40
User
Lombardia
43
41
User
Lombardia
43
42
Pietro Giani
Piemonte
43
43
Caragnano Giusy
Puglia
42
44
mentalmather
Emilia-Romagna
42
45
User
Lombardia
42
46
User
Veneto
41
47
User
Lazio
41
48
User
Veneto
41
49
User
Veneto
40
50
User
Lombardia
40
51
User
Lombardia
40
52
Marco Cipolletti
Puglia
40
53
m.guyon@hotmail.com
Lombardia
39
54
Jeremy Alvarez
Lombardia
39
55
Giuseppe Tringali
Sicilia
38
56
User
Campania
38
57
Cris Milano
Lombardia
38
58
Arturo Mariani
Puglia
37
59
User
Sicilia
37
60
mentalmather
Emilia-Romagna
36
61
Massimo Caliaro
Lombardia
36
62
Giovanni Passuello
Trentino-Alto Adige
36
63
User
Lombardia
36
64
Wyatt Swanson
Campania
36
65
User
Toscana
36
66
User
Lombardia
35
67
User
Lombardia
35
69
User
Lombardia
35
70
User
Lombardia
35
71
Nicola Lanzarone
Lombardia
35
72
Alessandro Naia
Lombardia
35
73
User
Lombardia
35
74
User
Sardegna
34
75
User
Lombardia
34
76
User
Sardegna
34
77
Florian W
Lombardia
34
78
User
Lazio
34
79
User
Veneto
33
80
User
Lazio
33
81
User
Lazio
32
82
Davide Accordini
Veneto
32
83
User
Piemonte
32
84
User
Lombardia
32
85
User
Lombardia
31
86
User
Lombardia
31
87
Sergio Iommi
Abruzzo
31
88
User
Lombardia
31
89
User
Puglia
31
90
User
Lombardia
30
91
User
Lombardia
30
92
User
Lombardia
30
93
User
Lombardia
30
94
User
Piemonte
30
95
User
Lombardia
30
96
User
Lombardia
30
97
User
Lombardia
30
98
User
Sardegna
29
99
Oliver K
Veneto
29
100
User
Lazio
29
101
User
Lombardia
29
102
Michele Celentano
Lazio
29
103
Giorgio Pesenti
Region
29
104
User
Campania
29
105
User
Sicilia
28
106
User
Lombardia
28
107
User
Liguria
28
108
User
Lombardia
28
109
User
Lombardia
28
110
User
Marche
28
111
User
Lazio
27
112
User
Lombardia
27
113
User
Lombardia
27
114
User
Marche
27
115
User
Piemonte
27
116
User
Piemonte
27
117
User
Veneto
27
118
Gaetano
Lazio
27
119
Fabrice Legrange
Toscana
27
120
User
Lombardia
27
121
User
Lombardia
26
122
User
Umbria
26
123
User
Lombardia
26
124
User
Puglia
26
125
User
Lombardia
26
126
User
Campania
26
127
User
Lazio
25
128
User
Lazio
25
129
User
Lombardia
25
130
User
Campania
25
131
User
Lombardia
25
132
Tom
Lombardia
24
133
User
Sicilia
24
134
User
Lombardia
24
135
User
Piemonte
24
136
User
Lombardia
23
137
bebop
Piemonte
23
138
Nicolò P.
