Leaderboard
Denmark
Denmark
Worldwide
Rank
Level
2
Erik M. L. Johannesen
Hovedstaden
88
7
User
Syddanmark
64
9
Ida Maria Soelje
Hovedstaden
62
10
Jesper Ramsing
Syddanmark
61
11
ole jensen
Hovedstaden
58
12
User
Hovedstaden
56
13
User
Midtjylland
53
14
User
Hovedstaden
52
15
Ixnatifual
Midtjylland
51
16
Alexander Mønster
Hovedstaden
51
17
User
Hovedstaden
50
18
User
Hovedstaden
50
19
User
Midtjylland
49
20
Hannesbo
Hovedstaden
49
21
Magnus Kanstrup
Sjælland
48
22
User
Midtjylland
47
23
User
Nordjylland
46
24
Teodor Andersen
Hovedstaden
46
25
User
Hovedstaden
46
26
User
Hovedstaden
46
27
Christian Alexander Yabar Mejer 2018e
Sjælland
46
28
Laura Gutiérrez
Hovedstaden
45
29
User
Nordjylland
45
30
Dyrlund
Hovedstaden
44
31
User
Nordjylland
44
32
User
Hovedstaden
44
33
User
Hovedstaden
44
34
Torben Musolf
Syddanmark
40
35
Laura Turner
Hovedstaden
39
36
User
Hovedstaden
39
37
User
Hovedstaden
39
38
Pedro Rodrigues
Syddanmark
38
39
User
Hovedstaden
38
40
User
Hovedstaden
38
41
User
Hovedstaden
38
42
User
Syddanmark
37
43
User
Midtjylland
37
44
User
Hovedstaden
36
45
User
Midtjylland
36
46
Asmus Rosendal
Midtjylland
36
47
User
Hovedstaden
36
48
User
Midtjylland
35
49
User
Hovedstaden
35
50
Iulia Prundurel
Hovedstaden
35
51
User
Sjælland
34
52
User
Midtjylland
34
53
User
Midtjylland
34
54
Logan
Nordjylland
33
55
User
Hovedstaden
33
56
User
Hovedstaden
32
57
User
Hovedstaden
32
58
Christian Mejer
Sjælland
32
59
User
Hovedstaden
32
60
Kasper Rasmussen
Nordjylland
31
61
User
Midtjylland
31
62
Achmel
Hovedstaden
30
63
User
Hovedstaden
30
64
User
Midtjylland
30
65
Danny Nygård Hansen
Midtjylland
30
66
User
Hovedstaden
29
67
User
Sjælland
28
68
Foxylock
Nordjylland
28
69
User
Nordjylland
28
70
User
Midtjylland
27
71
User
Hovedstaden
27
72
User
Hovedstaden
27
73
User
Region
27
74
Kasper
Syddanmark
26
75
User
Hovedstaden
26
76
User
Hovedstaden
26
77
User
Hovedstaden
26
78
Marin Jeppesen
Region
26
79
User
Hovedstaden
25
80
User
Sjælland
25
81
User
Hovedstaden
25
82
User
Hovedstaden
25
83
User
Region
25
84
marcus1x33@gmail.com
Hovedstaden
24
85
User
Syddanmark
24
86
User
Hovedstaden
24
87
User
Region
24
89
User
Hovedstaden
23
90
User
Syddanmark
23
91
User
Midtjylland
23
92
User
Hovedstaden
22
93
User
Midtjylland
22
94
Gustav Haugan
Hovedstaden
22
95
User
Syddanmark
22
96
User
Sjælland
22
97
User
Region
22
98
User
Hovedstaden
21
99
Ayushman Chaudhary - 8A
Nordjylland
21
100
User
Hovedstaden
21
101
User
Syddanmark
21
102
User
Hovedstaden
21
103
User
Midtjylland
21
104
User
Hovedstaden
21
105
Ida Marie Soelje
Hovedstaden
21
106
User
Hovedstaden
21
107
Finn Nørgaard-Pedersen
Hovedstaden
21
108
User
Syddanmark
21
109
User
Hovedstaden
21
110
User
Region
21
111
User
Region
21
112
User
Region
21
113
User
Hovedstaden
20
114
Rina
Hovedstaden
20
115
Lucazzz
Hovedstaden
20
116
User
Hovedstaden
20
117
User
Hovedstaden
20
118
User
Region
20
119
User
Region
20
