Leaderboard
Tennessee
Rank
Level
1
User
Johnson City
83
2
Davis Zhang
Nashville
81
3
User
Nashville
76
4
User
Nashville
68
5
Dennisson Sherrell
Chattanooga
66
6
Steven Johnson
Kingsport
63
7
User
Nashville
62
8
User
Nashville
56
9
User
Franklin
56
10
woo woo
Chattanooga
55
11
User
Knoxville
55
12
User
Chattanooga
54
13
User
Nashville
54
14
User
Nashville
54
15
User
Smyrna
51
16
User
Nashville
49
17
Troy Morrow
Nashville
48
18
User
Collierville
46
19
User
Nashville
46
20
User
Nashville
45
21
User
Nashville
44
22
User
Nashville
43
23
Binyad Qaradakhi
Smyrna
41
24
User
Franklin
41
25
User
Nashville
41
26
User
Smyrna
40
27
User
Franklin
39
28
User
Collierville
39
29
User
Smyrna
38
30
User
Nashville
38
31
User
Chattanooga
37
32
Brandon Gong
Collierville
37
33
User
Nashville
37
34
User
Nashville
36
35
User
Chattanooga
35
36
User
Nashville
34
37
User
Smyrna
33
38
User
Nashville
33
39
Yunyu Lin
Nashville
33
40
User
Nashville
33
41
Kavitha Rangampally
Collierville
33
42
User
Sewanee
32
43
User
Smyrna
32
44
User
Clarksville
32
45
User
Chattanooga
31
46
User
Smyrna
31
47
User
Nashville
31
48
User
Smyrna
30
49
User
Kingsport
29
50
Benjamin Litt
Nashville
29
51
User
Memphis
29
52
User
Smyrna
28
53
User
Oak Hill
28
54
User
Memphis
27
55
User
Memphis
27
56
User
Smyrna
27
57
User
Nashville
27
58
User
Nashville
27
59
M C
Gallatin
26
60
User
Nashville
26
61
User
Nashville
26
62
Ashwin Balaje
Franklin
26
63
User
Smyrna
25
64
User
Smyrna
25
65
User
Smyrna
25
66
User
Nashville
25
67
User
Nashville
25
68
User
Charleston
25
69
User
Mountain City
25
70
User
Nashville
25
71
User
Clarksville
25
72
User
Nashville
24
73
User
Smyrna
24
74
User
Smyrna
24
75
User
Franklin
24
76
User
Smyrna
24
77
Mary Phan
Memphis
24
78
User
Franklin
24
79
User
Nashville
24
80
User
Lewisburg
23
81
Jared Duda
Nashville
23
82
User
Hendersonville
23
83
User
Nashville
23
84
User
Murfreesboro
23
85
User
Smyrna
22
86
User
Franklin
22
87
User
Nashville
22
88
User
Clarksville
22
89
User
Middleton
22
90
User
Nashville
22
91
User
Jefferson City
22
92
User
Memphis
22
93
User
Benton
22
94
User
Smyrna
21
95
User
Smyrna
21
96
User
Smyrna
21
97
SpaceGerm
Mount Juliet
21
98
User
Pleasant View
21
99
User
Smyrna
20
100
User
Smyrna
20
101
User
Clarksville
20
102
User
Collierville
20
103
User
Memphis
20
104
User
Smyrna
19
105
User
Smyrna
19
106
User
Smyrna
19
107
User
Clarksville
19
108
User
Clarksville
19
109
User
Memphis
19
110
User
La Vergne
18
111
User
Franklin
18
112
User
Nashville
18
113
User
Smyrna
17
114
User
Smyrna
17
115
User
Smyrna
17
116
User
Smyrna
17
117
User
Nashville
17
118
User
Nashville
17
119
User
Chattanooga
17
120
User
Nashville
17
121
User
Smyrna
16
122
User
Smyrna
16
123
User
Smyrna
16
124
User
Smyrna
16
125
User
Smyrna
16
126
User
Gallatin
16
127
User
Nashville
16
128
User
Memphis
16
129
Ehab Mansoor
Knoxville
16
130
User
Nashville
16
131
User
Knoxville
16
132
User
Knoxville
16
133
User
White House
15
134
User
Chattanooga
15
135