Puglia
23
139
User
Lazio
23
140
User
Lombardia
23
141
User
Campania
23
142
User
Lombardia
23
143
Nacho S
Emilia-Romagna
23
144
User
Lombardia
23
145
User
Sardegna
23
146
User
Piemonte
23
147
User
Lazio
23
148
User
Sardegna
22
149
User
Campania
22
150
civilizedbug
Lombardia
22
151
User
Lombardia
22
152
User
Toscana
22
153
User
Lazio
22
154
Pingable
Lazio
22
155
User
Lombardia
22
156
User
Lombardia
21
157
User
Lazio
21
158
User
Campania
21
159
User
Toscana
21
160
User
Lombardia
21
161
User
Lombardia
21
162
User
Piemonte
21
163
User
Lazio
21
164
User
Lombardia
21
165
Aelita Schaeffer
Piemonte
20
166
User
Lombardia
20
167
User
Lombardia
20
168
User
Lombardia
20
169
User
Campania
20
170
User
Lombardia
20
171
User
Lombardia
20
172
User
Lombardia
20
173
User
Campania
20
174
User
Lombardia
20
175
User
Campania
20
176
User
Lombardia
20
177
User
Lombardia
20
178
User
Lazio
19
179
User
Lombardia
19
180
User
Lombardia
19
181
User
Sardegna
19
182
Alessio Parolari
Lombardia
19
183
User
Marche
19
184
User
Lazio
19
185
User
Lombardia
19
186
User
Sardegna
19
187
David Thomas
Veneto
19
188
zahraranjbar
Lazio
19
189
Chimm K
Lombardia
19
190
User
Lazio
19
191
User
Sardegna
19
192
User
Lombardia
18
193
User
Veneto
18
194
User
Lombardia
18
195
User
Lombardia
18
196
User
Lombardia
18
197
User
Lombardia
18
198
User
Lombardia
18
199
User
Lazio
18
200
User
Lombardia
18
201
User
Lazio
18
202
User
Lombardia
18
203
User
Piemonte
18
204
User
Lombardia
18
205
User
Piemonte
18
206
User
Lombardia
18
207
User
Lazio
17
208
User
Piemonte
17
209
User
Lombardia
17
210
User
Lombardia
17
211
User
Lombardia
17
212
Matteo Magni
Lombardia
17
213
User
Lombardia
17
214
Anto
Sardegna
17
215
Alessandro Anderson
Lombardia
17
216
User
Lombardia
17
217
User
Veneto
17
218
User
Lombardia
17
219
Gabriele
Campania
17
220
User
Lazio
17
221
User
Campania
17
222
User
Lombardia
17
223
User
Calabria
17
224
User
Lombardia
16
225
Tom Verlaine
Lazio
16
226
Nathaniel Corley
Toscana
16
227
User
Lombardia
16
228
User
Veneto
16
229
User
Lombardia
16
230
User
Campania
16
231
User
Lombardia
16
232
User
Lombardia
16
233
User
Lombardia
16
234
Antonio
Lazio
16
235
User
Piemonte
16
236
Happypig
Toscana
16
237
User
Lombardia
15
238
User
Lombardia
15
239
User
Lombardia
15
240
User
Lombardia
15
241
User
Lombardia
15
242
User
Lombardia
15
243
User
Lombardia
15
244
Reyromy
Lombardia
15
245
User
Lombardia
15
246
Simone
Lombardia
15
247
User
Toscana
15
248
User
Lombardia
15
250
User
Lombardia
15
251
User
Lazio
15
252
Riccardo Ronzani
Veneto
15
253
User
Lombardia
15
254
VINCENZO
Puglia
15
255
Roberto
Lombardia
14
256
User
Lombardia
14
257
User
Piemonte
14
258
User
Lombardia
14
259
User
Lombardia
14
261
Anna Caturano
Lazio
14
262
User
Lazio
14
263
User
Lombardia
14
264
André Ferretti
Piemonte
14
266
User
Lombardia
14
267
Ondiola
Sardegna
14
268
Luciano Grermorio
Lombardia
14
269
User
Toscana
14
270
User
Lombardia
14
271
User
Puglia
14
272
User
Lazio
14
273
User
Piemonte
14
274
User
Piemonte
14
275
User
Sicilia
14
