Leaderboard
Colorado
Rank
Level
1
Ryan Zhang
Fort Collins
88
2
Ellen Khat
Aurora
85
3
User
Westminster
81
4
Cody Goldman
Boulder
71
5
User
Boulder
66
6
User
Denver
58
7
User
Fort Collins
56
8
Bosssssssss
Englewood
54
9
User
Castle Rock
53
10
David Sprague
Denver
49
11
User
Denver
49
12
Mage of Antura
Lakewood
46
13
User
Castle Rock
46
14
User
Boulder
46
15
mad genius
Louisville
44
16
Aba
Castle Rock
42
17
User
Denver
42
18
User
Denver
41
19
Damon Achey
Colorado Springs
39
20
User
Denver
39
21
User
Denver
38
22
Kyle Kirkby
Fort Collins
38
23
User
Castle Rock
37
24
Abdullah Al Izzi
Denver
36
25
User
Colorado Springs
36
26
User
Longmont
35
27
User
Denver
33
28
Fred Yarger
Denver
32
29
User
Denver
32
30
User
Denver
31
31
Student Micah Kilburn
Castle Rock
31
32
Student Cooper Roberts
Castle Rock
31
33
User
Boulder
31
34
User
Greenwood Village
30
35
User
Aurora
30
36
Daniel Talero
Boulder
30
37
User
Denver
30
38
User
Denver
29
39
User
Denver
29
40
User
Denver
28
41
Julian Harig
Littleton
28
42
Kathryn Sprague
Denver
27
43
Gabriela Carcasson
Fort Collins
27
44
Brandon Watson
Gypsum
27
45
User
Highlands Ranch
27
46
User
Aurora
27
47
User
Aurora
26
48
Neil Carlson
Colorado Springs
26
49
User
Arvada
26
50
User
Aurora
26
51
Student Konnor Ramirez
Castle Rock
25
52
User
Denver
24
53
User
Denver
24
54
User
U.S. Air Force Academy
23
55
User
Parker
23
56
User
Castle Rock
23
57
Cecil Macgregor
Boulder
22
58
User
Denver
22
59
User
Boulder
22
60
User
Denver
22
61
User
Castle Rock
22
62
User
Castle Rock
22
63
User
Castle Rock
22
64
User
Boulder
22
65
User
Parker
22
66
User
Colorado Springs
21
67
User
Denver
21
68
User
Denver
21
69
User
Denver
21
70
User
Fort Lupton
21
71
User
Castle Rock
21
72
User
Fort Collins
21
73
User
Castle Rock
21
74
User
Littleton
21
75
User
Boulder
21
76
User
Broomfield
21
77
User
Commerce City
20
78
Josh Leigh
Greeley
20
79
User
Denver
20
80
User
Castle Rock
20
81
Nikhila Punuru
Denver
19
82
User
Lakewood
19
83
User
Castle Rock
19
84
User
Boulder
19
85
User
Gypsum
18
86
Amari Valentine
Pueblo
18
87
User
Pagosa Springs
18
88
User
Ken Caryl
18
89
User
Boulder
18
90
User
Denver
18
91
User
Fowler
17
92
User
Denver
17
93
User
Centennial
17
94
User
Lafayette
17
95
User
Castle Rock
17
96
Student Tanner Fitzsimons
Castle Rock
17
97
User
Aurora
16
98
Grace Regnier
Cherry Hills Village
16
99
User
Denver
16
100
User
Arvada
16
101
User
Boulder
16
102
User
Denver
16
103
User
U.S. Air Force Academy
16
104
User
Boulder
16
105
User
Denver
15
106
User
Denver
15
107
User
Thornton
15
108
User
Castle Rock
15
109
User
Denver
15
110
Student Kaleid Headley
Castle Rock
15
111
Student Vitorio Dante
Castle Rock
15
112
User
Denver
15
113
User
Denver
14
114
jay shah
Broomfield
14
115
User
Stonegate
14
116
User
Castle Rock
14
117
User
Denver
13
118
User
Greenwood Village
13
119
User
Boulder
13
120
User
Denver
13
121
User
Denver
13
122
User
Fowler
12
123
User
Lakewood
12
124
User
Denver
12
125
User
Grand Junction
12
126
User
Vail
12
127
User
Aurora
12
128
Jacob Hough
Denver
12
129
User
Denver
12
130
User
Denver
12
131
User
Denver
12
132
User
Fowler
11
133
User
Boulder
11
134
User
Denver
11
135
User
Denver
11
136
User
Air Force Academy
11
137
User
Denver
11
138
Grace Wilson
Fort Collins
10
139
User
Ponderosa Park
10
140
User
Castle Rock
10
141
User
Castle Rock
10
142
User
Denver
10
143
User
Fowler
9
144
User
Holly
9
145
User
Colorado Springs
9
146
User
Broomfield
9
147
User
Thornton
9
148
User
Denver
9
149
User
Denver
9
150
User
Denver
9
151
User
Aurora
9
152
User
Greeley
9
153
User
Castle Rock
9
154
User
Broomfield
9
155
User
Colorado Springs
9
156
User
Denver
9
157
User
Holly
8
158
User
Wheat Ridge
8
159
User
Denver
8
160
Nicholas Heil
Denver
8
161
User
Denver
8
162
User
Thornton
8
163
User
Gypsum
8
164
User
Limon
8
165
User
Denver
8
166
User
Denver
8
167
User
Denver
8
168
User
Denver
8
169
User
Denver
8
170
User
Denver
8
171
User
Colorado Springs
8
172
User
Denver
8
173
User
Holly
7
174
User
Aurora
7
175
User
Longmont
7
176
User
Fort Collins
7
177
User
Fort Collins
7
178
User
Denver
7
179
User
Denver
7
180
User
Lakewood
7
181
User
Denver
7
182
User
Lakewood
7
183
Curtis Turner
Colorado Springs
7
184
User
Castle Rock
7
185
User
Castle Rock
7
186
User
Castle Rock
7
187
User
Castle Rock
7
188
User
Castle Rock
7
189
User
Boulder
7
190
User
Holly
6
191
Noah Lievers
Denver
6
192
User
Colorado Springs
6
193
User
Boulder
6
194
User
Colorado Springs
6
195
User
Boulder
6
196
User
Fort Collins
6
197
Nick P.
Parker
6
198
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Denver
6
199
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Loveland
6
200
User
Denver
6
201
User
Colorado Springs
6
202
User
Boulder
6
203
User
Denver
6
204
User
Denver
6
205
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Berthoud
6
206
User
Denver
6
207
User
Castle Rock
6
208
User
Littleton
6
209
User
Denver
6
210
User
Denver
6
211
User
Parker
6
212
User
Gunnison
6
213
User
Fowler
5
214
User
Fowler
5
215
User
Holly
5
216
User
Fowler
5
217
User
Colorado Springs
5
218
User
Laporte
5
219
Jokester gotten real good hah you thought
Denver
5
220
User
Fort Collins
5
221
User
Centennial
5
222
User
Denver
5
223
User
Denver
5
224
User
Denver
5
225
User
Parker
5
226
User
Boulder
5
227
User
Denver
5
228
User
Denver
5
229
User
Castle Rock
5
230
User
Castle Rock
5
231
User
Castle Rock
5
232
User
Colorado Springs
5
233
User
Colorado Springs
5
234
User
Air Force Academy
5
235
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Denver
5
236
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Conifer
4
237
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Fowler
4
238
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Fowler
4
239
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La Junta
4
240
User
Fowler
4
241
User
Fowler
4
242
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Denver
4
243
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Ken Caryl
4
244
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Lakewood
4
245
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Denver
4
246
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Lakewood
4
247
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Lakewood
4
248
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Commerce City
4
249
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Denver
4
250
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Fort Collins
4
251
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Denver
4
252
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Denver
4
253
User
Loveland
4
254
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Castle Rock
4
255
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Castle Rock
4
256
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Castle Rock
4
257
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Castle Rock
4
258
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Castle Rock
4
259
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Woodland Park
4
260
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Denver
4
261
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Boulder
4
262
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Fowler
3
263
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Durango
3
264