User
Chattanooga
15
136
User
Nashville
15
137
User
Cleveland
15
138
User
Lakesite
15
139
kyle Allman
Smyrna
15
140
User
Smyrna
15
141
User
Smyrna
15
142
User
Smyrna
15
143
User
Smyrna
15
144
User
Clarksville
15
145
User
Nashville
15
146
User
Nashville
15
147
User
Franklin
15
148
Angela Cordova
Cleveland
15
149
User
Murfreesboro
15
150
User
Nashville
15
151
User
Memphis
15
152
User
Franklin
15
153
User
Nashville
15
154
User
Knoxville
15
155
User
Nashville
14
156
User
Smyrna
14
157
User
Smyrna
14
158
Andrew Brodginski
Collierville
14
159
User
Nashville
14
160
User
Chattanooga
14
161
Kamiyah
Nashville
13
162
User
Smyrna
13
163
User
Smyrna
13
164
User
Smyrna
13
165
User
Smyrna
13
166
User
Smyrna
13
167
User
Smyrna
13
168
Camden
Smyrna
13
169
User
Smyrna
13
170
User
Smyrna
13
171
User
Bell Buckle
13
172
User
Nashville
13
173
User
Nashville
13
174
User
Memphis
13
175
User
Murfreesboro
13
176
User
Memphis
13
177
User
Nashville
13
178
User
Nashville
13
179
User
Pulaski
13
180
User
Chattanooga
13
181
User
Nashville
13
182
User
Knoxville
13
183
User
Chattanooga
12
184
User
Nashville
12
185
User
Smyrna
12
186
User
Smyrna
12
187
User
Smyrna
12
188
User
Smyrna
12
189
User
Smyrna
12
190
User
Smyrna
12
191
User
Smyrna
12
192
User
Smyrna
12
193
User
Smyrna
12
194
User
Smyrna
12
195
User
Mount Juliet
12
196
User
Smyrna
12
197
User
Smyrna
12
198
User
Smyrna
12
199
User
Smyrna
12
200
User
Smyrna
12
201
User
Smyrna
12
202
User
Clarksville
12
203
User
Nashville
12
204
User
Memphis
12
205
User
Nashville
12
206
User
City
12
207
User
Franklin
12
208
User
Knoxville
12
209
User
Knoxville
12
210
User
Nashville
11
211
User
Smyrna
11
212
User
Smyrna
11
213
User
Smyrna
11
214
User
Smyrna
11
215
User
Smyrna
11
216
User
Smyrna
11
217
User
Smyrna
11
218
User
Bell Buckle
11
219
Odin Sebring
Clarksville
11
220
User
Franklin
11
221
User
Clarksville
11
222
User
City
11
223
User
Hendersonville
11
224
User
Nashville
11
225
User
Knoxville
11
226
User
Nashville
11
227
User
Bloomingdale
10
228
User
Nashville
10
229
User
Smyrna
10
230
User
Smyrna
10
231
User
Smyrna
10
232
User
Smyrna
10
233
savannah carroll
Tullahoma
10
234
User
Smyrna
10
235
User
Smyrna
10
236
User
Smyrna
10
237
User
Smyrna
10
238
User
Nashville
10
239
User
Nashville
10
240
User
Nashville
10
241
User
Nashville
10
242
User
Franklin
10
243
User
Nashville
10
244
User
Memphis
10
245
User
Chattanooga
9
246
User
Smyrna
9
247
User
Smyrna
9
248
User
Smyrna
9
249
User
Smyrna
9
250
User
Smyrna
9
251
User
Smyrna
9
252
User
Smyrna
9
253
User
Smyrna
9
254
User
Smyrna
9
255
User
Maryville
9
256
User
Sparta
9
257
User
Farragut
9
258
User
Bartlett
9
259
Khalil Guy
Nashville
9
260
User
Clarksville
9
261
User
Dickson
9
262
User
Nashville
9
263
Kaleb Ruddle
Columbia
8
264
Chill Cyndaquil
Winchester
8
265
User
Clarksville
8
266
User
Nashville
8
267
User
Nashville
8
268
User
Smyrna
8
269
User
Smyrna
8
270
User
Smyrna
8
271
User
Smyrna
8
272
User
Smyrna
8
273
User
Smyrna
8
274
User
Smyrna
