Leaderboard
Canada
Canada
Worldwide
Rank
Level
1
Tyler Chiu
Ontario
99
2
User
Ontario
97
3
Eggma Eggma
Ontario
88
4
Elliot Wong
Quebec
87
5
User
Alberta
84
6
User
Ontario
83
7
Tyler Chiu
Ontario
82
8
User
Alberta
82
9
Yi Shi
Ontario
82
10
User
Alberta
81
11
User
Ontario
81
12
User
Ontario
81
13
Jeong Augustine
British Columbia
80
15
rdu
Ontario
80
18
User
Ontario
78
19
Renee Weng
British Columbia
77
20
User
Quebec
76
21
K. Miranda
Ontario
73
22
Joey Hotz
Ontario
71
23
jeong minji is better than jeong augustine
British Columbia
70
24
IPlaiRoblox
Quebec
69
25
User
Ontario
69
26
Aiko Fujita
Quebec
69
27
Raiyana Shams
Alberta
68
28
User
Ontario
68
29
User
Quebec
67
30
Yichen Sun
Quebec
67
31
User
Ontario
66
32
Cindy Zou
Ontario
66
33
Lyss
Ontario
65
34
Renee Weng
British Columbia
65
35
User
Ontario
65
36
Cindy
Ontario
64
37
User
Ontario
64
38
Martin Tin
Ontario
64
39
User
Ontario
64
40
User
Ontario
63
41
Hudson Vance
Ontario
63
42
robin yoo
Quebec
62
43
Chris Christensen
Ontario
62
44
User
Ontario
62
45
Richard Williams
Quebec
61
46
User
Ontario
61
48
User
Ontario
61
49
Cayden Z
Alberta
61
51
User
Ontario
60
52
User
Ontario
60
53
User
Nova Scotia
60
55
User
Manitoba
59
57
Azmy Taha
Ontario
58
58
B.Arindam
British Columbia
58
59
Kennens
Ontario
58
60
User
Ontario
58
61
User
Alberta
57
62
User
Ontario
56
63
Matthew Hobbs
Ontario
55
64
User
Ontario
55
65
Khalisa Valani
Ontario
54
66
User
Ontario
54
67
User
Ontario
54
68
Yatin Kapur
Ontario
52
69
User
Manitoba
52
70
Mark Gravely
Ontario
52
72
Hao Hu
Ontario
52
73
Philippe
Quebec
52
74
User
Ontario
52
75
User
Ontario
52
76
User
Ontario
52
77
Antony Fournier
Quebec
52
78
User
Quebec
51
79
Samuel Zheng
Ontario
51
80
User
Ontario
51
81
Steven Craig
Quebec
51
82
User
Quebec
51
83
BenBen
Quebec
51
85
User
Ontario
51
86
Harjot Singh Parmar
Ontario
51
87
awesome mathman
Ontario
50
88
User
Ontario
50
89
User
Ontario
50
90
Ismaël
Ontario
50
91
Miniapple Emily
Quebec
50
92
User
Ontario
50
93
Tuong,Hoang
Quebec
50
95
User
Ontario
50
96
User
Alberta
50
97
Durraiz Ashraf
Ontario
49
99
User
Ontario
49
100
User
Ontario
48
101
Joyce
Ontario
48
102
User
Ontario
47
103
User
Quebec
47
104
JiNo Yoon
Quebec
47
105
User
Ontario
47
106
User
New Brunswick
47
107
User
Ontario
47
108
Lisa Hines
Ontario
46
109
User
Ontario
46
110
Mark
Ontario
46
111
User
Ontario
46
112
User
Ontario
46
113
User
Ontario
46
114
Kevin Kyle
Ontario
46
115
User
Ontario
46
116
Dominic Gapero
Alberta
46
117
User
Ontario
46
118
User
Ontario
46
119
User
Ontario
46
120
User
Ontario
46
121
User
Ontario
46
122
Stefan Pizzinato
Ontario
45
123
Darwin To
British Columbia
45
124
User
Quebec
45
125
User
Ontario
45
126
User
Ontario
45
127
NIKHIL PURI
British Columbia
45
129
User
Quebec
45
130
Jordan Tarasoff
Alberta
45
131
Katie
Ontario
45
132
User
Ontario
44
133
Jacob Fitzgerald
Nova Scotia
44
136
Meriem Saadaoui
Quebec
44
138
User
Ontario
44
139
User
Manitoba
44
140
jeong minji
British Columbia
44
141
Mohammed
Ontario
44
142
Heather M
British Columbia
44
143
User
Quebec
43
144
Cobra
Alberta
43
145
User
New Brunswick
43
146
User
Quebec
43
147
Omar Abukhodair
Alberta
43
148
User
Ontario
43
149
User
Ontario
43
150
User
