Leaderboard
Michigan
Rank
Level
1
User
Adrian
97
2
Anand Bahl
Ann Arbor
91
3
Hui Cai
Ann Arbor
84
4
User
Grass Lake Charter Township
67
5
User
Grass Lake Charter Township
59
6
User
Ann Arbor
59
7
User
Ann Arbor
55
8
User
Ann Arbor
55
9
User
Grass Lake Charter Township
53
10
User
East Lansing
53
11
User
Grass Lake Charter Township
52
12
User
Grass Lake Charter Township
52
13
User
Ann Arbor
52
14
User
Ann Arbor
52
15
User
Detroit
52
16
User
Ann Arbor
51
17
User
Ann Arbor
51
18
User
Parma
51
19
User
Grass Lake Charter Township
51
20
User
Grass Lake Charter Township
51
21
User
Ann Arbor
50
22
User
Ann Arbor
50
23
User
Ann Arbor
50
24
User
Ann Arbor
48
25
User
Grass Lake Charter Township
48
26
User
Ann Arbor
48
27
Jake Hart
Plymouth
46
28
User
Sterling Heights
45
29
User
Adrian
44
30
User
Ann Arbor
44
31
User
Ann Arbor
43
32
Carter Pryor
Ann Arbor
42
33
User
Rochester Hills
42
34
Sára Losonczy
Novi
39
35
User
Grass Lake Charter Township
39
36
User
Grass Lake Charter Township
38
37
Ben Stensen
Ann Arbor
38
38
User
Farmington Hills
38
39
User
Ann Arbor
38
40
User
Ann Arbor
38
41
User
Hesperia
37
42
Maria Dawson
Sterling Heights
37
43
Michael Levesque
Saline
37
44
User
Ann Arbor
35
45
User
Niles
35
46
Vanni T
Detroit
34
47
User
Grass Lake Charter Township
33
48
User
Ann Arbor
33
49
User
Ann Arbor
33
50
Continuous Peach
Ann Arbor
33
51
User
Ann Arbor
32
52
User
Dearborn
32
53
User
Ann Arbor
31
54
User
Ann Arbor
31
55
User
Marquette
30
56
maes
Blackman
30
57
KAT
Oscoda Township
30
58
Alex Zureick
West Bloomfield Township
30
59
User
Sterling Heights
29
60
Aliya Renee Khan
Northville
29
61
User
Georgetown Township
29
62
User
Ann Arbor
29
63
Saarah Alam
Plymouth
29
64
Alan Tilley
Traverse City
28
65
User
Ann Arbor
28
66
User
Ann Arbor
28
67
User
Ann Arbor
28
68
User
Sterling Heights
27
69
User
West Bloomfield Township
27
70
User
Detroit
27
71
User
Detroit
26
72
User
Petoskey
26
73
Shelagh Fehrenbach
Traverse City
26
74
User
Capac
26
75
Albert Cai
Ann Arbor
26
76
Ananya
Saginaw
26
77
User
Capac
25
78
User
Ann Arbor
25
79
User
Charlotte
25
80
KrisViana
Kentwood
25
81
User
Grass Lake Charter Township
24
82
User
Ann Arbor
24
83
Anshu ankita raj
Ann Arbor
24
84
User
Ann Arbor
23
85
User
Ann Arbor
23
86
User
Beechwood
23
87
David Postiff
York Charter Township
23
88
User
Detroit
23
89
User
Ann Arbor
23
90
User
Detroit
23
91
Alexa Williams-Lall
Ypsilanti Charter Township
22
92
User
Houghton
22
93
User
Fair Plain
22
94
User
Rochester Hills
22
95
User
Ann Arbor
22
96
Erin Glumb
Pinckney
22
97
User
Charlotte
22
98
User
Ann Arbor
22
99
User
Beechwood
21
100
User
Ann Arbor
21
101
User
Ann Arbor
21
102
Seminar
Homer
21
103
Jennifer Kusch
Saline
21
104
User
Capac
21
105
User
Beechwood
21
106
User
Dearborn
21
107
User
Ann Arbor
21
108
lbaroudi
Ann Arbor
21
109
User
Canton
21
110
User
Ann Arbor
21
111
László Losonczy
Novi
20
112
Nick Reymann
Detroit
20
113
User
Detroit
20
114
User
Lyon Charter Township
20
115
User
Lansing
19
116
User
Ferndale
19
117
User
Capac
19
118
User
Midland
19
119
User
Midland
