Leaderboard
Ohio
Rank
Level
1
EASTON SINGER
Solon
95
2
User
Solon
90
3
Cory Huelskamp
Sidney
84
4
LIGHTSTARE
Upper Arlington
80
5
Jon Yoon
Columbus
78
6
King of the Hill
Blue Ash
67
7
ROBERT BATCHLET
Wooster
66
8
Abhinav Goel
Montgomery
60
9
User
Blue Ash
55
10
Emanuel Melaku
Reynoldsburg
54
11
User
Columbus
52
12
Mason Greene
Jerry City
51
13
Kevin Zhu
Barboursville
50
14
User
Sidney
49
15
Realism The Future
Boardman
49
16
Andrew Zhou
Montgomery
48
17
User
Beckett Ridge
48
18
Nathaniel Day
Cincinnati
46
19
User
Boardman
46
20
User
Cleveland
45
21
Nick M
Columbus
45
22
User
Columbus
45
23
Jonathan Nash
Reynoldsburg
44
24
User
Cleveland
44
25
User
Blue Ash
42
26
User
Forest Park
42
27
User
New Albany
42
28
User
Solon
41
29
Rohith Venkataraman
Cincinnati
41
30
Zerun Zhao
Montgomery
40
31
User
Cincinnati
40
32
User
Columbus
39
33
User
Montgomery
39
34
User
Blue Ash
39
35
User
Lyndhurst
39
36
Darren Kaiser
Sidney
37
37
User
Cleveland
37
38
User
Cincinnati
36
39
User
Withamsville
36
40
User
Columbus
36
41
Divey
Sidney
35
42
Siddharth Vikram
Blue Ash
35
43
User
Pataskala
35
44
Chenxi Zhou
Columbus
35
45
User
Sidney
34
46
User
Wooster
34
47
User
Mason
34
48
User
Upper Arlington
33
49
User
Oxford
33
50
Sean Griffin
Reynoldsburg
33
51
User
Westlake
32
52
User
Cleveland
31
53
ANDREW THOMAS
Pataskala
30
54
User
Westlake
30
55
Nicholas Johnson
Lewis Center
29
56
User
Columbus
29
57
JEREMY BAILEY
Orrville
29
58
User
Oxford
29
59
User
New Albany
28
60
Jeremy Arcus-Goldberg
New Albany
28
61
User
Defiance
28
62
User
Twinsburg
28
63
User
Cleveland
28
64
User
Solon
27
65
User
Mason
27
66
Kristen Schutte
Blue Ash
27
67
User
Pataskala
27
68
User
Jerry City
27
69
User
Beachwood
27
70
User
Cleveland
27
71
User
Columbus
26
72
User
Columbus
26
73
User
Sharonville
26
74
User
Westlake
26
75
Sofiya Lozovskiy
Brecksville
26
76
User
Reynoldsburg
26
77
User
Oxford
26
78
User
Brooklyn
26
79
User
Cleveland
26
80
User
Blue Ash
25
81
User
Springfield
25
82
User
Wooster
25
83
User
Findlay
25
84
User
Oxford
25
85
User
Reynoldsburg
25
86
User
Cincinnati
25
87
User
Brooklyn
25
88
User
Fairfield
24
89
User
Columbus
24
90
User
Blue Ash
24
91
User
Sidney
24
92
Alex P
Cedarville
24
93
User
Columbus
24
94
User
Dayton
24
95
User
Kent
23
96
Erice
Finneytown
23
97
User
Centerville
23
98
User
Columbus
23
99
User
Akron
23
100
User
Cincinnati
23
101
User
Defiance
23
102
User
Columbus
23
103
User
Brooklyn
23
104
User
Cincinnati
22
105
User
Sylvania
22
106
User
Columbus
22
107
User
Cincinnati
22
108
User
Beachwood
22
109
Josh Kelly
Columbus
21
110
User
Columbus
21
111
orlando wyman
Ashland
21
112
User
Indian Hill
21
113
User
Columbus
21
114
User
Cincinnati
21
115
User
Cleveland
21
116
User
Cleveland
21
117
User
Cleveland
21
118
Rachelle Hiestand
Sidney
20
119
User
Springfield
20
120
Brandon Engster
Warren
20
121
User
Reynoldsburg
20
122
User
Cleveland
20
123
User
Casstown
19
124
User
City
19
125
User
Columbus
19
126
User
Columbus
19
127
User
Reynoldsburg
19
128
User
Olmsted Falls
19
129
User
Upper Sandusky