8
275
User
Smyrna
8
276
User
Smyrna
8
277
User
Smyrna
8
278
User
Smyrna
8
279
User
Smyrna
8
280
User
Smyrna
8
281
User
Smyrna
8
282
User
Smyrna
8
283
User
Smyrna
8
284
User
Smyrna
8
285
User
Smyrna
8
286
User
Nashville
8
287
User
Tullahoma
8
288
User
Nashville
8
289
User
Dickson
8
290
User
Murfreesboro
8
291
User
Nashville
8
292
User
Collierville
8
293
User
Nashville
8
294
User
Nashville
8
295
User
Nashville
7
296
User
Memphis
7
297
User
Smyrna
7
298
User
Smyrna
7
299
andres frailes
Smyrna
7
300
User
Smyrna
7
301
User
Nashville
7
302
User
Smyrna
7
303
User
Smyrna
7
304
User
Smyrna
7
305
User
Smyrna
7
306
User
Smyrna
7
307
User
Smyrna
7
308
User
Franklin
7
309
User
Smyrna
7
310
User
Smyrna
7
311
User
Smyrna
7
312
User
Nashville
7
313
User
Smyrna
7
314
User
Smyrna
7
315
User
Smyrna
7
316
User
Bell Buckle
7
317
User
Nashville
7
318
User
Nashville
7
319
User
Dickson
7
320
User
Clarksville
7
321
User
Hohenwald
7
322
User
Maryville
7
323
User
Knoxville
7
324
User
Brentwood
7
325
User
Nashville
6
326
Ariel Smith
Chattanooga
6
327
Heidi
Nashville
6
328
User
Brentwood
6
329
User
Smyrna
6
330
User
Smyrna
6
331
User
Smyrna
6
332
User
Smyrna
6
333
User
Smyrna
6
334
User
Smyrna
6
335
User
Smyrna
6
336
User
Smyrna
6
337
User
Smyrna
6
338
User
Smyrna
6
339
User
Smyrna
6
340
User
Smyrna
6
341
User
Smyrna
6
342
User
Smyrna
6
343
User
Smyrna
6
344
User
Smyrna
6
345
Marco
Smyrna
6
346
User
Smyrna
6
347
User
Smyrna
6
348
User
Smyrna
6
349
User
Smyrna
6
350
User
Smyrna
6
351
User
Smyrna
6
352
User
Nashville
6
353
Clairmarie Stotts
Memphis
6
354
User
Nashville
6
355
User
Nashville
6
356
User
Nashville
6
357
User
Shelbyville
6
358
User
Memphis
6
359
User
Memphis
6
360
User
Collierville
6
361
User
Charleston
6
362
User
Murfreesboro
6
363
User
Nashville
6
364
User
Memphis
6
365
User
Collierville
6
366
User
Columbia
5
367
User
Lafayette
5
368
User
Smyrna
5
369
User
Smyrna
5
370
User
Smyrna
5
371
User
Smyrna
5
372
User
Smyrna
5
373
User
Smyrna
5
374
User
Smyrna
5
375
User
Smyrna
5
376
User
Smyrna
5
377
User
Nashville
5
378
User
Smyrna
5
379
User
Smyrna
5
380
User
Smyrna
5
381
User
Smyrna
5
382
User
Smyrna
5
383
User
Smyrna
5
384
User
Smyrna
5
385
User
Smyrna
5
386
User
Smyrna
5
387
User
Kingsport
5
388
User
Nashville
5
389
User
Smyrna
5
390
User
Smyrna
5
391
User
Smyrna
5
392
User
Smyrna
5
393
User
Bell Buckle
5
394
User
Nashville
5
395
User
Sevierville
5
396
User
Memphis
5
397
User
Farragut
5
398
User
Livingston
5
399
User
Murfreesboro
5
400
User
Clarksville
5
401
User
Columbia
5
402
User
Collierville
5
403
User
Chapel Hill
5
404
User
Nashville
5
405
User
Eagleville
5
406
User
Charleston
5
407
User
Charleston
5
408
User
Kingsport
5
409
User
Pleasant View
5
410
User
Nashville
4
411
L
Dunlap
4
412
User
Chattanooga
4
413
User
Nashville
4
414
User
Gallatin
4
415
User
Smyrna
4
416
User
Smyrna
4
417
User
Smyrna
4
418
User
Smyrna
4
419
User
Smyrna
4
420
User
Smyrna
4
421
User
Smyrna
4
422
User
Smyrna
4
423
User
Nashville