19
120
User
Charlotte
19
121
User
Troy
19
122
User
Alpena
18
123
User
Charter Township Of Clinton
18
124
User
South Lyon
18
125
User
Capac
18
126
User
Midland
18
127
User
Midland
18
128
Max
Ann Arbor
18
129
User
Plymouth
18
130
User
Ann Arbor
18
131
User
Oak Park
17
132
User
Ann Arbor
17
133
User
Ann Arbor
17
134
ISAIAH POTTER
Charlotte
17
135
User
Ann Arbor
17
136
User
Ann Arbor
17
137
User
Wyoming
16
138
User
Ann Arbor
16
139
User
Rochester Hills
16
140
User
Midland
16
141
User
Beechwood
16
142
User
Ann Arbor
16
143
User
Ann Arbor
16
144
User
Ann Arbor
16
145
User
Ann Arbor
16
146
User
Ann Arbor
15
147
User
Detroit
15
148
User
Milan
14
149
User
Meridian Charter Township
14
150
Deena Shah
Canton
14
151
User
Detroit
14
152
User
Midland
14
153
User
Ann Arbor
14
154
User
Ann Arbor
14
155
User
Ann Arbor
14
156
User
Charter Township Of Clinton
14
157
User
Ann Arbor
14
158
User
Ann Arbor
14
159
User
Charlotte
14
160
Patricia Collard
Delhi Charter Township
14
161
Advik Dogra
Troy
13
162
User
Ann Arbor
13
163
User
Caro
13
164
User
Grand Rapids
13
165
User
Ann Arbor
13
166
User
Midland
13
167
User
Charlotte
13
168
User
Romulus
13
169
User
Southfield
13
170
User
Plymouth
13
171
User
East Lansing
12
172
Deja
Detroit
12
173
User
Harrison Charter Township
12
174
User
Rochester Hills
12
175
User
Ann Arbor
12
176
User
Ann Arbor
12
177
User
Ann Arbor
12
178
User
Novi
12
179
User
Dearborn Heights
12
180
User
Ann Arbor
12
181
User
Troy
12
182
User
Detroit
11
183
Ryan Meister
Marquette
11
184
User
Rochester Hills
11
185
User
East Lansing
11
186
Milo Pappas
Ann Arbor
11
187
User
Detroit
11
188
User
Warren
11
189
User
Allendale Charter Township
11
190
Bobby b
Ann Arbor
11
191
User
Beechwood
11
192
User
Southfield
11
193
User
Sterling Heights
11
194
User
Genesee Township
11
195
User
Ann Arbor
10
196
User
Westland
10
197
User
Georgetown Township
10
198
User
Coldwater
10
199
User
Ann Arbor
10
200
User
Portage
10
201
User
Grass Lake Charter Township
10
202
User
Ann Arbor
10
203
User
Saline
10
204
User
Royal Oak
10
205
User
Detroit
9
206
User
Ann Arbor
9
207
User
Houghton
9
208
User
Novi
9
209
User
Midland
9
210
User
Grand Rapids
9
211
User
South Lyon
9
212
User
Ann Arbor
9
213
User
Rochester Hills
9
214
User
Capac
9
215
User
Midland
9
216
User
Ann Arbor
9
217
User
Charlotte
9
218
User
Ann Arbor
9
219
User
Farmington Hills
9
220
User
Brighton
9
221
User
Farmington Hills
8
222
User
Macomb
8
223
User
Detroit
8
224
sk
Vestaburg
8
225
User
Kalamazoo
8
226
User
East Lansing
8
227
User
Lansing
8
228
User
Marquette
8
229
User
Marquette
8
230
User
Coldwater
8
231
Mercedes
Midland
8
232
User
Midland
8
233
User
Midland
8
234
User
Ann Arbor
8
235
User
Midland
8
236
User
Charlotte
8
237
User
Ann Arbor
8
238
User
Grand Rapids
8
239
User
Detroit
8
240
User
Detroit
7
241
User
Clinton Township
7
242
Retard
Commerce Charter Township
7
243
User
Detroit
7
244
User
Jackson
7
245
User
Farmington Hills
7
246
User
Holland
7
247
User
Ann Arbor
7
248
User
Beechwood
7
249
User
Capac
7
250
User
Midland
7
251
User
Detroit
7
