Leaderboard
Pakistan
Pakistan
Worldwide
Rank
Level
1
User
Sindh
85
2
User
Sindh
83
3
User
Panjāb
72
4
Gagoru Bhaiyya
Sindh
69
5
Malik Muhammad Kashif Saeed
Islāmābād
61
6
User
Panjāb
57
7
Fatima Batool
Sindh
51
8
Zaaid Ahmed
Islāmābād
44
9
Ammad Ali
Sindh
43
10
User
Panjāb
43
11
User
Panjāb
39
13
Hajira Zaman
Sindh
38
14
User
Islāmābād
38
15
User
Panjāb
37
16
User
Sindh
36
17
User
Islāmābād
30
18
User
Panjāb
29
19
User
Sindh
29
20
User
Panjāb
29
21
User
Islāmābād
27
22
User
Panjāb
27
23
User
Panjāb
26
24
mariam raza
Panjāb
25
25
User
Sindh
25
26
User
Panjāb
25
27
Shahid Anwer
Islāmābād
25
28
User
Sindh
24
29
User
Panjāb
22
30
User
Islāmābād
22
31
User
Panjāb
22
32
User
Sindh
21
34
User
Sindh
21
35
User
Sindh
20
37
User
Sindh
19
38
User
Panjāb
19
39
malik muhammad kashif saeed
Islāmābād
19
40
User
Sindh
18
41
User
Sindh
18
42
User
Sindh
18
43
User
Sindh
18
44
User
Sindh
17
45
User
Panjāb
17
46
Muhammad Razi
Panjāb
17
47
User
Panjāb
17
48
User
Sindh
17
49
User
Sindh
17
50
User
Sindh
17
51
User
Panjāb
16
52
User
Islāmābād
16
53
User
Panjāb
16
54
User
Panjāb
15
55
Ali Hanif
Sindh
15
56
User
Sindh
15
57
User
Islāmābād
15
58
User
Panjāb
15
59
User
Panjāb
15
60
User
Sindh
15
61
User
Panjāb
15
62
User
Panjāb
15
63
User
Sindh
14
64
User
Panjāb
14
65
User
Islāmābād
14
66
User
Sindh
14
67
User
Panjāb
14
68
User
Sindh
14
69
User
Sindh
13
70
PAK gamming
Sindh
13
71
User
Sindh
13
72
User
Panjāb
13
74
User
Sindh
13
75
User
Sindh
13
76
Palwasha Faizan
Sindh
13
77
Omar Khan
Panjāb
13
78
User
Sindh
13
79
Aribah Noman
Sindh
13
80
User
Sindh
13
81
User
Sindh
13
82
User
Panjāb
12
83
User
Panjāb
12
84
User
Panjāb
12
85
User
Panjāb
12
86
User
Islāmābād
12
87
User
Sindh
12
88
User
Sindh
12
89
User
Sindh
12
90
User
Sindh
12
91
User
Sindh
12
92
User
Sindh
12
93
User
Panjāb
12
94
User
Panjāb
12
95
User
Panjāb
11
96
User
Sindh
11
97
User
Panjāb
11
98
User
Sindh
11
99
User
Panjāb
11
100
User
Panjāb
11
101
User
Panjāb
11
102
User
Sindh
11
103
User
Sindh
11
104
User
Panjāb
11
105
User
Islāmābād
11
106
User
Islāmābād
11
107
User
Sindh
11
108
Faraz
Islāmābād
11
109
User
Sindh
11
110
User
Panjāb
11
111
User
Panjāb
11
112
User
Sindh
10
113
User
Sindh
10
114
User
Islāmābād
10
115
User
Sindh
10
116
User
Sindh
10
117
User
Sindh
10
118
User
Panjāb
10
119
User
Panjāb
10
120
User
Panjāb
10
121
User
Panjāb
10
122
User
Panjāb
10
123
User
Sindh
10
124
User
Panjāb
10
125
User
Islāmābād
10
126
User
Panjāb
10
127
madiha kamal
Islāmābād
10
128
User
Sindh
10
129
User
Panjāb
10
130
User
Islāmābād
10
131
User
Panjāb
10
132
User
Panjāb
10
133
User
Sindh
10
134
User
Sindh
10
135
User
Panjāb
10
136
User
Panjāb
10
137
User
Sindh
10
138
User
Islāmābād
10
139
User
Sindh
10
140
rehman
Islāmābād
9
141
User
Panjāb
9
142
User
Panjāb
9
143
User
Sindh
9
144
User
Islāmābād
9
145
User
Panjāb
9
146
User
Sindh
9
147
User
Panjāb
9
148
User
Sindh
9
149
User
Panjāb
9
150
User
Sindh
9
151
User
Panjāb
9
152
User
Sindh
9
153
User
Panjāb
9
154
User
Panjāb
9
155
User
Sindh
9
156
User
Panjāb
9
157
User
Sindh
9
158
User
Sindh
9
159
User
Panjāb
9
160
User
Sindh
9
161
User
Sindh
8
162
User
Sindh
8
163
User
Sindh
8
