Leaderboard
Australia
Australia
Worldwide
Rank
Level
1
Callan
New South Wales
100
2
i got 99 problems but mathematics ain't one
Queensland
99
3
User
Victoria
99
6
Shikamaru Nara
New South Wales
96
7
niber
Queensland
94
8
Henry Yapeter
Western Australia
93
9
Tyson Field
Victoria
93
10
Bob Smith
New South Wales
92
11
User
Queensland
91
13
Terry Zhong
New South Wales
90
16
Mark Bergman
New South Wales
89
17
marcus dyson
Western Australia
86
18
David Guo
New South Wales
86
21
User
Victoria
84
22
Alberto Piva
New South Wales
84
26
Crispin Zou
New South Wales
83
28
Elle C
Victoria
82
29
Francois
Victoria
81
30
Jakmac
Victoria
81
31
Kevin Trang
New South Wales
81
33
Wang Ziyuan
New South Wales
81
34
Crack Eggs
Victoria
80
35
User
Victoria
80
37
User
Victoria
78
38
Ken Fang
New South Wales
78
39
Alexander Pham
New South Wales
78
40
User
Queensland
78
41
Crazynumnums
Queensland
77
42
ID ZHAO
Victoria
76
44
Andrew
South Australia
75
45
David Malik
New South Wales
75
46
User
Victoria
75
48
Alex LIANG
Victoria
74
51
IncMan7 . .