Ontario
43
151
Shahid Ivan the Great
Quebec
42
152
User
Ontario
42
153
Donald Moodger
Quebec
42
154
Corey
Quebec
42
155
yfradkin98@gmail.com
Ontario
42
156
User
Ontario
42
157
User
Alberta
42
159
User
Ontario
42
160
User
Quebec
42
162
tender
Ontario
42
163
User
Ontario
42
164
Matthew Taylor
Ontario
41
165
Jack Sunhaoya
Quebec
41
167
User
Alberta
41
168
User
Ontario
41
169
User
Quebec
41
170
User
Ontario
41
171
User
Manitoba
41
172
User
Alberta
41
173
Ethan
Ontario
41
174
User
Ontario
41
175
User
Ontario
41
176
User
Ontario
41
177
User
Ontario
40
178
Joshua Lawrence
Quebec
40
179
Balazs Somfalvi
Alberta
40
180
User
Ontario
40
181
Vanessa Sementilli
Quebec
40
182
Sanskriti Sharma
British Columbia
40
183
User
Quebec
40
184
User
Ontario
40
185
User
Alberta
40
186
User
Ontario
40
187
User
Ontario
40
188
User
Ontario
40
189
Balazs Somfalvi
Alberta
40
190
User
Quebec
40
191
User
Ontario
39
192
Brady Metherall
Ontario
39
193
User
Ontario
39
194
User
Ontario
39
195
User
Ontario
39
196
User
Ontario
39
197
Nolan MacLean
Alberta
39
198
User
Ontario
39
199
User
Ontario
39
200
User
Ontario
39
201
Sam C
Ontario
39
202
William Zhang
British Columbia
39
203
User
Ontario
39
204
shezshade
Ontario
39
206
User
Alberta
38
208
User
Quebec
38
209
Stephane
Quebec
38
210
User
Ontario
38
211
Edward Ji - Ellengale PS (1376)
Ontario
38
212
User
Ontario
38
213
User
Ontario
38
214
User
Alberta
38
215
Madison Buchberger
Alberta
38
216
Greg
Quebec
38
217
User
Ontario
38
218
User
Quebec
37
219
Hoang Khanh Tu Duong
Quebec
37
220
User
Ontario
37
221
User
Ontario
37
222
User
Ontario
37
224
User
Quebec
37
225
David Bedayo
Alberta
37
227
User
Alberta
36
229
Miranda
Ontario
36
230
Nicholas Plamondon
Ontario
36
232
User
Quebec
36
233
Cindy Zou
Ontario
36
234
User
Ontario
36
235
User
Ontario
36
237
User
Ontario
36
238
User
Quebec
36
239
User
Ontario
36
240
User
Ontario
36
241
User
Quebec
36
242
User
Ontario
36
243
User
Ontario
36
244
User
Ontario
35
245
User
Ontario
35
246
Thomas
British Columbia
35
247
User
Ontario
35
248
Hugeldorf
Ontario
35
249
User
Ontario
35
250
User
Ontario
35
251
User
Ontario
35
252
Mike Kusyk
Manitoba
35
253
User
Ontario
35
254
User
Ontario
35
256
Nathan Dyck
Manitoba
35
257
User
Ontario
35
259
Art Yu
British Columbia
35
260
User
Quebec
35
262
User
Ontario
34
263
Tian Lai Li
Quebec
34
264
User
Ontario
34
265
User
Ontario
34
266
User
Ontario
34
268
Josh Craig
Quebec
33
269
User
Alberta
33
270
User
Ontario
33
271
User
Ontario
33
272
User
Alberta
33
273
Rex VS Gaming
Quebec
33
274
User
Ontario
33
275
User
Ontario
33
277
User
Ontario
33
278
User
Alberta
33
279
User
Ontario
33
281
User
Quebec
33
282
Eyes Mojave
Ontario
33
283
User
Ontario
33
284
Renee Weng
British Columbia
32
286
User
Alberta
32
287
User
Ontario
32
288
User
Alberta
32
289
User
Ontario
32
290
iPad
Quebec
32
291
User
Ontario
32
292
User
Alberta
32
293
User
Ontario
32
294
Ramtin Shekari
Ontario
32
295
User
Ontario
32
297
User
Alberta
32
298
katherineee
Ontario
32
299
User
Ontario
32
300
User
Ontario
31
301
User
Ontario
31
302
User
Ontario
31
303
User
Ontario
31
304
User
Ontario
31
305
User
Ontario
31
306
User
Alberta
31
307
User
Ontario
31
308
User
Ontario
