Leaderboard
China
China
Worldwide
Rank
Level
1
Aarush Sahoo
Region
98
2
James TSANG
Region
95
3
scjennywong
Region
94
4
Shanon
Region
92
5
User
Region
88
6
fox
Region
88
7
User
Region
87
8
Sandy Chung
Region
84
9
User
Region
82
10
Harsh vardhan
Region
81
11
User
Region
78
12
User
Region
75
13
Pak Bun Ko
Region
75
14
rahma safraoui
Region
74
15
CTY
Region
72
16
User
Region
69
17
Amanda Margo
Region
68
18
User
Region
66
19
User
Region
62
20
reloan
Region
62
21
User
Region
62
22
User
Region
60
23
User
Region
60
24
User
Region
60
25
User
Region
54
26
lol
Region
53
27
User
Region
52
28
User
Region
52
29
User
Region
52
30
User
Region
51
31
sid somani
Region
51
32
User
Region
50
33
User
Region
50
34
dont fix my gauge
Beijing
50
35
HappyNewYear
Region
50
36
User
Region
49
37
User
Region
49
38
User
Region
48
39
User
Region
47
40
Yi Ke Hu
Zhejiang
47
41
User
Region
46
42
User
Region
46
43
User
Region
46
44
Velvet thunder
Region
46
45
Ymmt
Region
45
46
george geroge
Region
45
47
User
Region
45
48
User
Region
45
49
User
Guangdong
44
50
User
Region
43
51
User
Region
43
52
User
Region
41
53
Peter
Region
39
54
User
Region
39
55
User
Region
39
56
User
Region
38
57
User
Region
38
58
User
Region
37
59
User
Region
36
60
K
Region
36
61
User
Guangdong
35
62
User
Region
34
63
User
Region
34
64
Alistair Sexton
Shanghai
34
65
User
Region
33
66
User
Region
33
67
arcross4682
Region
33
68
User
Region
33
69
User
Region
32
70
User
Guangdong
32
71
Zhou Fan
Shanghai
31
72
User
Region
31
73
User
Region
31
74
User
Region
30
75
User
Region
29
76
cestquilepapa
Region
28
77
User
Region
28
78
User
Region
27
79
User
Region
27
80
User
Beijing
27
81
User
Beijing
26
82
User
Region
26
83
User
Region
26
84
User
Region
26
85
User
Shanghai
25
86
Felix Wong
Region
25
87
Ian Brookhart
Region
25
88
Josh Brookhart
Region
24
89
User
Region
24
90
User
Region
24
91
User
Region
24
92
User
Region
24
93
User
Region
23
94
User
Region
23
95
General Jake
Shanghai
23
96
User
Region
23
97
User
Region
23
98
Zhenglei Shi
Beijing
23
99
Ka Ming Sit
Region
22
100
User
Region
22
101
User
Region
22
102
User
Region
22
103
User
Region
22
104
User
Region
21
105
User
Region
21
106
User
Region
21
107
Ollie Wall
Region
20
108
User
Region
20
109
User
Region
20
110
User
Region
19
111
User
Region
19
112
User
Region
19
113
James Baghurst
Region
18
114
User
Region
18
115
User
Region
18
116
User
Region
18
117
Tony Park
Region
18
118
shenglan zhang
Region
18
119
User
Guangdong
18
120
User
Region
18
121
User
Region
17
122
User
Region
17
123
User
Region
17
124
User
Region
17
125
User
Region
17
126
User
Region
17
127
Matthew Lau
Region
16
128
User
Region
16
129
User
Region
16
130
User
Region
16
131
User
Region
16
132
User
Region
16
133
User
Region
16
134
User
Region
16
135
User
Guangdong
15
136
User
Region
15
137
User
Region
15
138
User
Region
15
139
User
Region
15
140
User
Region
15
141
aa
Region
15
142
User
Region
14
143
User
Region
14
144
User
Region
14
145
User
Region
14
146
User
Region
14
147
User
Region
14
148
User
Region
14
149
User
Region
14
150
User
Region
14
151
User
Region
14
152
User
Region
14
153
Thane Rigwald
Region
14
154
James Chin
Region
14
155
User
Region
14
156
User
Region
13