4
424
User
Smyrna
4
425
User
Smyrna
4
426
User
Smyrna
4
427
User
Smyrna
4
428
User
Smyrna
4
429
User
Smyrna
4
430
User
Smyrna
4
431
User
Smyrna
4
432
User
Smyrna
4
433
User
Smyrna
4
434
User
Smyrna
4
435
Clairmarie
Memphis
4
436
User
Smyrna
4
437
sam
Smyrna
4
438
User
Clarksville
4
439
User
Memphis
4
440
User
Franklin
4
441
User
Smyrna
4
442
User
Smyrna
4
443
User
Smyrna
4
444
User
Smyrna
4
445
User
Smyrna
4
446
User
Smyrna
4
447
User
Smyrna
4
448
User
Smyrna
4
449
User
Smyrna
4
450
User
Smyrna
4
451
User
Mount Juliet
4
452
User
Brentwood
4
453
User
Collierville
4
454
User
Lebanon
4
455
User
Clarksville
4
456
Spencer Michaels
Nashville
4
457
User
Chattanooga
4
458
User
Memphis
4
459
User
Memphis
4
460
User
Nashville
4
461
User
Nashville
4
462
User
Nashville
4
463
User
Clarksville
3
464
User
Memphis
3
465
User
Nashville
3
466
Ashley Lepard
Chattanooga
3
467
User
Maryville
3
468
User
Signal Mountain
3
469
User
Memphis
3
470
User
Memphis
3
471
User
Chattanooga
3
472
User
Smyrna
3
473
User
Smyrna
3
474
User
Smyrna
3
475
carlitos
Smyrna
3
476
User
Smyrna
3
477
User
Smyrna
3
478
User
Smyrna
3
479
User
Smyrna
3
480
User
Smyrna
3
481
User
Smyrna
3
482
User
Smyrna
3
483
User
Smyrna
3
484
User
Smyrna
3
485
User
Smyrna
3
486
User
Smyrna
3
487
User
Smyrna
3
488
User
Smyrna
3
489
User
Nashville
3
490
User
Smyrna
3
491
User
Murfreesboro
3
492
User
Collierville
3
493
User
Millington
3
494
User
Spring Hill
3
495
User
Memphis
3
496
User
Sweetwater
3
497
User
Clarksville
3
498
User
Nashville
3
499
User
Coopertown
3
500
User
Millington
3
501
User
Hendersonville
3
502
User
Nashville
3
503
User
Nashville
3
504
User
Memphis
3
505
User
Smithville
3
506
Marianne Jour
Nashville
3
507
User
Knoxville
3
508
User
Nashville
3
509
User
Hendersonville
3
510
User
Manchester
3
511
User
Nashville
3
512
User
Nashville
2
513
User
Nashville
2
514
User
Knoxville
2
515
User
Smyrna
2
516
User
Smyrna
2
517
User
Smyrna
2
518
User
Smyrna
2
519
User
Smyrna
2
520
User
Smyrna
2
521
User
Smyrna
2
522
User
Smyrna
2
523
User
Smyrna
2
524
User
Smyrna
2
525
User
Smyrna
2
526
User
Smyrna
2
527
User
Smyrna
2
528
User
Smyrna
2
529
User
Smyrna
2
530
User
Smyrna
2
531
User
Smyrna
2
532
User
Smyrna
2
533
User
Smyrna
2
534
User
Smyrna
2
535
User
Smyrna
2
536
User
Smyrna
2
537
User
Smyrna
2
538
User
Smyrna
2
539
User
Smyrna
2
540
User
Smyrna
2
541
User
Smyrna
2
542
User
Sevierville
2
543
User
Oak Ridge
2
544
User
Knoxville
2
545
User
Chattanooga
2
546
User
Memphis
2
547
User
Algood
2
548
User
Oak Ridge
2
549
User
Collierville
2
550
User
Collierville
2
551
User
Nashville
2
552
User
Cosby
2
553
User
Collierville
2
554
Sophie Ramseur
Chattanooga
2
555
User
Franklin
2
556
User
Nashville
2
557
User
Nashville
2
558
User
Maryville
2
559
User
Memphis
2
560
User
Chattanooga
2
561
User
Memphis
2
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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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