276
User
Lombardia
14
277
User
Piemonte
14
278
User
Campania
13
279
User
Piemonte
13
280
User
Marche
13
281
User
Campania
13
282
User
Toscana
13
284
User
Campania
13
285
User
Lombardia
13
287
User
Puglia
13
288
User
Campania
13
289
User
Abruzzo
13
290
User
Lombardia
13
291
User
Lombardia
13
292
Lorenzo
Campania
13
294
User
Lombardia
13
295
User
Lombardia
13
296
User
Lombardia
13
297
User
Lombardia
13
299
User
Veneto
13
300
User
Lombardia
13
301
User
Lombardia
13
302
Zhou Erika
Toscana
13
303
User
Lombardia
13
304
User
Campania
12
305
User
Campania
12
306
User
Lombardia
12
307
User
Calabria
12
308
User
Lazio
12
309
User
Lombardia
12
310
User
Campania
12
311
User
Piemonte
12
312
User
Sardegna
12
313
User
Lombardia
12
314
User
Veneto
12
315
User
Veneto
12
316
User
Campania
12
317
User
Lombardia
12
318
User
Lombardia
12
319
User
Lombardia
12
320
User
Campania
12
321
User
Lazio
12
323
User
Lombardia
12
324
Daniele Nerstini
Piemonte
12
325
User
Lombardia
12
326
User
Veneto
12
327
User
Lombardia
12
328
User
Campania
12
329
Emeraldvv Vertex
Sardegna
12
330
User
Lombardia
12
331
Marco
Campania
12
332
Simone Bombelli
Lombardia
12
333
User
Lombardia
12
334
User
Lombardia
12
335
Name Surname
Campania
12
336
User
Veneto
11
337
User
Veneto
11
338
User
Toscana
11
339
User
Lombardia
11
340
User
Lombardia
11
341
User
Veneto
11
342
User
Lombardia
11
343
Andrea
Lombardia
11
344
User
Lombardia
11
345
User
Calabria
11
346
User
Campania
11
347
User
Lombardia
11
348
User
Piemonte
11
350
User
Lazio
11
353
User
Toscana
11
354
User
Lombardia
11
355
User
Lombardia
11
356
User
Lazio
11
357
User
Lombardia
11
358
User
Liguria
11
359
User
Puglia
11
360
User
Lombardia
11
361
User
Toscana
11
362
User
Lombardia
11
363
User
Lazio
11
364
User
Lombardia
11
365
Brian Dervishi
Lombardia
11
366
User
Lombardia
11
367
User
Lombardia
11
368
User
Sardegna
11
369
User
Sicilia
11
370
User
Lombardia
11
371
User
Puglia
11
372
User
Basilicata
10
373
User
Lombardia
10
374
Francesco Boraso
Veneto
10
375
User
Puglia
10
376
User
Puglia
10
377
User
Piemonte
10
378
Greta Grimaldi
Lombardia
10
379
User
Lazio
10
380
User
Puglia
10
381
User
Puglia
10
382
User
Puglia
10
383
User
Lombardia
10
384
User
Campania
10
385
User
Lombardia
10
386
User
Campania
10
387
User
Sicilia
10
388
MarcoO
Lazio
10
389
User
Lombardia
10
390
User
Lazio
10
391
User
Lombardia
10
392
Adriano Vlad
Piemonte
10
393
User
Lazio
10
394
mentalmather
Emilia-Romagna
10
395
Pablo Andre Alvarado Leonardo
Lombardia
10
396
User
Sicilia
9
397
User
Lombardia
9
398
anna marca
Lombardia
9
399
Michał Gren
Lombardia
9
400
User
Lazio
9
401
User
Puglia
9
402
User
Puglia
9
403
User
Toscana
9
404
User
Puglia
9
405
User
Lombardia
9
406
GryZzlw
Lombardia
9
407
User
Lombardia
9
408
User
Toscana
9
409
User
Campania
9
410
User
Lombardia
9
411
User
Lombardia
9
412
User
Lombardia
9
413
User
Lombardia
9
414
Giacomo Mazzotti
Lombardia
9
415
User
Campania
9
416
Paolo Marini
Lombardia
9
417
User
Lombardia
9
418
User
Piemonte
9
419
User
Veneto
9
420
User