164
User
Panjāb
8
165
User
Panjāb
8
166
User
Sindh
8
167
Ali Shan
Panjāb
8
168
User
Panjāb
8
169
User
Islāmābād
8
170
KTX ANNAN
Sindh
8
171
User
Sindh
8
172
User
Panjāb
8
173
Muhammad Ali
Islāmābād
8
174
User
Islāmābād
8
175
User
Panjāb
8
176
User
Sindh
8
177
User
Panjāb
8
178
User
Panjāb
8
179
User
Panjāb
8
180
User
Sindh
8
181
User
Sindh
8
182
User
Panjāb
8
183
User
Sindh
8
184
User
Islāmābād
8
185
User
Sindh
8
186
Fahd
Islāmābād
8
187
User
Sindh
8
188
User
Sindh
8
189
User
Panjāb
8
190
User
Islāmābād
8
191
User
Sindh
8
192
User
Sindh
8
193
User
Sindh
8
194
User
Panjāb
8
195
User
Sindh
8
196
User
Sindh
7
197
User
Panjāb
7
198
User
Sindh
7
199
User
Sindh
7
200
User
Panjāb
7
201
User
Islāmābād
7
202
User
Panjāb
7
204
User
Islāmābād
7
205
User
Panjāb
7
206
User
Panjāb
7
207
User
Panjāb
7
208
User
Panjāb
7
209
User
Panjāb
7
210
User
Balochistan
7
211
Farhaj Ahmed
Sindh
7
212
User
Panjāb
7
213
User
Panjāb
7
214
User
Islāmābād
7
215
User
Panjāb
7
216
User
Panjāb
7
217
User
Sindh
7
218
User
Sindh
7
219
User
Sindh
7
220
User
Panjāb
7
222
User
Panjāb
7
223
User
Panjāb
7
225
User
Panjāb
7
226
User
Panjāb
7
227
User
Panjāb
7
228
User
Sindh
7
229
User
Sindh
7
230
User
Sindh
7
231
User
Sindh
7
232
User
Sindh
7
233
User
Sindh
7
234
User
Sindh
6
235
User
Islāmābād
6
236
User
Panjāb
6
237
User
Sindh
6
238
Aaseya
Panjāb
6
239
User
Panjāb
6
240
User
Panjāb
6
241
User
Panjāb
6
242
User
Sindh
6
243
User
Panjāb
6
244
User
Sindh
6
245
User
Sindh
6
246
User
Panjāb
6
247
User
Islāmābād
6
248
User
Sindh
6
249
User
Sindh
6
250
User
Panjāb
6
251
User
Islāmābād
6
252
User
Panjāb
6
253
User
Panjāb
6
254
freckles
Islāmābād
6
255
User
Islāmābād
6
256
User
Sindh
6
257
User
Sindh
6
258
User
Panjāb
6
259
User
Panjāb
6
260
User
Panjāb
6
261
User
Panjāb
6
262
User
Panjāb
6
263
User
Sindh
6
264
User
Panjāb
6
265
User
Sindh
6
266
User
Sindh
6
267
User
Sindh
6
268
User
Panjāb
6
270
User
Sindh
6
271
User
Sindh
6
272
User
Sindh
6
273
User
Panjāb
5
274
User
Sindh
5
275
User
Sindh
5
276
User
Islāmābād
5
277
User
Sindh
5
278
User
Panjāb
5
279
User
Sindh
5
280
User
Sindh
5
281
User
Panjāb
5
282
User
Panjāb
5
284
User
Sindh
5
285
User
Islāmābād
5
286
Haya Khan
Sindh
5
287
User
Sindh
5
288
User
Panjāb
5
289
User
Panjāb
5
291
User
Panjāb
5
292
User
Islāmābād
5
293
User
Panjāb
5
294
User
Islāmābād
5
295
User
Islāmābād
5
296
User
Islāmābād
5
297
User
Panjāb
5
298
User
Panjāb
5
299
User
Sindh
5
300
User
Islāmābād
5
301
User
Islāmābād
5
302
User
Panjāb
5
303
User
Sindh
5
304
User
Sindh
5
305
Ahmed AbuBakr
Panjāb
5
306
User
Panjāb
5
307
User
Sindh
5
308
User
Sindh
5
309
User
Islāmābād
5
310
User
Panjāb
5
311
User
Sindh
5
312
User
Islāmābād
5
313
User
Panjāb
5
314
User
Sindh
5
315
User
Sindh
5
316
User
Sindh
5
318
User
Panjāb
5
319
User
Sindh
5
320
User
Islāmābād
5
321
User
Sindh
5
322
aaima adil
Sindh
5
323
User
Sindh
5
324
User
Panjāb
5
325
User
Panjāb
5
326
User
Panjāb
5
327
User
Panjāb
5
328
User
Panjāb
5
329
User
Islāmābād
5
330
User
Islāmābād
5
331
User
Sindh
5
332
User
Islāmābād
5
333
User
Sindh
4
334
User
Sindh
4
335
User
Sindh
4
336
User
Sindh
4
337
User
Panjāb
4
338
User
Sindh