120
User
Hovedstaden
19
121
User
Syddanmark
19
122
Rebekka Lykke
Hovedstaden
19
123
User
Hovedstaden
19
124
Mikkel Risgaard
Syddanmark
19
125
User
Hovedstaden
19
126
User
Hovedstaden
19
127
User
Region
19
128
User
Hovedstaden
18
129
User
Hovedstaden
18
130
User
Hovedstaden
18
131
User
Hovedstaden
18
132
User
Hovedstaden
18
133
User
Midtjylland
18
134
User
Nordjylland
18
135
User
Region
18
136
User
Region
18
137
User
Hovedstaden
17
138
User
Sjælland
17
139
User
Syddanmark
17
140
User
Hovedstaden
17
141
User
Hovedstaden
17
142
User
Syddanmark
17
143
User
Hovedstaden
17
144
User
Hovedstaden
17
145
User
Region
17
146
User
Region
17
147
User
Region
17
148
User
Sjælland
16
149
User
Midtjylland
16
150
August Soelje
Hovedstaden
16
151
User
Hovedstaden
16
152
User
Hovedstaden
16
153
User
Hovedstaden
16
154
User
Hovedstaden
15
155
User
Hovedstaden
15
156
User
Syddanmark
15
157
Tommy Carlsen
Hovedstaden
15
158
User
Midtjylland
15
159
User
Midtjylland
15
160
User
Nordjylland
15
161
User
Nordjylland
15
162
User
Hovedstaden
15
163
User
Midtjylland
15
164
User
Syddanmark
15
165
User
Region
15
166
User
Hovedstaden
14
167
User
Sjælland
14
168
fishcannon gaming
Hovedstaden
14
169
User
Hovedstaden
14
170
User
Midtjylland
14
171
User
Hovedstaden
14
172
User
Syddanmark
14
173
User
Midtjylland
14
174
User
Hovedstaden
13
175
User
Syddanmark
13
176
User
Hovedstaden
13
177
Caroline CC
Hovedstaden
13
178
User
Hovedstaden
13
179
User
Hovedstaden
13
180
User
Midtjylland
13
181
User
Hovedstaden
13
182
User
Syddanmark
13
183
User
Nordjylland
13
184
User
Region
13
185
User
Hovedstaden
12
186
User
Hovedstaden
12
187
User
Syddanmark
12
188
User
Hovedstaden
12
189
User
Hovedstaden
12
190
User
Sjælland
12
191
User
Hovedstaden
12
192
Mergus
Syddanmark
12
193
User
Hovedstaden
12
194
User
Hovedstaden
12
195
User
Hovedstaden
12
196
User
Hovedstaden
12
197
User
Hovedstaden
12
198
User
Hovedstaden
12
199
User
Hovedstaden
12
200
User
Midtjylland
12
201
User
Midtjylland
12
202
User
Syddanmark
12
203
User
Hovedstaden
12
204
User
Hovedstaden
11
205
User
Syddanmark
11
206
User
Hovedstaden
11
207
User
Hovedstaden
11
208
User
Hovedstaden
11
209
Karl Levinsen
Hovedstaden
11
210
User
Midtjylland
11
211
User
Hovedstaden
11
212
Lasse Andersen
Syddanmark
11
213
Johan Sørensen
Hovedstaden
11
214
User
Midtjylland
11
215
User
Hovedstaden
11
216
User
Hovedstaden
11
217
User
Hovedstaden
11
218
User
Region
11
219
User
Midtjylland
10
220
User
Syddanmark
10
221
User
Hovedstaden
10
222
User
Nordjylland
10
223
User
Hovedstaden
10
224
User
Hovedstaden
10
225
User
Hovedstaden
10
226
User
Hovedstaden
10
227
User
Hovedstaden
10
228
User
Hovedstaden
10
229
User
Hovedstaden
10
230
User
Syddanmark
10
231
User
Hovedstaden
10
232
User
Hovedstaden
10
233
User
Region
10
234
User
Region
10
235
User
Sjælland
9
236
User
Hovedstaden
9
237
User
Hovedstaden
9
238
User
Midtjylland
9
239
User
Hovedstaden
9