252
User
Ann Arbor
7
253
User
West Bloomfield Township
7
254
User
Ann Arbor
7
255
User
Charlotte
7
256
User
Charlotte
7
257
Mara Rada
Troy
7
258
User
Detroit
7
259
User
Plymouth
7
260
User
Mason
7
261
User
Troy
7
262
User
Detroit
6
263
User
Detroit
6
264
User
Garfield Township
6
265
User
New Baltimore
6
266
User
Troy
6
267
User
Kalamazoo
6
268
User
Dearborn
6
269
User
Ann Arbor
6
270
User
Grass Lake Charter Township
6
271
User
Capac
6
272
User
Capac
6
273
User
Capac
6
274
User
Capac
6
275
User
Capac
6
276
User
Midland
6
277
User
Midland
6
278
User
Midland
6
279
User
Midland
6
280
User
Midland
6
281
User
Troy
6
282
User
Charlotte
6
283
User
Charlotte
6
284
User
Ann Arbor
6
285
User
Midland
6
286
User
Clay Township
6
287
User
Kent County
6
288
User
Detroit
5
289
User
Edmore
5
290
User
York Charter Township
5
291
User
Ann Arbor
5
292
User
Detroit
5
293
User
Marquette
5
294
User
Detroit
5
295
User
Detroit
5
296
User
Ann Arbor
5
297
User
Ann Arbor
5
298
User
Ionia
5
299
User
Shelby Charter Township
5
300
User
East Lansing
5
301
User
Warren
5
302
User
Saint Joseph
5
303
User
Capac
5
304
User
Midland
5
305
User
Forest Hills
5
306
User
Charlotte
5
307
User
Charlotte
5
308
User
Charlotte
5
309
Patricia Collard
Charlotte
5
310
User
Farmington Hills
5
311
User
Ann Arbor
5
312
User
Detroit
5
313
Crispin Contreras
East Lansing
5
314
User
Whitehall
5
315
User
Okemos
5
316
Merit Amadi
Dearborn
5
317
User
Detroit
4
318
User
Detroit
4
319
User
Livonia
4
320
User
Harrison
4
321
User
Detroit
4
322
User
Royal Oak
4
323
User
Auburn Hills
4
324
User
Grass Lake Charter Township
4
325
User
Ann Arbor
4
326
User
Rochester Hills
4
327
User
Saginaw
4
328
User
Taylor
4
329
User
Marquette
4
330
User
Ann Arbor
4
331
Scott Halasy
Ferndale
4
332
User
Ypsilanti
4
333
User
Grand Rapids
4
334
User
Ann Arbor
4
335
foggy
Independence Charter Township
4
336
User
Grand Rapids
4
337
Ryan
Commerce Charter Township
4
338
User
Ann Arbor
4
339
User
Holland
4
340
User
Capac
4
341
User
Midland
4
342
User
Pontiac
4
343
User
Texas Charter Township
4
344
User
Beechwood
4
345
User
Beechwood
4
346
User
Beechwood
4
347
User
Midland
4
348
User
Beechwood
4
349
User
Beechwood
4
350
User
Midland
4
351
User
Dansville
4
352
User
Midland
4
353
User
Charlotte
4
354
User
Charlotte
4
355
User
Detroit
4
356
User
Ann Arbor
4
357
User
Rochester
4
358
Dawn
Port Huron
4
359
User
Auburn Hills
4
360
User
Hancock
4
361
User
Ann Arbor
3
362
User
Grand Rapids
3
363
User
Shelby Charter Township
3
364
User
Edmore
3
365
User
Dansville
3
366
User
Rochester Hills
3
367
User
Grand Rapids
3
368
User
Rockford
3
369
User
Southfield
3
370
User
Detroit
3
371
User
Richland
3
372
User
Georgetown Township
3
373
User
Ann Arbor
3
374
Jackson
Ann Arbor
3
375
User
Ann Arbor
3
376
User
Roseville
3
377
User
Ann Arbor
3
378
User
Ann Arbor
3
379
User
Ann Arbor
3
380
User
Garfield Township
3
381
User
Ann Arbor
3
382
User
Petoskey
3
383
User
Beechwood
3
384
User
Ann Arbor
3
385
User
Coldwater
3
386
User
Coldwater
3
387
User
Charter Township Of Clinton
3
388
User
Dexter
