Leaderboard
Australia
Australia
Worldwide
Rank
Level
1
i got 99 problems but mathematics ain't one
Queensland
99
2
User
Victoria
99
3
Shikamaru Nara
New South Wales
96
4
Tyson Field
Victoria
93
6
User
Queensland
91
15
Terry Zhong
New South Wales
83
16
Elle C
Victoria
82
17
Jakmac
Victoria
81
18
Clinton Johnston
Queensland
81
19
Kevin Trang
New South Wales
81
21
Wang Ziyuan
New South Wales
81
22
Crispin Zou
New South Wales
80
24
David Malik
New South Wales
75
25
User
Victoria
75
29
marcus dyson
Western Australia
69
30
Mars Ultor
Queensland
69
32
marcus dyson
Western Australia
68
38
Yuta Masuda
South Australia
66
39
Amal Reji
Queensland
66
40
User
Victoria
66
43
Darien Winarso
New South Wales
65
45
Andrew
South Australia
64
46
User
Queensland
64
47
Lachlan Taylor
New South Wales
64
49
Callum
New South Wales
63
51
Semyon Furman
Victoria
62
52
User
Queensland
61
55
Prijwal
Queensland
60
56
User
Victoria
59
57
Brendan Fulton
New South Wales
59
58
AGC
Victoria
59
59
User
Queensland
59
60
Amal
Queensland
59
61
Ben Murrie
New South Wales
57
64
User
Victoria
56
67
User
Victoria
56
68
User
Queensland
55
71
User
Victoria
55
75
GARRY Giltendez
Queensland
54
80
Rezkiller
New South Wales
52
82
User
Victoria
51
84
Lerrrrm
Queensland
51
85
User
Queensland
51
86
Michael Cafra
Victoria
51
89
Terry Hong
New South Wales
50
90
User
Victoria
50
91
Kyle Bishop
New South Wales
50
92
Phillip Halloran
Queensland
49
95
Jim Vella
Queensland
48
98
User
Victoria
48
101
Kyle Rosa
Victoria
48
102
User
Victoria
48
104
User
Queensland
47
106
User
Queensland
47
107
Yash
Victoria
47
108
User
Victoria
47
109
Steven James Nicholas
Western Australia
47
110
Namitha Jacob
Queensland
46
113
User
Victoria
46
117
User
Victoria
45
120
User
Victoria
45
125
User
Queensland
45
126
User
Queensland
45
128
User
Victoria
44
129
Jorjia
New South Wales
44
130
User
Victoria
44
132
User
Victoria
44
133
SCP 35
Victoria
44
136
User
Victoria
43
137
User
Queensland
43
139
Cody Bellgrove
Victoria
42
142
User
Queensland
42
148
User
Queensland
42
149
Bruce Wayne
Queensland
42
150
User
Queensland
42
152
Shane
Queensland
41
153
User
Queensland
41
156
Will Udovenya
Victoria
41
157
User
Queensland
41
158
User
Victoria
41
160
User
Queensland
41
161
User
Victoria
41
162
User
Victoria
41
165
Priscilla Chen
New South Wales
41
166
User
Victoria
41
168
User
Queensland
41
169
User
Victoria
40
170
User
Queensland
40
175
User
Victoria
40
177
Nathan Trinh
New South Wales
40
179
Gerard Dyson
Western Australia
40
181
Saifuddin kamran
New South Wales
40
182
User
Queensland
40
184
User
Victoria
39
186
Shane C
New South Wales
39
187
User
Victoria
39
188
User
Victoria
39
189
User
Victoria
39
191
User
Queensland
39
192
User
Victoria
39
193
7mad
Victoria
39
194
Mumud Santun
Victoria
38
201
User
Victoria
38
202
User
Queensland
38
203
User
Victoria
38
204
Liya Joseph
South Australia
37
207
Harrison Jin
New South Wales
37
208
User
Victoria
37
211
User
Victoria
37
212
User
Queensland
37
214
KingZac
Western Australia
37
215
Curtis
South Australia
37
216
Kiriakos Pampalis
New South Wales
37
217
User
Victoria
37
220
sandraa
Western Australia
37
224
User
Victoria
36
226
dofu rex
New South Wales
36
228
User
Victoria
36
229
User
Queensland
36
230
Bo Feng
Victoria
36
232
User
Queensland
36
233
User
Victoria
36
235
Gidon Kessler
New South Wales
35
236
wilson
New South Wales
35
238
User
Queensland
35
242
Zawad Aziz
Queensland