240
User
Hovedstaden
9
241
User
Hovedstaden
9
242
User
Syddanmark
9
243
mikkel.vigel@hotmail.com
Hovedstaden
9
244
User
Hovedstaden
9
245
User
Sjælland
9
246
User
Hovedstaden
9
247
User
Hovedstaden
9
248
User
Midtjylland
9
249
User
Midtjylland
9
250
User
Hovedstaden
8
251
User
Syddanmark
8
252
User
Hovedstaden
8
253
User
Syddanmark
8
254
User
Syddanmark
8
255
User
Hovedstaden
8
256
User
Midtjylland
8
257
User
Hovedstaden
8
258
User
Hovedstaden
8
259
User
Hovedstaden
8
260
User
Hovedstaden
8
261
User
Hovedstaden
8
262
User
Hovedstaden
8
263
User
Hovedstaden
8
264
User
Hovedstaden
8
265
User
Nordjylland
8
266
User
Region
8
267
User
Hovedstaden
7
268
Ozzy
Midtjylland
7
269
User
Midtjylland
7
270
Max Hansen
Syddanmark
7
271
User
Midtjylland
7
272
User
Midtjylland
7
273
User
Hovedstaden
7
274
User
Hovedstaden
7
275
User
Midtjylland
7
276
User
Hovedstaden
7
277
User
Hovedstaden
7
278
User
Hovedstaden
7
279
User
Hovedstaden
7
280
User
Hovedstaden
7
281
User
Syddanmark
7
282
User
Hovedstaden
7
283
User
Hovedstaden
7
284
User
Midtjylland
7
285
User
Hovedstaden
7
286
Philip Olsson
Hovedstaden
7
287
User
Hovedstaden
7
288
User
Hovedstaden
7
289
User
Hovedstaden
7
290
User
Region
7
291
User
Region
7
292
User
Region
7
293
User
Hovedstaden
6
294
User
Hovedstaden
6
295
User
Hovedstaden
6
296
User
Syddanmark
6
297
User
Midtjylland
6
298
User
Hovedstaden
6
299
User
Nordjylland
6
300
User
Hovedstaden
6
301
User
Nordjylland
6
302
User
Midtjylland
6
303
User
Midtjylland
6
304
User
Sjælland
6
305
User
Hovedstaden
6
306
User
Syddanmark
6
307
User
Hovedstaden
6
308
User
Hovedstaden
6
309
User
Region
6
310
User
Region
6
311
User
Region
6
312
User
Region
6
313
User
Hovedstaden
5
314
Ricky
Hovedstaden
5
315
Torben Musolf
Syddanmark
5
316
User
Hovedstaden
5
317
User
Hovedstaden
5
318
User
Hovedstaden
5
319
User
Midtjylland
5
320
User
Midtjylland
5
321
User
Hovedstaden
5
322
User
Nordjylland
5
323
User
Midtjylland
5
324
User
Hovedstaden
5
325
User
Hovedstaden
5
326
User
Syddanmark
5
327
User
Sjælland
5
328
User
Syddanmark
5
329
User
Midtjylland
5
330
User
Hovedstaden
5
331
User
Midtjylland
5
332
User
Midtjylland
5
333
User
Hovedstaden
5
334
User
Midtjylland
5
335
User
Hovedstaden
5
336
User
Syddanmark
5
337
User
Syddanmark
5
338
User
Midtjylland
5
339
User
Nordjylland
5
340
User
Hovedstaden
5
341
User
Hovedstaden
5
342
Marco Simmini
Hovedstaden
5
343
User
Hovedstaden
5
344
User
Hovedstaden
5
345
User
Syddanmark
5
346
User
Hovedstaden
4
347
User
Hovedstaden
4
348
User
Hovedstaden
4
349
User
Hovedstaden
4
350
User
Syddanmark
4
351
User
Syddanmark
4
352
User
Nordjylland
4
353
User
Hovedstaden
4
354
User
Midtjylland
4
355
User
Syddanmark
4
356
User
Hovedstaden
4
357
User
Midtjylland
4
358
User
Syddanmark
4
359
User
Hovedstaden
4
360
User
Hovedstaden
4
361
User
Hovedstaden
4
362
User
Hovedstaden
4
363
User
Hovedstaden
4
364
User
Nordjylland
4
365
User
Midtjylland