31
309
User
Ontario
31
310
User
Ontario
31
311
User
New Brunswick
31
312
Marcella
Ontario
31
313
User
Ontario
31
314
User
Manitoba
31
315
User
Ontario
31
316
User
Ontario
31
317
User
Ontario
31
318
User
Ontario
31
319
User
Ontario
31
320
User
Quebec
31
322
User
Ontario
31
323
Ramya Venkataraman
Ontario
30
324
User
Ontario
30
325
User
Ontario
30
326
User
Ontario
30
327
User
Quebec
30
328
ahmedshaikh626
Newfoundland and Labrador
30
329
User
Ontario
30
330
User
Ontario
30
331
User
Ontario
30
334
Froylanne Paril
Alberta
30
335
Rene Tung
Quebec
30
336
SSR RAO
Manitoba
30
337
Mahendirakumar Krishnakumar
British Columbia
30
338
User
Alberta
30
339
User
Ontario
30
340
User
Ontario
30
341
User
Ontario
29
342
User
Ontario
29
343
User
Ontario
29
344
User
Ontario
29
345
Jim Kerry
Ontario
29
347
User
Ontario
29
348
User
Ontario
29
349
User
Quebec
29
350
User
Ontario
29
351
User
Quebec
29
352
John Walsh
Ontario
29
353
User
Ontario
29
354
User
Ontario
29
355
User
Quebec
29
356
User
Alberta
29
357
User
Manitoba
29
359
User
Ontario
29
360
User
Ontario
29
362
User
Quebec
29
363
User
Ontario
29
364
User
Ontario
29
365
User
Alberta
29
366
User
Ontario
28
367
User
Quebec
28
369
User
Alberta
28
370
User
Alberta
28
371
User
Manitoba
28
372
User
Ontario
28
373
User
Ontario
28
374
User
Ontario
28
375
User
Ontario
28
376
User
Ontario
28
377
User
Quebec
28
378
User
Ontario
28
379
User
Ontario
28
380
User
Ontario
28
381
User
Ontario
28
383
Ryan dai
Ontario
28
384
User
Ontario
28
385
User
Ontario
28
386
User
Ontario
28
387
User
Quebec
28
388
User
Manitoba
28
389
User
Quebec
28
390
Alex Zed
Quebec
28
391
User
Quebec
28
392
Mini Store
Ontario
28
393
User
Ontario
28
394
User
Ontario
28
396
User
Alberta
28
397
User
Ontario
28
398
User
Ontario
28
399
User
Ontario
28
400
User
Manitoba
28
402
User
Manitoba
28
404
MathVader
Quebec
27
406
Sahil Jamani
Alberta
27
407
User
Ontario
27
408
User
Manitoba
27
409
User
Nova Scotia
27
410
Tori J
Saskatchewan
27
411
User
Ontario
27
412
User
Ontario
27
413
User
Ontario
27
415
Rodela Afnam
Ontario
27
416
User
Ontario
27
417
User
Ontario
27
418
User
Ontario
27
419
User
Alberta
27
420
User
Nova Scotia
27
421
User
Alberta
27
422
User
Ontario
27
423
User
Nova Scotia
27
424
User
Ontario
27
425
User
Ontario
27
426
User
Ontario
27
427
User
Alberta
27
428
User
New Brunswick
27
429
User
Ontario
27
430
User
Manitoba
27
431
User
Quebec
27
432
User
Manitoba
27
433
User
Alberta
27
434
Kevin-Joseph Sioquin
Quebec
26
435
User
Quebec
26
436
User
Quebec
26
437
User
Alberta
26
438
Ibrahima Diallo
Ontario
26
439
Ricardo Chang
Ontario
26
440
Ethan C3285
Alberta
26
441
Chuck Duford
Quebec
26
442
Alec van
Quebec
26
443
User
Nova Scotia
26
444
User
Ontario
26
445
youm
Quebec
26
446
User
Quebec
26
447
User
Ontario
26
449
User
Ontario
26
450
User
Ontario
26
451
User
Ontario
26
453
User
Ontario
26
455
User
Ontario
26
456
Quantum
Ontario
26
457
User
Ontario
26
458
User
Manitoba
26
459
User
Saskatchewan
26
460
User
Quebec
26
461
User
Ontario
26
462
User
Alberta
26
463
User
Ontario
26
464
User
Ontario
26
466
User
Alberta
26
467
User
Ontario
26
470
User
Alberta
26
471
User
Alberta
26
472
User
Ontario
26
473
User
Nova Scotia
26