4
339
User
Sindh
4
340
User
Panjāb
4
341
User
Sindh
4
342
User
Sindh
4
343
User
Sindh
4
344
User
Islāmābād
4
345
User
Sindh
4
346
User
Sindh
4
347
User
Sindh
4
348
Huzaifa Rashid
Sindh
4
349
User
Sindh
4
350
User
Panjāb
4
351
User
Panjāb
4
352
User
Panjāb
4
353
User
Sindh
4
354
Yousuf Khawaja
Panjāb
4
355
User
Sindh
4
356
User
Panjāb
4
357
User
Sindh
4
358
Hina Tahir
Sindh
4
359
User
Sindh
4
360
User
Sindh
4
361
User
Panjāb
4
362
User
Islāmābād
4
363
User
Panjāb
4
364
Syed Shah
Islāmābād
4
365
User
Sindh
4
366
User
Sindh
4
367
User
Sindh
4
368
User
Panjāb
4
369
User
Islāmābād
4
370
User
Sindh
4
371
User
Islāmābād
4
372
User
Sindh
4
373
User
Sindh
4
374
User
Panjāb
4
375
User
Panjāb
4
376
User
Islāmābād
4
377
Saad Akhtar
Sindh
4
378
User
Sindh
4
380
User
Sindh
4
381
User
Sindh
4
382
User
Panjāb
4
383
User
Islāmābād
4
384
User
Sindh
4
385
Sadia Mamji
Sindh
4
386
User
Sindh
4
387
User
Sindh
4
388
User
Islāmābād
4
389
User
Sindh
4
390
User
Panjāb
4
391
User
Sindh
4
392
User
Sindh
4
393
User
Panjāb
4
394
User
Sindh
4
395
User
Sindh
4
396
User
Sindh
4
397
User
Sindh
4
398
User
Islāmābād
4
399
User
Sindh
4
400
User
Sindh
3
402
User
Sindh
3
403
User
Panjāb
3
404
User
Islāmābād
3
405
User
Sindh
3
406
User
Sindh
3
407
User
Sindh
3
408
User
Sindh
3
409
User
Panjāb
3
410
User
Islāmābād
3
411
User
Panjāb
3
412
User
Sindh
3
413
User
Sindh
3
414
User
Sindh
3
415
User
Panjāb
3
416
User
Sindh
3
417
User
Sindh
3
418
User
Islāmābād
3
419
User
Panjāb
3
420
User
Islāmābād
3
421
User
Sindh
3
422
User
Islāmābād
3
423
User
Sindh
3
424
User
Sindh
3
425
User
Sindh
3
426
User
Panjāb
3
427
User
Panjāb
3
428
User
Sindh
3
429
User
Panjāb
3
430
User
Sindh
3
431
User
Sindh
3
432
User
Sindh
3
433
User
Sindh
3
434
User
Panjāb
3
435
User
Sindh
3
436
User
Islāmābād
3
437
User
Panjāb
3
438
User
Panjāb
3
439
User
Panjāb
3
440
User
Islāmābād
3
441
User
Panjāb
3
442
User
Sindh
3
443
User
Panjāb
3
444
User
Panjāb
3
445
User
Sindh
3
446
User
Islāmābād
3
447
User
Islāmābād
3
448
User
Islāmābād
3
449
User
Panjāb
3
450
User
Panjāb
3
451
User
Panjāb
3
452
User
Panjāb
3
453
User
Sindh
3
454
User
Sindh
3
455
User
Panjāb
3
456
User
Sindh
3
457
User
Sindh
3
458
User
Sindh
3
459
User
Sindh
3
460
User
Panjāb
3
461
User
Sindh
3
462
User
Sindh
3
463
User
Sindh
3
464
User
Sindh
3
465
User
Sindh
3
466
User
Panjāb
3
467
User
Sindh
3
468
User
Islāmābād
3
469
User
Sindh
3
470
User
Sindh
3
471
User
Panjāb
3
472
User
Sindh
3
473
User
Sindh
3
474
User
Panjāb
3
475
User
Sindh
3
476
User
Panjāb
3
477
User
Sindh
3
478
User
Azad Kashmir
3
479
User
Panjāb
3
480
User
Panjāb
3
481
User
Sindh
3
482
User
Sindh
3
483
User
Panjāb
3
484
User
Sindh
3
485
User
Sindh
3
486
User
Sindh
3
487
User
Sindh
3
488
User
Sindh
3
489
User
Islāmābād
3
490
User
Sindh
3
491
User
Sindh
3
492
User
Sindh
3
493
User
Sindh
3
495
User
Panjāb
3
496
User
Sindh
3
497
User
Sindh
3
498
User
Sindh
3
499
User
Sindh
3
500
User
Panjāb
3
501
User
Sindh
3
502
User
Panjāb
3
503
User
Azad Kashmir
2
504
User
Sindh
2
505
User
Panjāb
2
506
User
Islāmābād
2
507
User
Panjāb
2
508
User
Sindh
2
509
User
Islāmābād
2
510
hi
Sindh
2
511
Baryal Sher