4
366
User
Hovedstaden
4
367
User
Hovedstaden
4
368
User
Midtjylland
4
369
User
Hovedstaden
4
370
User
Region
4
371
User
Region
4
372
User
Region
4
373
User
Region
4
374
User
Region
4
375
User
Region
4
376
User
Region
4
377
User
Region
4
378
User
Hovedstaden
3
379
User
Sjælland
3
380
User
Hovedstaden
3
381
jrk
Syddanmark
3
382
User
Hovedstaden
3
383
User
Hovedstaden
3
384
User
Hovedstaden
3
385
User
Midtjylland
3
386
User
Hovedstaden
3
387
User
Hovedstaden
3
388
User
Hovedstaden
3
389
User
Hovedstaden
3
390
User
Hovedstaden
3
391
User
Hovedstaden
3
392
User
Nordjylland
3
393
User
Hovedstaden
3
394
User
Hovedstaden
3
395
User
Syddanmark
3
396
User
Syddanmark
3
397
User
Hovedstaden
3
398
User
Hovedstaden
3
399
User
Hovedstaden
3
400
User
Midtjylland
3
401
User
Hovedstaden
3
402
User
Hovedstaden
3
403
User
Hovedstaden
3
404
User
Hovedstaden
3
405
User
Region
3
406
User
Region
3
407
KIm Lungay
Region
3
408
User
Midtjylland
2
409
User
Hovedstaden
2
410
User
Syddanmark
2
411
User
Hovedstaden
2
412
Emil Brandt
Syddanmark
2
413
User
Hovedstaden
2
414
User
Midtjylland
2
415
User
Midtjylland
2
416
User
Nordjylland
2
417
User
Syddanmark
2
418
User
Midtjylland
2
419
User
Hovedstaden
2
420
User
Hovedstaden
2
421
User
Nordjylland
2
422
User
Midtjylland
2
423
maryama hashi
Sjælland
2
424
User
Hovedstaden
2
425
User
Hovedstaden
2
426
User
Hovedstaden
2
427
User
Hovedstaden
2
428
User
Nordjylland
2
429
User
Nordjylland
2
430
User
Syddanmark
2
431
User
Hovedstaden
2
432
User
Syddanmark
2
433
User
Hovedstaden
2
434
User
Hovedstaden
2
435
User
Sjælland
2
436
User
Hovedstaden
2
438
User
Midtjylland
2
439
User
Sjælland
2
440
User
Hovedstaden
2
441
User
Hovedstaden
2
442
User
Hovedstaden
2
443
User
Hovedstaden
2
444
User
Midtjylland
2
445
User
Syddanmark
2
446
User
Hovedstaden
2
447
User
Midtjylland
2
448
User
Syddanmark
2
449
User
Midtjylland
2
450
User
Hovedstaden
2
451
User
Hovedstaden
2
452
User
Hovedstaden
2
453
User
Hovedstaden
2
454
User
Syddanmark
2
455
User
Hovedstaden
2
456
User
Syddanmark
2
457
User
Hovedstaden
2
458
User
Midtjylland
2
459
User
Hovedstaden
2
460
User
Hovedstaden
2
461
User
Hovedstaden
2
462
User
Hovedstaden
2
463
User
Midtjylland
2
464
User
Midtjylland
2
465
User
Hovedstaden
2
466
User
Region
2
467
User
Region
2
468
User
Region
2
469
User
Region
2
470
User
Region
2
471
User
Region
2
472
User
Region
2
473
User
Region
2
474
User
Region
2
475
User
Hovedstaden
1
476
User
Midtjylland
1
477
User
Sjælland
1
478
User
Syddanmark
1
479
User
Hovedstaden
1
480
User
Syddanmark
1
481
Na el
Midtjylland
1
482
User
Midtjylland
1
483
User
Hovedstaden
1
484
User
Syddanmark
1
485
User
Hovedstaden
1
486
User
Sjælland
1
487
User
Hovedstaden
1
488
User
Midtjylland
1
489
User
Nordjylland
1
490
User
Midtjylland
1
491
User
Syddanmark
1
492
User
Hovedstaden
1
493
User
Syddanmark
1
494
Emil Lykke Grann
Midtjylland
1
495
Helga correia
Sjælland
1