Lombardia
9
421
User
Piemonte
9
422
Libreria Belvedere
Campania
9
423
User
Lombardia
9
424
User
Lazio
9
425
User
Lombardia
9
426
User
Lombardia
9
427
Veronica Pietrobon
Lombardia
9
428
User
Lombardia
9
429
User
Puglia
9
430
User
Veneto
9
431
User
Puglia
9
432
User
Campania
9
433
User
Lombardia
9
434
User
Lombardia
9
435
User
Lombardia
9
436
User
Campania
8
437
User
Puglia
8
438
User
Puglia
8
439
User
Lombardia
8
440
Marco
Campania
8
441
User
Lombardia
8
442
User
Lombardia
8
443
User
Lombardia
8
444
User
Lombardia
8
445
User
Campania
8
446
User
Campania
8
447
User
Lombardia
8
448
Gabriele Zigurella
Emilia-Romagna
8
449
User
Sardegna
8
450
User
Campania
8
451
User
Liguria
8
452
User
Puglia
8
454
User
Piemonte
8
455
User
Piemonte
8
456
User
Puglia
8
457
User
Toscana
8
458
User
Lombardia
8
459
User
Campania
8
460
User
Lazio
8
461
Ric
Calabria
8
462
User
Lombardia
8
463
User
Marche
8
464
User
Marche
8
465
User
Campania
8
466
User
Campania
8
468
User
Lombardia
8
469
User
Lombardia
8
471
User
Lombardia
8
472
User
Lazio
8
473
User
Lazio
8
474
User
Lombardia
8
475
User
Lombardia
8
476
User
Lombardia
8
477
User
Lombardia
8
478
User
Lombardia
8
479
User
Sicilia
8
480
User
Piemonte
8
481
User
Marche
8
482
User
Sicilia
8
483
User
Lombardia
8
484
User
Lombardia
8
485
User
Lazio
8
486
User
Lombardia
8
487
User
Lombardia
8
488
User
Lazio
8
490
User
Campania
7
491
User
Lombardia
7
492
User
Lazio
7
493
User
Lombardia
7
494
User
Lazio
7
495
User
Lombardia
7
496
User
Liguria
7
497
User
Campania
7
498
Pedro Pedrito
Lombardia
7
499
User
Piemonte
7
500
User
Lombardia
7
501
User
Campania
7
502
User
Lazio
7
503
User
Lazio
7
504
User
Lombardia
7
505
User
Toscana
7
506
User
Campania
7
507
User
Lombardia
7
508
User
Lazio
7
509
User
Lombardia
7
510
User
Lombardia
7
511
User
Liguria
7
512
diana aparo
Lazio
7
513
richard abrokwa
Emilia-Romagna
7
514
User
Lazio
7
515
User
Lombardia
7
516
User
Campania
7
517
Eduard Sokoli
Toscana
7
518
User
Lombardia
7
519
User
Lombardia
7
520
User
Lombardia
7
521
User
Lombardia
7
522
User
Campania
7
523
User
Lombardia
7
525
User
Lazio
7
526
User
Lombardia
7
527
User
Lombardia
7
528
User
Lombardia
7
529
User
Lazio
7
530
User
Lombardia
7
531
User
Piemonte
7
532
User
Lazio
7
533
User
Lazio
7
534
User
Puglia
7
535
User
Campania
7
536
User
Campania
7
538
Maya
Veneto
7
539
User
Campania
7
541
User
Campania
7
542
User
Lombardia
7
543
User
Campania
7
544
User
Lombardia
7
545
Stefano Morabito
Lazio
7
546
User
Lombardia
7
547
User
Lazio
7
548
User
Campania
7
549
User
Campania
6
550
User
Lombardia
6
551
User
Campania
6
552
User
Lombardia
6
553
User
Puglia
6
554
User
Lombardia
6
555
Marco Masini
Lombardia
6
556
User
Campania
6
557
User
Lombardia
6
558
User
Campania
6
559
Luca Luciani
Lombardia
6
560
User
Lazio
6
561
User
Lombardia
6
562
Juan Diego Donneys
Lombardia
6
563
User
Campania
6
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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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