35
244
Da Icy Kid
Victoria
35
245
Adrian Mac
Victoria
35
249
Daniel Rothschild
Victoria
35
251
User
Victoria
35
252
User
Queensland
35
254
User
Victoria
34
255
User
Queensland
34
257
User
Queensland
34
259
User
Victoria
34
260
User
Queensland
34
262
User
Victoria
34
263
User
Victoria
34
264
Francois Petit
Victoria
33
266
Juuso
New South Wales
33
267
User
Queensland
33
268
Colleen McCasker
Queensland
33
271
User
Victoria
33
272
Mat Furlong
New South Wales
33
273
V Syd
New South Wales
33
274
User
Victoria
33
275
User
Victoria
33
276
User
Tasmania
33
277
User
Victoria
33
279
User
Victoria
33
281
User
Queensland
32
282
User
Victoria
32
284
John Kam
Victoria
32
286
User
Victoria
32
287
Aaron Ravi
Western Australia
32
291
User
Victoria
32
292
User
Victoria
32
293
User
Victoria
32
295
User
Queensland
32
296
User
Queensland
32
297
User
Queensland
32
298
User
Victoria
32
299
Thomas Kammermann
Western Australia
31
300
User
Queensland
31
301
User
Victoria
31
302
User
Queensland
31
306
User
Queensland
31
307
User
Queensland
31
308
ABCDEFGHIJKLMNOP QRSTUVWXYZ
Queensland
31
311
User
Tasmania
31
312
Riley Crawford
Queensland
31
314
Sohothin
Victoria
31
316
User
Tasmania
31
317
User
Victoria
31
318
User
Queensland
31
320
User
Victoria
31
321
Asad Khan
South Australia
30
322
User
Victoria
30
323
Kenneth Huang
New South Wales
30
325
Andy Little
Queensland
30
328
Oscar Woodland Grove
Western Australia
30
331
User
Victoria
30
333
User
Victoria
30
335
User
Victoria
30
336
User
Queensland
30
338
User
Victoria
30
339
Nathan Briffa
Victoria
30
340
User
Queensland
30
341
User
Victoria
29
346
Jeff Normandy
New South Wales
29
348
User
Queensland
29
349
User
Victoria
29
350
User
Victoria
29
351
User
Queensland
29
357
User
Victoria
29
358
User
Victoria
29
359
User
Queensland
29
360
Benjamin White
Victoria
29
362
User
Victoria
28
363
Rick
Queensland
28
364
Olivia Piatanesi
South Australia
28
367
User
Victoria
28
368
User
Queensland
28
369
Igor Radchenko
Victoria
28
370
Vishal Khanna
Victoria
28
375
User
Victoria
28
376
Arpon Sarker
Queensland
28
377
User
Queensland
28
378
Josie Jo
Tasmania
28
379
User
Queensland
28
381
User
Queensland
28
383
johann trader135
South Australia
28
385
User
Queensland
28
386
Damon Sara
Region
28
388
Scott Bartlett
Victoria
28
390
User
Queensland
28
391
User
Victoria
28
393
User
Queensland
28
394
User
Victoria
27
395
User
Victoria
27
396
User
Victoria
27
398
The Destroyer
New South Wales
27
399
Henry C
New South Wales
27
400
User
Victoria
27
405
M. K.
Victoria
27
407
User
Victoria
27
409
kate
Queensland
27
411
User
Victoria
27
412
User
Queensland
27
413
User
Queensland
27
416
User
Victoria
27
417
User
Victoria
27
418
User
Queensland
27
420
User
Victoria
27
421
User
Queensland
27
422
User
Victoria
27
424
User
Victoria
27
426
User
Queensland
27
429
User
Victoria
26
430
User
Queensland
26
431
User
Victoria
26
434
User
Victoria
26
435
User
Victoria
26
436
User
Queensland
26
437
User
Victoria
26
440
Abdullah Karim
New South Wales
26
441
User
Queensland
26
442
ATROCITOUS
Victoria
26
445
Reach
Victoria
26
446
User
Victoria
26
447
User
Queensland
26
448
User
Victoria
26
449
User
Victoria
26
450
undefined
New South Wales
26
454
User
Queensland
26
455
User
Queensland
26
456
User
Victoria
26
461
User
Victoria
26
462
Jackson Turner
Victoria
26
463
User
Queensland
26
465
User
Victoria
25
468
Kurt Enkera
Queensland
25
469
Elvis nguyen
New South Wales