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A Web App for Mathematics Training

Do you want to be fast at mental math? Many people do, but the options for doing the necessary exercises are simply too cumbersome for all but the most dedicated of trainees. In physical fitness, many people are interested in training their bodies but allocating the time, energy, and money for it is a significant obstacle. Likewise, lugging around books and whatnot for math practice is a threshold that just doesn't meet the standards of modern life.

Training yourself to be skilled at mental math needs to be quick and convenient. mathtrainer.org is a web app that works in your browser rather than a program you have to download and install on your computer or phone. This allows users to try and use the app without having to install new software. As a web app, updates are also much simpler. There is no need to download endless updates—the website will always be the most current version.

You can access a web app from any device connected to the internet and equipped with a web browser, including smartphones, tablets, and desktop computers. Moreover, you are free to use whichever browser you prefer, including Google Chrome, Safari, Firefox, and others. Google Chrome is the recommended browser for the best maths training since it tends to lead the pack in supporting the latest web technologies.

Math Trainer is designed to offer a similar experience regardless of what you’re using to access it, whether it be Android, iOS, Windows, or another operating system. Though an on-screen touch keyboard will appear on mobile devices, you may prefer to use the app on a desktop with a keyboard. Hopefully the advantages of a web app for convenient mathematics training are apparent.

Another part of making the app easy to use is eliminating the need for signing up and logging in. Users can get started with their math training as soon as they click the start button on this page. After progressing to higher levels in the app, your progress is automatically saved so long as your return to the site through the same browser.


What's a Mental Math Tip?

A mental math tip is a sequence of steps that can be taken to solve a math problem in your head. Click the arrow below to see an example for the following problem:

÷
984
3
328

A tip like this one is available for every problem in Math Trainer, so there's always help if you get stuck. With enough practice, you'll be able to predict what the tips will say—you'll have learned mental math!


Get Better at Mental Math

The ability to quickly perform mental calculations offers advantages in certain circumstances. But even without applications, getting better at mental math is a great way to stimulate one’s mind. It develops better number sense and intuition for quantifying the world around us. Practicing mental calculation will strengthen your foundation for learning more advanced maths topics.

Nonetheless, the tangible benefits of improving at mental math are many. It is certainly expected that educated people are able to do simple arithmetic without having to pull out a calculator. An inability to do so may reflect poorly on you, while being well-practiced in mental calculation will leave your contemporaries impressed. In many scientific and technical circles, mental math ability is even more highly regarded.