Western Australia
71
53
Shyam Maharaj
New South Wales
70
55
marcus dyson
Western Australia
69
56
Mars Ultor
Queensland
69
59
Deepak RJ
Western Australia
68
60
Namitha Jacob
Queensland
68
64
Nirav Keshava
New South Wales
67
65
Thomas Kammermann
Western Australia
67
69
User
Victoria
66
70
Yuta Masuda
South Australia
66
71
Amal Reji
Queensland
66
72
User
Victoria
66
75
bobothy bobson
New South Wales
65
76
Mumud Santun
Victoria
65
78
Taeyeong Kim
Victoria
65
79
Darien Winarso
New South Wales
65
80
User
Victoria
64
82
User
Victoria
64
83
User
Queensland
64
88
User
Queensland
64
89
Trung Le
Victoria
63
91
Callum
New South Wales
63
92
Ethan Nguyen
New South Wales
62
94
Semyon Furman
Victoria
62
98
User
Queensland
61
101
User
Victoria
60
102
Prijwal
Queensland
60
103
User
Victoria
59
105
User
Victoria
59
106
Brendan Fulton
New South Wales
59
107
AGC
Victoria
59
108
User
Queensland
59
109
Amal
Queensland
59
113
Rezkiller
New South Wales
58
115
User
Queensland
58
119
Peter A
Victoria
57
120
Ben Murrie
New South Wales
57
123
User
Victoria
56
125
User
Queensland
56
126
User
Victoria
56
129
User
Victoria
56
130
User
Queensland
55
133
Wilson Jiang
New South Wales
55
134
Zawad Aziz
Queensland
55
136
LKT
Victoria
55
137
User
Queensland
55
140
User
Victoria
55
143
User
Victoria
54
144
Nugget
New South Wales
54
149
GARRY Giltendez
Queensland
54
150
User
Queensland
53
151
User
Victoria
53
152
User
Victoria
53
158
It’s me Pad
New South Wales
52
159
User
Victoria
52
160
User
Victoria
52
167
Harsh Patel
Victoria
51
168
User
Victoria
51
169
User
Victoria
51
170
User
Queensland
51
171
User
Victoria
51
173
Shahar Eyal
Western Australia
51
175
User
Victoria
51
177
Lerrrrm
Queensland
51
178
User
Queensland
51
179
Michael Cafra
Victoria
51
180
User
Victoria
50
181
User
Victoria
50
184
User
Victoria
50
186
User
Victoria
50
188
Terry Hong
New South Wales
50
189
User
Victoria
50
191
User
Victoria
49
194
User
Victoria
49
195
Phillip Halloran
Queensland
49
199
User
Victoria
48
200
User
Victoria
48
201
User
Queensland
48
202
ShadowRange
New South Wales
48
205
Jim Vella
Queensland
48
207
User
Victoria
48
210
Kyle Rosa
Victoria
48
211
User
Victoria
48
212
Rick
Queensland
47
214
User
Tasmania
47
217
User
Victoria
47
220
User
Victoria
47
223
User
Queensland
47
225
User
Queensland
47
226
Yash
Victoria
47
227
User
Victoria
47
228
Steven James Nicholas
Western Australia
47
229
User
Victoria
46
230
User
Victoria
46
231
User
Victoria
46
232
LKT
Victoria
46
236
User
Victoria
46
241
User
Victoria
45
243
User
Victoria
45
244
Random Person
Victoria
45
245
User
Victoria
45
247
User
Victoria
45
248
User
Victoria
45
255
User
Queensland
45
256
User
Queensland
45
258
User
Victoria
44
260
User
Victoria
44
261
User
Victoria
44
267
User
Victoria
44
268
Jorjia
New South Wales
44
269
User
Victoria
44
271
User
Victoria
44
272
SCP 35
Victoria
44
275
Aadi A
Tasmania
43
276
Kristian Rados
Victoria
43
277
User
Victoria
43
278
User
Queensland
43
279
User
Victoria
43
280
User
Queensland
43
282
User
Victoria
43
283
User
Queensland
43
287
Liya Joseph
South Australia
42
288
Posaunekaiser
Victoria
42
289
Cody Bellgrove
Victoria
42
292
User
Queensland
42
298
User
Queensland
42
299
Bruce Wayne
Queensland
42
301
Eric Sciberras
Victoria
41
302
User
Victoria
41
303
User
Victoria
41
305
Sue tucker
New South Wales
41
308
Shane
Queensland
41
309
User
Queensland
41
312
Will Udovenya
Victoria
41
313
User
Queensland
41
314
User
Victoria
41
316
User
Queensland
41
317
User
Victoria
41
318
User
Victoria
41
321
Priscilla Chen
New South Wales
41
322
User
Victoria
41
324
User
Queensland
41
326
User
Victoria
40
327
Jake Demasi
South Australia
40
328
User
Victoria
40
331
User
Victoria
40
332
User
Victoria
40
333
User
Queensland
40
338
User
Victoria
40
340
Nathan Trinh
New South Wales
40
342
Gerard Dyson
Western Australia
40
344
Saifuddin kamran
New South Wales
40
345
User
Queensland
40
352
User
Victoria
39
354
User
Queensland
39
355
User
Victoria
39
357
Shane C
New South Wales
39
358
User
Victoria
39
359
User
Victoria
39
360
User
Victoria
39
362
User
Queensland
39
363
User
Victoria
39
364
7mad
Victoria
39
365
User
Queensland
38
366
User
Queensland
38
367
User
Victoria
38
370
User
Victoria
38
371
User
Victoria
38
374
User
Queensland
38
381
User
Victoria
38
382
User
Queensland
38
383
User
Victoria
38
385
User