157
User
Region
13
158
User
Guangdong
13
159
User
Region
13
160
User
Region
13
161
User
Region
13
162
Julius
Shanghai
13
163
User
Region
13
164
User
Region
13
165
User
Region
13
166
User
Region
13
167
User
Region
13
168
User
Region
12
169
User
Region
12
170
User
Region
12
171
User
Region
12
172
User
Region
12
173
Megan Tse
Region
12
174
User
Region
12
175
User
Region
12
176
User
Region
12
177
User
Region
12
178
User
Region
12
179
User
Region
12
180
User
Region
12
181
User
Region
12
182
1C (10) Ng Sze Hon
Region
12
183
User
Region
12
184
User
Region
12
185
User
Region
12
186
Jino Lee
Region
11
187
Yuchan Shim
Region
11
188
User
Region
11
189
User
Region
11
190
User
Region
11
191
Matthijs De Lange
Region
11
192
User
Region
11
193
User
Region
11
194
User
Region
11
195
User
Region
11
196
User
Region
11
197
User
Region
11
198
User
Region
11
199
User
Region
11
200
Peter Wong
Guangdong
11
201
User
Region
11
202
User
Region
11
203
User
Region
11
204
User
Region
11
205
Rhys Ng
Region
11
206
User
Jiangsu
10
207
User
Region
10
208
User
Beijing
10
209
User
Region
10
210
User
Region
10
211
User
Region
10
212
User
Region
10
213
User
Region
10
214
rdfyli@gmail.com
Region
10
215
User
Region
10
216
User
Region
10
217
User
Region
10
218
User
Region
10
219
User
Region
10
220
Patrick Omey
Region
10
221
User
Region
10
222
User
Region
9
223
User
Region
9
224
User
Region
9
225
User
Region
9
226
User
Region
9
227
User
Region
9
228
User
Region
9
229
Callista Ring
Region
9
230
User
Region
9
231
User
Region
9
232
User
Region
9
233
User
Region
9
234
Omar Crutchley
Region
9
235
User
Region
9
236
User
Region
9
237
User
Region
9
238
User
Region
9
239
User
Region
9
240
User
Region
9
241
User
Region
9
242
User
Region
9
243
TONY TIN-CIANG LO
Region
9
244
User
Region
9
245
Павел Липаев
Region
9
246
User
Region
9
247
User
Region
9
248
User
Region
9
249
User
Region
9
250
User
Region
9
251
User
Region
9
252
bimal316
Region
9
253
User
Region
9
254
User
Region
8
255
Aisling McCarron
Region
8
256
User
Region
8
257
User
Region
8
258
User
Region
8
259
User
Region
8
260
amber
Region
8
261
User
Region
8
262
User
Region
8
263
User
Region
8
264
User
Region
8
265
User
Region
8
266
User
Region
8
267
User
Region
8
268
Desmond Leung
Region
8
269
User
Region
8
270
User
Region
8
271
User
Region
8
272
User
Region
8
273
User
Region
8
274
Jerry (Yudingkun) Tan
Chongqing
8
275
User
Region
7
276
User
Region
7
277
User
Region
7
278
User
Region
7
279
User
Region
7
280
User
Chongqing
7
281
User
Shanghai
7
282
User
Region
7
283
User
Region
7
284
User
Region
7
285
User
Region
7
286
User
Region
7
287
User
Region
7
288
User
Region
7
289
User
Region
7
290
User
Liaoning
7
291
User
Region
7
292
User
Region
7
293
User
Region
7
294
User
Region
7
295
User
Region
7
296
User
Region
7
297
Jerome L Tan
Region
7
298
User
Region
7
299
User
Region
7
300
User
Region
7
301
User
Region
7
302
User
Region
7
303
User
Region
7
304
User
Region
7
305
User
Region
7
306
Henry Liu
Region
7
307
User
Region
7
308
User
Region
7
309
User
Region
7
310
User
Guangdong
7
311
User
Region
7
312
Munisha Sharma
Region
6
313
User
Region
6
314
User
Region
6
315
User
Fujian
6
316
User
Region
6
317
Hugo So
Region
6
318