For students, mental calculation speed will often have a direct impact on math and science test scores. At all grade levels, it is not sufficient to know how to solve math problems when tests have a time limit on them. The highest-scoring test takers are able to answer questions both correctly and efficiently. Improving mental math skills will only benefit a student’s academic career.

Calculating the solution to an arithmetic problem in your head is often faster than pulling out a device to tell you the answer. For example, figuring out how much to tip a server at a restaurant is a straightforward arithmetic problem that many people are unable to perform without a calculator. By training your brain to solve basic math problems, you can save time in situations like these.

Mental math can also be relied upon when calculation devices are not available. Even with the conveniences of modern life, we occasionally find ourselves without access to our cell phones or other capable devices. A mind skilled in mental math is always available to you.

Finally, getting better at mental math enables a quick estimate and sanity check on results obtained from calculators. While computers are extremely reliable at solving math problems, there is always the risk of incorrectly inputting the problem to the computer. By getting better at mental mathematics, you will develop an intuition for whether the results of calculators make sense.

In fact, the ability to estimate is often sufficient to avoid using calculators altogether. While the use of computers is widespread, estimation is an increasingly valued skill in many industries. There are many situations where complex math will eventually be required, but a preliminary estimate is needed quickly. A major boost to productivity!


Use a Math Trainer

Mental math ability is a lot like physical fitness training. You may be out of shape in the beginning, but with diligent training you can and will improve. Initially you might not enjoy the exercise, but you will reap significant rewards for your effort. As you become more fit, you’ll begin to enjoy the activity much more. If you are serious about it, your mental calculation fitness could become a source of energy, galvanizing you to face the challenges of life with enthusiasm.

In physical training, you break down the fibers in your muscles during a workout session. Your muscles actually sustain tiny tears during resistance training exercises. While you rest afterwards, your body repairs the damage, rebuilding the fibers thicker and stronger.

A similar process is believed to occur for cognitive tasks. A 2016 study found "extensive evidence that brain-training interventions improve performance on the trained tasks".1 Therefore you can expect training your brain to answer mental math questions will lead to improved performance over time.

In the context of physical fitness, a "trainer" often refers to a trained professional who guides the workout and recovery process. Personal trainers are tasked with assessing a trainee's level of ability, prescribing an exercise regimen, and offering feedback as the training goes along. The word "trainer" could also refer to a system that automates the role of a personal trainer. Many aerobic exercise machines today offer interactive training programs with feedback and analysis of a user's performance.

A math trainer is needed for optimal math fitness. Like in physical fitness, the trainer should be compatible with users at a variety of skill levels and should guide them to the next level. It should give an accurate assessment of a user's strengths and weakness, as well as offer helpful feedback on where to focus one's efforts. Learning the ropes of mental maths with a math trainer should be a seamless, rewarding journey to ever-greater abilities.


Mental Calculation

Mental calculation, or mental math, is performing arithmetical calculations without the aid of tools or supplies. As opposed to using a calculator or pencil and paper, mental math is performed entirely in one’s head.

People use mental calculation when computation aides are not available, when it is faster to do so, or when they wish to practice, show off, or participate in mental math competitions. Most people perform basic mental calculation using elementary arithmetic on a daily basis. An inability to calculate mentally is a serious obstacle to many common tasks.

In U.S. schools, mental calculation is taught only for the most elementary arithmetic, such as single-digit addition and multiplication of two numbers between 0 and 12. To solve addition problems involving multiple digits, you are taught to add columns of digits from right to left, carrying the tens digit if the column sum exceeds 9. For example, how would you approach this addition problem?

Example of two-digit
addition problem

If you were trained like many of us were, you’d add the right column to obtain 12. Since that’s two digits, you’d write the 2 under the right column and carry the 10 by writing a 1 above the left column. Finally, you’d add the two tens digits and the carried 1 to obtain the answer, 52.