Queensland
37
387
prasidha
New South Wales
37
392
Harrison Jin
New South Wales
37
393
User
Victoria
37
396
User
Victoria
37
397
User
Queensland
37
399
KingZac
Western Australia
37
400
Curtis
South Australia
37
401
Kiriakos Pampalis
New South Wales
37
402
User
Victoria
37
405
sandraa
Western Australia
37
406
Nola
Queensland
36
407
User
Queensland
36
408
BC
Victoria
36
410
Luke Wanless
New South Wales
36
411
User
Victoria
36
415
User
Victoria
36
417
dofu rex
New South Wales
36
419
User
Victoria
36
420
User
Queensland
36
421
Bo Feng
Victoria
36
423
User
Queensland
36
424
User
Victoria
36
427
Matthew
Victoria
35
428
Alexander Etges
Queensland
35
431
User
Victoria
35
432
Gidon Kessler
New South Wales
35
433
User
Queensland
35
438
Adrian Mac
Victoria
35
442
Daniel Rothschild
Victoria
35
444
User
Victoria
35
445
User
Queensland
35
447
Adem
Victoria
34
448
Maddy McBrearty
Victoria
34
451
Athiya Azeem
New South Wales
34
452
User
Queensland
34
453
Asad Khan
South Australia
34
454
User
Victoria
34
455
User
Queensland
34
457
User
Queensland
34
459
User
Victoria
34
460
User
Queensland
34
462
User
Victoria
34
463
User
Victoria
34
465
User
Victoria
33
466
User
Victoria
33
467
Martijn Muis
Western Australia
33
468
User
Victoria
33
469
User
Victoria
33
472
User
Victoria
33
473
User
Victoria
33
475
Juuso
New South Wales
33
476
User
Queensland
33
477
Colleen McCasker
Queensland
33
479
User
Victoria
33
480
Mat Furlong
New South Wales
33
481
V Syd
New South Wales
33
482
User
Victoria
33
483
User
Victoria
33
484
User
Tasmania
33
485
User
Victoria
33
487
User
Victoria
33
489
User
Victoria
32
490
User
Victoria
32
493
User
Queensland
32
495
John Kam
Victoria
32
497
User
Victoria
32
498
Aaron Ravi
Western Australia
32
502
User
Victoria
32
503
User
Victoria
32
504
User
Victoria
32
506
User
Queensland
32
507
User
Queensland
32
508
User
Queensland
32
509
User
Victoria
31
511
OMG
Victoria
31
512
User
Victoria
31
513
User
Queensland
31
514
User
Victoria
31
515
User
Queensland
31
519
User
Queensland
31
520
ABCDEFGHIJKLMNOP QRSTUVWXYZ
Queensland
31
523
User
Tasmania
31
524
Riley Crawford
Queensland
31
526
Sohothin
Victoria
31
528
User
Tasmania
31
529
User
Victoria
31
530
User
Queensland
31
532
User
Victoria
31
533
im the best and humble
Queensland
30
534
DaIcyKid foot
Victoria
30
535
Joelle Thrush
Queensland
30
536
User
Victoria
30
538
Tomthy
New South Wales
30
539
User
Victoria
30
540
Kenneth Huang
New South Wales
30
542
Andy Little
Queensland
30
545
Oscar Woodland Grove
Western Australia
30
548
User
Victoria
30
550
User
Victoria
30
552
User
Victoria
30
553
User
Queensland
30
555
User
Victoria
30
556
Nathan Briffa
Victoria
30
557
User
Queensland
29
559
User
Queensland
29
560
User
Victoria
29
562
fin bob
New South Wales
29
563
User
Victoria
29
564
User
Victoria
29
567
User
Queensland
29
568
Lachlan Taylor
New South Wales
29
569
User
Victoria
29
573
Jeff Normandy
New South Wales
29
574
User
Queensland
29
575
User
Victoria
29
576
User
Victoria
29
577
User
Queensland
29
582
User
Victoria
29
583
User
Queensland
29
584
Benjamin White
Victoria
29
586
Vincent Phoummathep
New South Wales
28
588
Liam godly
Queensland
28
590
User
Victoria
28
592
Zee Gee
New South Wales
28
593
David Houghton
New South Wales
28
595
Lazy Cow
Victoria
28
597
User
Queensland
28
598
Olivia Piatanesi
South Australia
28
601
User
Victoria
28
602
User
Queensland
28
603
Igor Radchenko
Victoria
28
604
Vishal Khanna
Victoria
28
609
User
Victoria
28
610
Arpon Sarker
Queensland
28
611
User
Queensland
28
612
Josie Jo
Tasmania
28
613
User
Queensland
28
615
User
Queensland
28
617
johann trader135
South Australia
28
619
User
Queensland
28
620
Damon Sara
Region
28
622
Scott Bartlett
Victoria
28
624
User
Queensland
28
625
User
Victoria
28
628
User
Victoria
27
630
Polly
Queensland
27
632
User
Queensland
27
633
User
Queensland
27
634
User
Victoria
27
635
User
Victoria
27
636
User
Victoria
27
638
The Destroyer
New South Wales
27
639
Henry C
New South Wales
27
640
User
Victoria
27
645
M. K.
Victoria
27
647
User
Victoria
27
649
kate
Queensland
27
651
User
Victoria
27
652
User
Queensland
27
653
User
Queensland
27
656
User
Victoria
27
657
User
Victoria
27
658
User
Queensland
27
660
User
Victoria
27
661
User
Queensland
27
662
User
Victoria
27
664
User
Victoria
27
666
User
Queensland
27
669
User
Victoria
26
671