User
Region
6
319
User
Region
6
320
User
Shanghai
6
321
User
Region
6
322
User
Region
6
323
User
Region
6
324
User
Region
6
325
User
Region
6
326
User
Region
6
327
Daniel Cooper
Region
6
328
User
Region
6
329
User
Region
6
330
Kibaek Lee
Region
6
331
User
Region
6
332
User
Region
6
333
User
Region
6
334
User
Region
6
335
User
Jiangsu
6
336
User
Shanghai
6
337
User
Region
6
338
User
Region
6
339
User
Region
6
340
User
Region
6
341
User
Region
6
342
User
Region
6
343
User
Region
6
344
User
Region
6
345
User
Region
6
346
User
Region
6
347
User
Region
6
348
User
Region
6
349
User
Region
6
350
User
Region
6
351
User
Region
6
352
User
Region
5
353
User
Region
5
354
User
Region
5
355
User
Region
5
356
User
Region
5
357
User
Region
5
358
User
Region
5
359
User
Region
5
360
User
Region
5
361
User
Region
5
362
User
Region
5
363
JoyceKo
Region
5
364
User
Region
5
365
User
Region
5
366
User
Shanghai
5
367
User
Region
5
368
User
Region
5
369
User
Region
5
370
User
Region
5
371
User
Region
5
372
Angel Chen
Region
5
373
User
Region
5
374
User
Region
5
375
User
Region
5
376
User
Region
5
377
User
Region
5
378
User
Region
5
379
User
Anhui
5
380
User
Region
5
381
User
Region
5
382
User
Region
5
383
User
Region
5
384
User
Region
5
385
St
Region
5
386
User
Region
5
387
User
Region
5
388
User
Region
5
389
User
Region
5
390
User
Region
5
391
User
Region
5
392
User
Region
5
393
User
Region
5
394
User
Region
5
395
Seo Yun JEONG
Region
5
396
Ian Brookhart
Region
5
397
User
Region
4
398
User
Region
4
399
User
Region
4
400
User
Region
4
401
User
Region
4
402
User
Region
4
403
User
Region
4
404
User
Region
4
405
User
Region
4
406
User
Region
4
407
User
Region
4
408
User
Region
4
409
User
Region
4
410
User
Region
4
411
User
Region
4
412
User
Region
4
413
User
Jiangsu
4
414
User
Region
4
415
User
Region
4
416
User
Region
4
417
User
Region
4
418
User
Region
4
419
User
Region
4
420
User
Region
4
421
User
Region
4
422
User
Region
4
423
User
Region
4
424
User
Region
4
425
User
Region
4
426
User
Shanghai
4
427
User
Region
4
428
User
Region
4
429
User
Region
4
430
User
Region
4
431
User
Region
4
432
User
Region
4
433
User
Region
4
434
User
Region
4
435
User
Region
4
436
User
Region
4
437
User
Region
4
438
User
Region
4
439
User
Region
3
440
User
Region
3
441
Ailey Wong
Region
3
442
User
Region
3
443
User
Region
3
444
User
Region
3
445
User
Zhejiang
3
446
User
Region
3
447
asd
Region
3
448
User
Region
3
449
Yangdon Sersang
Region
3
450
User
Region
3
451
User
Region
3
452
User
Region
3
453
User
Region
3
454
User
Region
3
455
User
Guangdong
3
456
User
Region
3
457
User
Region
3
458
User
Region
3
459
User
Region
3
460
User
Region
3
461
User
Region
3
462
User
Region
3
463
User
Region
3
464
User
Region
3
465
User
Region
3
466
User
Region
3
467
User
Region
3
468
User
Region
3
469
User
Region
3
470
peper bubi
Region
3
471
User
Region
3
472
Amaia King
Region
3
473
User
Region
3
474
User
Region
3
475
User
Region
3
476
User
Region
3
477
User
Region
3
478
User
Region
3
479
User
Region
3
480
Ivan Lai
Region
3
481
User
Region
3
482
User
Region
3
483
User
Region
3
484
User
Region
2
485
User
Region
2
486
User
Region
2
487
User
Region
2
488
User
Region
2
489
User