To solve an addition problem mentally, it’s best to add the columns from left to right. In our example, you could add the tens digit of the second number, 30, to the first number, 14, to obtain 44. This is easier than the full problem because you’re just doing one mental calculation and tacking on the 4 from the 14 as the singles digit. Then you’d add the remaining ones digit of the second number, 8, to 44 to arrive at the answer, 52.

Which approach seems simpler to you? Can you do the first approach without pulling out a pencil and paper? It turns out the same advantages of left-to-right addition apply to much larger numbers as well. It’s unlikely that difficult addition problems can be solved right to left without needing to write it all out, which of course is more time consuming.

Mental math should be distinguished from the memorization of math facts such as multiplication tables. A foundation of memorized answers to simple math problems will make mental math easier, but performing maths in your head requires both memorized facts and the manipulation of numbers and operations to solve problems. This combination of skill and memory allows us to solve far more complex math questions than can be answered with readily-memorized math facts.

Many mental math tricks are specific to particular numbers or types of problems, usually dependent on the base of the number system used. In the decimal numeral system, for example, it is trivially easy to multiply by 10—just add a 0 to the end of the number. This mental math trick wouldn’t work in the hexadecimal numeral system, though, because the base is 16 instead of 10.

Therefore mental calculation is the ability to manipulate complex arithmetic problems in such a way that they can be resolved using simple memorized math facts.


Arithmetic

Arithmetic is the branch of mathematics concerning basic number operations: addition, subtraction, multiplication, and division. As kids, we are taught to do arithmetic because real-world math problems depend on a mastery of elementary arithmetic. Higher-level study of arithmetic and the integers, or whole numbers, is known as number theory.

Though the math kids initially study is arithmetic, the word is rarely used in this context anymore. Originally it comes from the Greek arithmos, meaning “number”. It has however been included in the “three Rs” of elementary Western education: reading, writing, and arithmetic.

There is evidence prehistoric humans were using arithmetic as hunter-gatherers. Archaeologists have uncovered a tally stick, believed to be over 20,000 years old, which may exhibit the earliest known sequences of prime numbers. An understanding of prime numbers, which are only divisible by themselves and the number 1, requires knowledge of the operation in arithmetic known as division.

From tally marks came base-10 numerals such as those used in Egypt over 5,000 years ago. Number systems based on 10 probably arose because humans have ten “digits” as fingers on their hands (or toes on their feet). A later advance in arithmetic was positional notation, which allowed the same symbols to represent different magnitudes depending on their position in the written number. These numeral systems allowed complex arithmetic to be communicated, recorded, and applied to the challenges faced by our ancestors.

The basic operation of arithmetic is addition. It combines two or more numbers into one, the sum of the terms. The terms can be added in any order, which is known as the commutative property of arithmetic. On a number line, the sum of two numbers is the total distance from zero covered by both numbers.

The inverse arithmetical operation of addition is subtraction. It finds the difference between two numbers. Subtraction is not commutative because the order of the numbers determines whether the answer is positive or negative. On a number line, the difference between two numbers is the distance between their positions.

A second basic operation of arithmetic is multiplication, which scales a number by another number. This second number is called a factor. Like addition, multiplication is commutative—you can change the order of the factors and still get the same answer. Multiplication on a number line can be viewed as adding the first number a total number of times equal to the second factor.

Finally, division is an arithmetical operation that is essentially the inverse of multiplication. Rather than scaling a number, it is divided into a number of pieces equal to the second number. Dividing by the number 0 is not defined in arithmetic because dividing something into zero pieces is impossible.

Basic arithmetic allows us to evaluate the answers to an unlimited number of mathematical expressions. Arithmetical expressions can be purely mathematical, as in 2 + 2, or they can represent quantities in the physical world, such as two items plus two more. Understanding the laws of arithmetic is tremendously powerful.


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