19
130
Brady Ayers
Orrville
18
131
User
Loveland
18
132
User
Cincinnati
18
133
User
Columbus
18
134
User
Pataskala
18
135
User
Parma
18
136
User
Columbus
18
137
Griffin Waddell
Reynoldsburg
18
138
User
Cincinnati
18
139
kate
Cleveland
18
140
User
Cleveland
18
141
User
North Olmsted
18
142
User
Blue Ash
17
143
User
Marion
17
144
Alex Fazioli
Columbiana
17
145
D S
Clyde
17
146
Ethan Ferkins (Student)
Upper Arlington
17
147
Orlando Wyman
Westlake
17
148
User
Athens
17
149
User
Columbus
17
150
User
Pickerington
17
151
User
Dayton
16
152
User
Blue Ash
16
153
User
Strongsville
16
154
User
Fairborn
16
155
User
Barnesville
16
156
User
Columbus
16
157
User
New Franklin
16
158
User
Monroe
16
159
User
City
16
160
User
City
16
161
User
Reynoldsburg
16
162
User
Fairfield
16
163
User
Cleveland
16
164
User
Columbus
15
165
User
Blue Ash
15
166
User
Blue Ash
15
167
User
Sidney
15
168
User
Ravenna
15
169
User
Oxford
15
170
User
Cedarville
15
171
User
Columbus
15
172
User
Beaverdam
15
173
User
Columbus
14
174
Colin Battson
Blue Ash
14
175
User
Cincinnati
14
176
User
Shaker Heights
14
177
User
Beckett Ridge
14
178
User
Columbus
14
179
User
Reynoldsburg
14
180
User
Bluffton
14
181
User
Toledo
14
182
User
Columbus
14
183
User
South Amherst
13
184
User
Sidney
13
185
User
Cleveland
13
186
Sabrina Woods
Saint Paris
13
187
User
Columbus
13
188
User
Painesville
13
189
User
Grove City
13
190
User
Centerville
13
191
User
Reynoldsburg
13
192
sunnysideup
Cleveland
13
193
User
Dublin
13
194
User
Norwood
12
195
PATRICK MILLS
Newark
12
196
User
Kent
12
197
User
Sidney
12
198
User
Columbus
12
199
User
Dublin
12
200
User
Columbus
12
201
Christian Landers
Columbus
12
202
User
Chardon
12
203
User
Cleveland
12
204
MJT
New Philadelphia
11
205
User
Blue Ash
11
206
User
Sidney
11
207
User
Cincinnati
11
208
User
Sidney
11
209
User
Parma
11
210
User
Akron
11
211
User
Cuyahoga Falls
11
212
User
Columbus
11
213
User
Mayfield Heights
11
214
User
Cleveland
11
215
User
Cleveland
11
216
User
Columbus
11
217
Donggeun Kim
Blue Ash
10
218
User
Middletown
10
219
User
Sidney
10
220
User
Sidney
10
221
User
Perrysburg
10
222
User
Lima
10
223
User
City
10
224
User
City
10
225
User
Reynoldsburg
10
226
User
Beachwood
10
227
User
Columbus
10
228
User
Columbus
9
229
DANIEL JONES
Columbus
9
230
User
Columbus
9
231
Ediomo Eshett
Gahanna
9
232
User
Celina
9
233
User
Cincinnati
9
234
User
Oxford
9
235
User
Toledo
9
236
User
Sidney
9
237
User
Sidney
9
238
User
Sidney
9
239
User
Reynoldsburg
9
240
User
Delta
9
241
User
Athens
9
242
User
Dayton
9
243
User
Gahanna
9
244
User
Reynoldsburg
9
245
User
Cincinnati
9
246
User
Painesville
9
247
User
Cleveland
8
248
User
Withamsville
8
249
User
Delaware
8
250
User
Sidney
8
251
User
Sidney
8
252
User
Sidney
8
253
User
Cincinnati
8
254
User
Columbus
8
255
User
Reynoldsburg
8
256
User
Columbus
8
257
User
Pickerington
8
258
User
Centerville
8
259
Carla Dearth
New London
8
260
User
Archbold
8
261
User
Wooster
8
262
User
Brunswick
8
263
User
Warren
8
264
User
Toledo
8
265
User