474
User
Ontario
26
476
User
Alberta
25
478
User
Ontario
25
479
User
Quebec
25
480
User
Alberta
25
482
sahil
Alberta
25
483
User
Ontario
25
484
User
Alberta
25
485
User
Alberta
25
486
User
Alberta
25
487
User
Ontario
25
488
User
Ontario
25
489
User
Ontario
25
490
User
Quebec
25
491
Jordan M
Ontario
25
492
Joshua
Ontario
25
493
User
Nunavut
25
494
User
Ontario
25
495
User
Ontario
25
496
User
Ontario
25
497
User
Manitoba
25
499
User
Quebec
25
500
User
Ontario
25
501
User
Alberta
25
502
User
Ontario
25
503
User
Ontario
25
504
User
Quebec
25
505
User
Ontario
25
506
User
Ontario
25
507
Cayden Studer
Alberta
25
508
User
Ontario
25
509
BBTester1 khan
Ontario
25
510
Isaac Jyrkama
Ontario
25
511
Rich
Quebec
25
513
User
Manitoba
25
514
zahra van gaalen
Ontario
25
516
User
Ontario
25
517
User
Ontario
25
518
User
Quebec
25
519
User
Ontario
25
521
User
Saskatchewan
25
522
Matthew Coull
Quebec
24
523
Matteo Carelli
Quebec
24
524
User
Ontario
24
525
User
Ontario
24
526
Alex Gillespie
British Columbia
24
527
User
Ontario
24
528
User
Ontario
24
529
User
Alberta
24
530
User
Alberta
24
531
Rowan Doerksen
Manitoba
24
532
User
Ontario
24
533
User
Ontario
24
534
User
Quebec
24
535
User
Ontario
24
536
Peter Harlow
Ontario
24
537
Krish Patel
Quebec
24
538
User
Nova Scotia
24
539
Reid Hull
Ontario
24
540
User
Alberta
24
541
Jett Bowler
Ontario
24
542
User
Ontario
24
543
User
Quebec
24
544
User
Ontario
24
545
User
Ontario
24
546
William B
British Columbia
24
547
User
Quebec
24
548
Justin MacDonald
Ontario
24
549
User
Manitoba
24
550
User
Nova Scotia
24
552
User
Ontario
24
553
User
Nova Scotia
24
555
User
Ontario
24
556
User
Ontario
24
557
Isaac Grant
Ontario
24
558
Zahra van Gaalen
Ontario
24
559
User
Alberta
24
560
User
Ontario
24
561
User
Ontario
24
562
User
Ontario
24
563
User
Ontario
24
564
User
Ontario
24
565
User
Alberta
24
566
User
Ontario
24
567
User
Ontario
23
568
User
Ontario
23
569
User
Alberta
23
570
User
Alberta
23
571
User
New Brunswick
23
572
Stan
Alberta
23
573
Ayushi Parajuli
Quebec
23
574
User
Ontario
23
575
justin
Alberta
23
576
Ismaël Robleh Habad
Ontario
23
577
User
Ontario
23
578
User
Nova Scotia
23
579
User
Alberta
23
580
User
Ontario
23
581
User
Quebec
23
582
User
Ontario
23
583
User
Ontario
23
585
User
Alberta
23
586
User
Ontario
23
587
User
Ontario
23
588
User
Alberta
23
589
User
Ontario
23
590
User
Ontario
23
591
User
Ontario
23
592
User
Ontario
23
593
Kyra G
Alberta
23
594
User
Ontario
23
595
Erin Toksoy
Ontario
23
596
User
Alberta
23
597
User
Ontario
23
598
Clover Joseph
Ontario
23
599
User
Quebec
23
600
User
Ontario
23
601
User
Alberta
23
602
User
Alberta
23
603
User
Nova Scotia
23
605
User
Alberta
23
606
User
Manitoba
23
607
User
Ontario
23
608
User
Ontario
23
609
User
Ontario
23
610
User
Ontario
23
611
User
Ontario
23
612
User
Manitoba
23
614
User
Ontario
23
615
User
Ontario
23
616
User
Ontario
23
617
User
Ontario
23
618
User
Quebec
23
619
Emily
Ontario
23
620
User
Ontario
23
621
Mango Ocean
British Columbia
23
623
User
Ontario
23
624
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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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