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Colorado Springs
3
265
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Colorado Springs
3
266
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Limon
3
267
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Denver
3
268
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Gypsum
3
269
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Colorado Springs
3
270
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Grand Junction
3
271
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Colorado Springs
3
272
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Parker
3
273
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Colorado Springs
3
274
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Denver
3
275
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Denver
3
276
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Thornton
3
277
User
Colorado Springs
3
278
User
Denver
3
279
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Rocky Ford
3
280
User
Denver
3
281
Student Tyler Garcia
Castle Rock
3
282
Student Aiden Hemsley
Castle Rock
3
283
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Castle Rock
3
284
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Castle Rock
3
285
Student Juliana Woodward
Castle Rock
3
286
User
Castle Rock
3
287
User
Castle Rock
3
288
User
Castle Rock
3
289
User
Castle Rock
3
290
Rebecca Herman
Colorado Springs
3
291
User
Denver
3
292
User
Fowler
2
293
User
Denver
2
294
User
Boulder
2
295
User
Colorado Springs
2
296
User
Thornton
2
297
User
Denver
2
298
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Denver
2
299
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Denver
2
300
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City
2
301
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Denver
2
302
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Battlement Mesa
2
303
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Lakewood
2
304
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Aurora
2
305
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Aurora
2
306
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Battlement Mesa
2
307
User
Parker
2
308
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Boulder
2
309
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Denver
2
310
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Denver
2
311
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Boulder
2
312
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Greeley
2
313
User
Fort Collins
2
314
User
Denver
2
315
Student Avery Hancock
Castle Rock
2
316
User
Denver
2
317
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Lafayette
2
318
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Denver
2
319
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Denver
2
320
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Denver
2
321
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322
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Castle Rock
2
323
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Castle Rock
2
324
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Castle Rock
2
325
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Castle Rock
2
326
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Castle Rock
2
327
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Castle Rock
2
328
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Castle Rock
2
329
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Castle Rock
2
330
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Castle Rock
2
331
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Castle Rock
2
332
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Castle Rock
2
333
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Castle Rock
2
334
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Castle Rock
2
335
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Castle Rock
2
336
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2
337
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2
338
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2
339
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2
340
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1
341
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1
342
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1
343
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1
344
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345
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1
346
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1
347
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1
348
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Rocky Ford
1
349
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Battlement Mesa
1
350
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Battlement Mesa
1
351
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Castle Rock
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A Web App for Mathematics Training

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

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

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

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

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


What's a Mental Math Tip?

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

÷
984
3
328

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


Get Better at Mental Math

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

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

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

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

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

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

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


Use a Math Trainer

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

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

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

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

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


Mental Calculation

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

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

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

Example of two-digit
addition problem

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

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

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

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

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

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


Arithmetic

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

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

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

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

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

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

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

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

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


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