Cincinnati
8
266
User
Greenville
7
267
User
Columbus
7
268
Avery
Finneytown
7
269
User
Blue Ash
7
270
User
Blue Ash
7
271
User
Stow
7
272
User
Sidney
7
273
User
Beckett Ridge
7
274
User
Perrysburg
7
275
User
Toledo
7
276
User
Cincinnati
7
277
User
Franklin
7
278
User
Clyde
7
279
User
Sylvania
7
280
User
Cleveland
7
281
User
Reynoldsburg
7
282
User
Reynoldsburg
7
283
User
Reynoldsburg
7
284
User
Columbus
7
285
User
Avon Lake
7
286
User
London
7
287
User
Athens
7
288
User
Twinsburg
7
289
User
Lebanon
7
290
User
Columbus
7
291
Sophia Wolf
North Royalton
6
292
User
Hilliard
6
293
User
Sidney
6
294
User
Sidney
6
295
User
Steubenville
6
296
User
New Richmond
6
297
User
Sidney
6
298
User
Sidney
6
299
User
Columbus
6
300
User
Struthers
6
301
User
City
6
302
User
City
6
303
User
Reynoldsburg
6
304
User
Columbus
6
305
User
Oxford
6
306
User
Columbus
6
307
User
Beckett Ridge
6
308
User
Cleveland
6
309
User
Reynoldsburg
6
310
Edward Deng
Cleveland Heights
5
311
Alyson Dildine
Columbus
5
312
User
Massillon
5
313
User
Columbus
5
314
User
Beachwood
5
315
User
Westerville
5
316
User
Elyria
5
317
User
Brewster
5
318
User
Sidney
5
319
User
Sidney
5
320
User
Sidney
5
321
User
Sidney
5
322
User
Toledo
5
323
User
Reynoldsburg
5
324
User
Reynoldsburg
5
325
User
Delaware
5
326
User
City
5
327
User
City
5
328
User
City
5
329
User
Columbus
5
330
User
Millville
5
331
User
Powell
5
332
User
Shaker Heights
5
333
User
Reynoldsburg
5
334
User
Gahanna
5
335
User
Reynoldsburg
5
336
User
Reynoldsburg
5
337
User
Brooklyn
5
338
User
Upper Sandusky
5
339
User
Portsmouth
5
340
User
Marietta
5
341
User
Beachwood
5
342
User
Oxford
5
343
User
Columbus
4
344
User
Newark
4
345
User
Lakewood
4
346
User
Middleburg Heights
4
347
User
Dayton
4
348
Shamini Gopinath
Sharonville
4
349
User
Lima
4
350
User
Columbus
4
351
User
Sidney
4
352
User
Columbus
4
353
User
Sidney
4
354
User
Sidney
4
355
User
Pataskala
4
356
User
Wetherington
4
357
User
Strongsville
4
358
User
Cleveland
4
359
User
Columbus
4
360
User
Cleveland Heights
4
361
User
Toledo
4
362
User
Shaker Heights
4
363
User
City
4
364
User
Reynoldsburg
4
365
User
Athens
4
366
User
Cincinnati
4
367
User
Mount Vernon
4
368
User
Mentor
4
369
User
North Royalton
4
370
User
Avon Lake
4
371
User
Reynoldsburg
4
372
User
Reynoldsburg
4
373
User
Reynoldsburg
4
374
User
Columbus
4
375
User
Grove City
4
376
User
Columbus
4
377
User
Dayton
4
378
User
Columbus
4
379
7th PVMS Alyson M Dildine
Columbus
3
380
User
Akron
3
381
Maddy simmons
Columbus
3
382
User
Strongsville
3
383
User
Richfield
3
384
User
Sylvania
3
385
Timofey Bessonov
Blue Ash
3
386
User
Strongsville
3
387
User
Massillon
3
388
User
Columbus
3
389
User
Sidney
3
390
User
Sidney
3
391
User
Sidney
3
392
User
Sidney
3
393
User
Sidney
3
394
User
Reynoldsburg
3
395
User
Beckett Ridge
3
396
User
Oxford
3
397
User
Norwood
3
398
User
Mansfield
3
399
User
Parma
3
400
User
Struthers
3
401
User
Broadview Heights
3
402
User
Broadview Heights
3
403
User
Broadview Heights
3
404
User
Mount Carmel
3
405
User