Shanghai
2
490
User
Region
2
491
User
Region
2
492
User
Region
2
493
User
Region
2
494
User
Shanghai
2
495
User
Region
2
496
User
Region
2
497
User
Region
2
498
User
Region
2
499
User
Region
2
500
User
Region
2
501
User
Region
2
502
User
Region
2
503
User
Region
2
504
User
Region
2
505
User
Region
2
506
User
Region
2
507
User
Region
2
508
User
Region
2
509
User
Region
2
510
User
Shanghai
2
511
User
Jiangsu
2
512
User
Region
2
513
User
Region
2
514
User
Region
2
515
User
Region
2
516
User
Zhejiang
2
517
User
Region
2
518
User
Region
2
519
User
Region
2
520
User
Region
2
521
User
Region
2
522
User
Region
2
523
User
Region
2
524
User
Region
2
525
User
Region
2
526
User
Region
2
527
User
Shaanxi
2
528
User
Region
2
529
User
Region
2
530
User
Region
2
531
User
Region
2
532
User
Region
2
533
User
Region
2
534
User
Region
2
535
User
Region
2
536
User
Region
2
537
User
Region
2
538
User
Region
2
539
User
Region
2
540
User
Region
2
541
User
Region
2
542
Tan Thien Luong
Region
2
543
User
Region
2
544
User
Region
2
545
User
Region
2
546
User
Region
2
547
User
Beijing
1
548
User
Region
1
549
User
Region
1
550
User
Region
1
551
User
Region
1
552
User
Region
1
553
User
Region
1
554
User
Beijing
1
555
User
Region
1
556
User
Region
1
557
User
Region
1
558
User
Region
1
559
User
Region
1
560
User
Region
1
561
User
Zhejiang
1
562
User
Region
1
563
User
Zhejiang
1
564
User
Region
1
565
User
Region
1
566
User
Shanghai
1
567
User
Hebei
1
568
User
Region
1
569
User
Region
1
570
User
Region
1
571
User
Region
1
572
User
Region
1
573
User
Zhejiang
1
574
User
Region
1
575
User
Region
1
576
Bernie Green
Region
1
577
User
Region
1
578
User
Region
1
579
User
Region
1
580
User
Region
1
581
User
Region
1
582
janice
Region
1
583
User
Region
1
584
User
Region
1
585
Robin Jiang
Region
1
586
TATANGELO Nicolas
Region
1
587
User
Region
1
588
User
Region
1
589
User
Beijing
1
590
User
Region
1
591
User
Region
1
592
User
Region
1
593
User
Yunnan
1
594
User
Region
1
595
User
Region
1
596
Karina
Region
1
597
User
Region
1
598
User
Region
1
599
User
Region
1
600
User
Anhui
1
601
User
Region
1
602
User
Shanghai
1
603
User
Region
1
604
User
Region
1
605
User
Region
1
606
User
Region
1
607
User
Region
1
608
User
Shanghai
1
609
User
Region
1
610
User
Region
1
611
User
Region
1
612
User
Region
1
613
User
Region
1
614
User
Region
1
615
User
Region
1
616
User
Jiangsu
1
617
User
Zhejiang
1
618
User
Region
1
619
User
Region
1
620
User
Shanghai
1
621
User
Region
1
622
User
Region
1
623
User
Region
1
624
User
Region
1
625
User
Region
1
626
Jian Lee
Region
1
627
User
Region
1
628
User
Region
1
629
User
Region
1
630
User
Region
1
631
User
Region
1
632
User
Region
1
633
User
Region
1
634
User
Region
1
635
User
Region
1
636
User
Region
1
637
User
Region
1
638
User
Region
1
639
User
Region
1
640
User
Region
1
641
User
Region
1
642
Ms. McCully
Region
1
643
User
Region
1
644
User
Region
1
645
User
Shanghai
1
646
User
Region
1
647
User
Region
1
648
User
Region
1
649
User
Region
1
650
User
Region
1
651
User
Region
1
652
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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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