User
Victoria
26
672
Lavra
Victoria
26
673
User
Victoria
26
676
Oliver Hoang
New South Wales
26
677
User
Queensland
26
679
User
Victoria
26
680
User
Queensland
26
681
User
Victoria
26
684
User
Victoria
26
685
User
Victoria
26
686
User
Queensland
26
687
User
Victoria
26
690
Abdullah Karim
New South Wales
26
691
User
Queensland
26
692
ATROCITOUS
Victoria
26
695
Reach
Victoria
26
696
User
Victoria
26
697
User
Queensland
26
698
User
Victoria
26
699
User
Victoria
26
700
undefined
New South Wales
26
704
User
Queensland
26
705
User
Queensland
26
706
User
Victoria
26
711
User
Victoria
26
712
Jackson Turner
Victoria
26
713
User
Queensland
26
715
User
Victoria
25
722
sanaz mozhdehifard
New South Wales
25
727
User
Victoria
25
730
Kurt Enkera
Queensland
25
731
Elvis nguyen
New South Wales
25
734
User
Victoria
25
737
User
Victoria
25
738
User
Queensland
25
739
User
Queensland
25
742
User
Victoria
25
743
User
Victoria
25
745
User
Queensland
25
752
User
Victoria
25
753
User
Victoria
25
754
User
Queensland
25
756
User
Victoria
25
757
User
Victoria
25
758
User
Victoria
25
759
User
Victoria
25
760
User
Victoria
25
764
DavidP
New South Wales
25
766
User
Queensland
25
767
User
Victoria
25
768
User
Queensland
25
769
User
Victoria
24
770
Grigory
Victoria
24
773
User
Victoria
24
774
Crack Eggs Some More
Victoria
24
775
Marco Ho
New South Wales
24
781
Malavika Sajimon
Victoria
24
782
User
Victoria
24
784
User
Victoria
24
787
Kenneth Chong
New South Wales
24
792
User
Victoria
24
793
User
Victoria
24
794
User
Queensland
24
795
Jerry Li
Victoria
24
798
User
Tasmania
24
799
Jordan Farrow
Victoria
24
802
User
Queensland
24
803
Patrick
Queensland
23
804
ericsuperman
Victoria
23
807
User
Victoria
23
808
Harrison WINDOW
Victoria
23
809
User
Queensland
23
810
Sajan Dangol
New South Wales
23
811
User
Victoria
23
817
User
Queensland
23
821
User
Queensland
23
822
User
Victoria
23
823
User
Victoria
23
824
User
Queensland
23
826
User
Queensland
23
827
Daniel L
New South Wales
23
831
User
Tasmania
23
832
User
Queensland
23
833
User
Queensland
23
835
User
Queensland
23
836
User
Queensland
23
838
User
Victoria
23
839
Dylan Brister
Queensland
23
840
User
Queensland
23
841
User
Queensland
23
842
Toni Smith
New South Wales
23
844
User
Victoria
23
846
User
Queensland
23
848
User
Victoria
22
850
User
Victoria
22
852
John Hartweger
Victoria
22
855
User
Victoria
22
859
User
Victoria
22
860
Connor Richards
Western Australia
22
862
Annika Schubert
Victoria
22
865
User
Victoria
22
866
User
Queensland
22
868
User
Victoria
22
869
User
Victoria
22
870
User
Victoria
22
871
User
Queensland
22
873
User
Victoria
22
874
User
Victoria
22
877
User
Queensland
22
879
takundarox
Western Australia
22
882
User
Victoria
22
884
Ryan Pham
Victoria
22
886
User
Queensland
22
887
Giorgio Aquino
Queensland
22
888
User
Victoria
22
889
User
Victoria
22
891
User
Victoria
22
892
User
Victoria
22
893
User
Queensland
22
894
User
Queensland
22
895
User
Queensland
22
896
User
Victoria
22
898
Hamish Grant
Queensland
22
899
User
Queensland
22
902
Charles Mangion
Australian Capital Territory
21
904
User
Victoria
21
905
User
Victoria
21
912
User
Victoria
21
913
User
Queensland
21
916
User
Victoria
21
918
User
Victoria
21
919
User
Queensland
21
923
User
Victoria
21
926
lemon juice
New South Wales
21
928
User
Queensland
21
935
User
Victoria
21
937
User
Victoria
21
938
User
Queensland
21
939
User
Victoria
21
940
User
Queensland
21
941
User
Victoria
21
942
Y. M.
New South Wales
21
943
User
Victoria
21
946
User
Victoria
21
949
User
Victoria
21
951
User
Victoria
21
952
Jin Ren Tay
Victoria
21
955
User
Queensland
21
958
User
Queensland
20
960
User
Victoria
20
961
User
Victoria
20
962
Maria
Victoria
20
963
User
Queensland
20
964
User
Victoria
20
966
User
Victoria
20
970
User
Victoria
20
971
User
Queensland
20
972
User
Victoria
20
974
User
Victoria
20
977
User
Queensland
20
978
User
Victoria
20
980
User
Queensland
20
981
User
Victoria
20
983
User
Victoria
20
987
User
Victoria
20
989
User
Queensland
20
990
User
Queensland
20
993
User
Victoria
20
994
User
Victoria
20
995
User
Victoria
20
997
User
Tasmania
20
999
Saisha Sethi
Victoria
20
1000
User
Victoria
20

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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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