3
389
User
Birch Run
3
390
User
Ann Arbor
3
391
User
Capac
3
392
User
Capac
3
393
User
Midland
3
394
User
Beechwood
3
395
User
Forest Hills
3
396
User
Livonia
3
397
User
Midland
3
398
User
Midland
3
399
User
Midland
3
400
User
Battle Creek
3
401
User
Ann Arbor
3
402
User
Ann Arbor
3
403
User
Midland
3
404
User
Midland
3
405
User
Midland
3
406
User
Ann Arbor
3
407
User
Ann Arbor
3
408
jadin
Charlotte
3
409
User
Charlotte
3
410
User
Charlotte
3
411
Kaylen Gangi
Charlotte
3
412
User
Detroit
3
413
User
Grand Rapids
3
414
User
Detroit
3
415
User
Ypsilanti
3
416
User
Dearborn Heights
3
417
User
Clinton Township
2
418
User
Detroit
2
419
User
Detroit
2
420
User
Detroit
2
421
User
Ann Arbor
2
422
User
Clinton Township
2
423
User
Capac
2
424
User
Capac
2
425
User
Sterling Heights
2
426
User
Detroit
2
427
Olivia Bowron
Ann Arbor
2
428
User
Township Of Hamburg
2
429
User
Ann Arbor
2
430
User
Midland
2
431
User
Traverse City
2
432
User
Jackson
2
433
User
Marquette
2
434
User
Dearborn
2
435
User
Taylor
2
436
User
Jackson
2
437
User
Rochester Hills
2
438
User
Midland
2
439
Valerie Chri
Ann Arbor
2
440
User
Union Charter Township
2
441
User
Grand Blanc Township
2
442
User
Ann Arbor
2
443
User
Kentwood
2
444
User
Sterling Heights
2
445
User
Parma
2
446
User
Coldwater
2
447
User
Coldwater
2
448
User
Coldwater
2
449
User
Georgetown Township
2
450
User
Ann Arbor
2
451
User
Coldwater
2
452
User
Coldwater
2
453
User
Coldwater
2
454
User
Coldwater
2
455
User
Coldwater
2
456
User
Livonia
2
457
User
Capac
2
458
User
Capac
2
459
User
Capac
2
460
User
Ann Arbor
2
461
User
Midland
2
462
User
Ann Arbor
2
463
User
Ann Arbor
2
464
User
Ann Arbor
2
465
User
Midland
2
466
User
Midland
2
467
Captain Moroni
Saint Johns
2
468
User
Detroit
2
469
User
Midland
2
470
User
Charlotte
2
471
User
Charlotte
2
472
User
Charlotte
2
473
User
Charlotte
2
474
User
Charlotte
2
475
User
Charlotte
2
476
User
Charlotte
2
477
User
Ann Arbor
2
478
User
Independence Charter Township
2
479
User
Farmington Hills
2
480
User
Ann Arbor
2
481
User
Ann Arbor
2
482
User
Ann Arbor
2
483
User
Novi
2
484
User
Plymouth
2
485
User
Zeeland
2
486
Lee
Detroit
2
487
User
East Lansing
2
488
User
Grand Blanc
2
489
User
Grand Rapids
2
490
User
Ann Arbor
2
491
User
Washtenaw County
2
492
User
Detroit
1
493
User
Detroit
1
494
User
Detroit
1
495
User
Detroit
1
496
User
Detroit
1
497
User
Detroit
1
498
User
Detroit
1
499
User
Detroit
1
500
User
Mount Pleasant
1
501
User
Detroit
1
502
User
Capac
1
503
User
Detroit
1
504
User
Ann Arbor
1
505
User
Dexter
1
506
User
Sterling Heights
1
507
User
Capac
1
508
User
Forest Hills
1
509
SimpleLivings
Detroit
1
510
Mike Wolf
East Jordan
1
511
User
Forest Hills
1
512
User
Meridian Charter Township
1
513
User
Battle Creek
1
514
Chante gary
Warren
1
515
Carole McVey
Rockford
1
516
gannon
Charlotte
1
517
Aarthi Gunasekaran
Holland
1
518
Markus Robinson
Inkster
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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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