25
472
User
Victoria
25
475
User
Victoria
25
476
User
Queensland
25
477
User
Queensland
25
480
User
Victoria
25
481
User
Victoria
25
483
User
Queensland
25
487
User
Victoria
25
491
User
Victoria
25
492
User
Victoria
25
493
User
Queensland
25
495
User
Victoria
25
496
User
Victoria
25
497
User
Victoria
25
498
User
Victoria
25
499
User
Victoria
25
503
User
Victoria
25
504
DavidP
New South Wales
25
506
User
Queensland
25
507
User
Victoria
25
508
User
Queensland
25
510
Malavika Sajimon
Victoria
24
511
User
Victoria
24
513
User
Victoria
24
516
Kenneth Chong
New South Wales
24
521
User
Victoria
24
522
User
Victoria
24
523
User
Queensland
24
524
Jerry Li
Victoria
24
527
User
Tasmania
24
528
Jordan Farrow
Victoria
24
531
User
Queensland
24
535
User
Queensland
23
536
User
Victoria
23
537
User
Victoria
23
538
User
Queensland
23
540
User
Queensland
23
541
Daniel L
New South Wales
23
545
User
Tasmania
23
546
User
Queensland
23
547
User
Queensland
23
549
User
Queensland
23
550
User
Queensland
23
552
User
Victoria
23
553
Dylan Brister
Queensland
23
554
User
Queensland
23
555
User
Queensland
23
556
Toni Smith
New South Wales
23
558
User
Victoria
23
560
User
Queensland
23
563
User
Victoria
22
566
User
Victoria
22
568
User
Victoria
22
569
Connor Richards
Western Australia
22
571
Annika Schubert
Victoria
22
574
User
Victoria
22
575
User
Queensland
22
577
User
Victoria
22
578
User
Victoria
22
579
User
Victoria
22
580
User
Queensland
22
582
User
Victoria
22
583
User
Victoria
22
586
User
Queensland
22
588
takundarox
Western Australia
22
591
User
Victoria
22
593
Ryan Pham
Victoria
22
595
User
Queensland
22
596
Giorgio Aquino
Queensland
22
597
User
Victoria
22
598
User
Victoria
22
600
User
Victoria
22
601
User
Victoria
22
602
User
Queensland
22
603
User
Queensland
22
604
User
Queensland
22
605
User
Victoria
22
607
Hamish Grant
Queensland
22
608
User
Queensland
22
611
User
Victoria
22
616
User
Victoria
21
617
User
Queensland
21
620
Shelby Lindrea
Victoria
21
621
User
Victoria
21
623
User
Victoria
21
624
User
Queensland
21
628
User
Victoria
21
631
lemon juice
New South Wales
21
633
User
Queensland
21
640
User
Victoria
21
642
User
Victoria
21
643
User
Queensland
21
644
User
Victoria
21
645
User
Queensland
21
646
User
Victoria
21
647
Y. M.
New South Wales
21
648
User
Victoria
21
651
User
Victoria
21
654
User
Victoria
21
656
User
Victoria
21
657
Jin Ren Tay
Victoria
21
660
User
Queensland
21
663
User
Victoria
20
667
User
Victoria
20
668
User
Victoria
20
676
User
Victoria
20
677
User
Queensland
20
680
User
Victoria
20
682
Manas Choudhury
New South Wales
20
685
User
Victoria
20
690
User
Queensland
20
691
User
Victoria
20
693
User
Queensland
20
694
User
Victoria
20
696
User
Victoria
20
700
User
Victoria
20
702
User
Queensland
20
703
User
Queensland
20
706
User
Victoria
20
707
User
Victoria
20
708
User
Victoria
20
710
User
Tasmania
20
712
Saisha Sethi
Victoria
20
713
User
Victoria
20
714
User
Victoria
19
716
User
Queensland
19
717
User
Victoria
19
718
User
Victoria
19
719
User
Victoria
19
720
Azal Naalband
Victoria
19
722
User
Victoria
19
723
User
Victoria
19
725
Yash A
New South Wales
19
728
Brendon
Victoria
19
729
User
Victoria
19
731
User
Queensland
19
734
User
Victoria
19
738
User
Victoria
19
741
User
Victoria
19
742
Demon Boi
Queensland
19
743
User
Victoria
19
744
User
Queensland
19
745
User
Queensland
19
751
michelle blair
Queensland
19
752
User
Victoria
