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

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

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

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

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

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


What's a Mental Math Tip?

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

÷
984
3
328

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


Get Better at Mental Math

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

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

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

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

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

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

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


Use a Math Trainer

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

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

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

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

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


Mental Calculation

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

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

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

Example of two-digit
addition problem

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

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

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

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

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

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


Arithmetic

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

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

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

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

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

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

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

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

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


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