Broadview Heights
3
406
User
City
3
407
User
City
3
408
User
Reynoldsburg
3
409
User
Reynoldsburg
3
410
User
Cleveland
3
411
User
North Olmsted
3
412
User
Columbus
3
413
User
Upper Sandusky
3
414
User
Cleveland
3
415
User
Cleveland
3
416
User
Columbus
3
417
User
Reynoldsburg
3
418
User
Wadsworth
3
419
User
Reynoldsburg
3
420
User
Reynoldsburg
3
421
User
Reynoldsburg
3
422
User
Columbus
3
423
User
Columbus
3
424
User
Vermilion
3
425
User
Columbus
2
426
User
Columbus
2
427
User
Oxford
2
428
User
Newark
2
429
User
Perrysburg
2
430
User
Columbus
2
431
User
Toledo
2
432
User
Cincinnati
2
433
User
Shaker Heights
2
434
User
Toledo
2
435
User
Montgomery
2
436
User
Blue Ash
2
437
User
Blue Ash
2
438
Priscilla Cianelli
Strongsville
2
439
User
North Canton
2
440
User
Coldstream
2
441
User
Columbus
2
442
User
Sidney
2
443
User
New Carlisle
2
444
User
Pataskala
2
445
User
Columbus
2
446
User
Sidney
2
447
User
Sidney
2
448
User
Sidney
2
449
User
Sidney
2
450
User
Sidney
2
451
User
Solon
2
452
User
Hamilton
2
453
User
Columbus
2
454
User
Upper Arlington
2
455
User
Franklin
2
456
User
Columbus
2
457
User
Newark
2
458
User
Delaware
2
459
User
Lakewood
2
460
User
Cedarville
2
461
User
Reynoldsburg
2
462
User
Massillon
2
463
User
Broadview Heights
2
464
User
Akron
2
465
User
Cleveland
2
466
User
Struthers
2
467
User
Columbus
2
468
User
Lordstown
2
469
User
Toledo
2
470
User
Cincinnati
2
471
User
North Jackson
2
472
User
Reynoldsburg
2
473
Andrey Ganin
Sylvania
2
474
User
Reynoldsburg
2
475
User
Milford
2
476
User
Reynoldsburg
2
477
User
Reynoldsburg
2
478
User
Cleveland Heights
2
479
User
Cleveland
2
480
User
Cincinnati
2
481
User
Hudson
2
482
User
Springfield
2
483
User
Alliance
2
484
User
Fairfield
2
485
User
Reynoldsburg
2
486
User
Reynoldsburg
2
487
User
Mason
2
488
User
Reynoldsburg
2
489
User
Reynoldsburg
2
490
User
Willow Wood
2
491
User
Aurora
2
492
User
Cincinnati
2
493
User
Columbus
2
494
User
Barberton
2
495
User
Cleveland
2
496
User
Cincinnati
1
497
User
Springboro
1
498
User
Columbus
1
499
User
Wilmington
1
500
User
Cleveland
1
501
User
Blue Ash
1
502
User
The Plains
1
503
User
Massillon
1
504
User
Forest Park
1
505
User
Cleveland
1
506
User
Cincinnati
1
507
User
Zanesville
1
508
User
Centerville
1
509
User
Newark
1
510
User
New Albany
1
511
User
Columbus
1
512
User
Cleveland
1
513
User
Gahanna
1
514
User
Strongsville
1
515
Beth Balaski
Canton
1
516
User
Canton
1
517
User
Strongsville
1
518
User
Louisville
1
519
yusuf
Strongsville
1
520
Dacey Leonard
Strongsville
1
521
Catherine Bradburne
Findlay
1
522
Joseph Auteberry
Broadview Heights
1
523
Brady A
Wooster
1
524
Paige Gentile
Richfield
1

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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.

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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.

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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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