19
758
User
Queensland
18
759
User
Queensland
18
760
User
Victoria
18
761
User
Queensland
18
762
Blythe Frederick
Queensland
18
763
User
Queensland
18
764
User
Queensland
18
765
JELLY CAT LIKES JELLY
South Australia
18
766
Angus Underdown
South Australia
18
768
User
Victoria
18
769
User
Queensland
18
770
Jasmine
New South Wales
18
771
User
Queensland
18
775
User
Queensland
18
776
User
Victoria
18
777
User
Victoria
18
779
User
Victoria
18
780
User
Victoria
18
782
User
Queensland
18
784
User
Queensland
18
786
User
Queensland
18
791
User
Victoria
18
793
Sargun Sohal
New South Wales
17
794
Jodie MacVean
New South Wales
17
795
Harry Wetherall
Victoria
17
800
User
Queensland
17
801
User
Queensland
17
803
User
Victoria
17
804
User
Victoria
17
811
Raymond Z
New South Wales
17
812
User
Victoria
17
813
jonathanfei
New South Wales
17
816
Aron Schwartz
Western Australia
17
818
User
Queensland
17
822
User
Victoria
17
823
User
Victoria
17
824
User
Queensland
17
825
User
Queensland
17
827
Ted Belcher
Queensland
17
830
User
Victoria
17
833
User
Victoria
17
834
User
Victoria
17
835
User
Victoria
17
837
James
New South Wales
16
838
Christabel
New South Wales
16
839
Anura Poh
Victoria
16
841
User
Queensland
16
842
User
Victoria
16
843
User
Victoria
16
846
User
Victoria
16
848
User
Victoria
16
849
User
Queensland
16
851
User
Queensland
16
852
User
Queensland
16
854
Josh Carson
Western Australia
16
855
User
Victoria
16
856
Ruari
Victoria
16
857
User
Queensland
16
859
User
Queensland
16
860
User
Queensland
16
861
User
Victoria
16
865
amir kasra jafari
New South Wales
16
866
ULTRALINK
Western Australia
16
867
User
Victoria
16
873
User
Victoria
16
878
User
Victoria
16
881
User
Victoria
16
883
User
Victoria
16
884
User
Victoria
16
885
User
Victoria
16
886
User
Victoria
16
887
User
Victoria
16
889
User
Queensland
16
891
User
Victoria
15
893
User
Queensland
15
895
User
Victoria
15
896
User
Queensland
15
898
User
Queensland
15
900
User
Victoria
15
901
User
Victoria
15
909
User
Victoria
15
910
User
Victoria
15
915
User
Victoria
15
917
User
Tasmania
15
918
User
Victoria
15
920
User
Victoria
15
922
User
Victoria
15
923
User
Queensland
15
924
User
Victoria
15
925
User
Victoria
15
928
User
Queensland
15
931
User
Victoria
15
933
User
Queensland
15
935
User
Queensland
14
940
User
Victoria
14
941
Cameron Holdsworth
Western Australia
14
942
Marco Ho
Queensland
14
944
Raymond Song
New South Wales
14
945
lmaccarone
Queensland
14
946
User
Victoria
14
948
User
Queensland
14
949
User
Victoria
14
950
Sam
Victoria
14
953
User
Victoria
14
954
User
Queensland
14
958
User
Queensland
14
960
Antonia Casey
Victoria
14
962
Masumi Hailey
South Australia
14
963
User
Victoria
14
964
Kosta
Queensland
14
966
User
Queensland
14
967
User
Queensland
14
968
User
Victoria
14
974
Lorraine Sanares
Victoria
14
976
User
Victoria
14
977
User
Victoria
14
980
User
Victoria
14
982
User
Victoria
14
983
User
Victoria
14
986
User
Victoria
14
990
User
Victoria
14
992
User
Queensland
14
993
User
Queensland
14
998
User
Queensland
13
999
User
Victoria
13

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A Web App for Mathematics Training

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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:

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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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