Khaībar Pakhtūnkhwā
2
512
User
Panjāb
2
513
User
Sindh
2
514
User
Sindh
2
515
User
Panjāb
2
516
User
Sindh
2
517
User
Sindh
2
518
User
Sindh
2
519
User
Sindh
2
520
User
Sindh
2
521
User
Panjāb
2
522
User
Panjāb
2
523
User
Sindh
2
524
User
Sindh
2
525
User
Islāmābād
2
526
User
Panjāb
2
527
User
Panjāb
2
528
User
Sindh
2
529
User
Sindh
2
530
User
Sindh
2
531
User
Sindh
2
532
User
Panjāb
2
533
User
Islāmābād
2
534
User
Sindh
2
535
User
Sindh
2
536
User
Sindh
2
537
User
Sindh
2
538
User
Sindh
2
539
User
Sindh
2
540
User
Sindh
2
541
User
Balochistan
2
542
User
Sindh
2
543
User
Islāmābād
2
544
User
Sindh
2
546
User
Sindh
2
547
User
Panjāb
2
548
User
Sindh
2
549
User
Sindh
2
550
User
Sindh
2
551
User
Sindh
2
552
User
Sindh
2
553
User
Islāmābād
2
554
User
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A Web App for Mathematics Training

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

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

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

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

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


What's a Mental Math Tip?

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

÷
984
3
328

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


Get Better at Mental Math

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

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

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

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

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

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

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


Use a Math Trainer

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

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

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

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

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


Mental Calculation

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

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

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

Example of two-digit
addition problem

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

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

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

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

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

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


Arithmetic

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

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

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

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

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

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

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

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

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


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