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