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1
User
Nyland
92
2
a b
Nyland
92
3
User
Nyland
91
4
User
Nyland
91
5
User
Birkaland
82
6
Noora Haapajarvi
Nyland
75
7
Adele S
Nyland
71
8
User
Nyland
71
9
User
Nyland
70
10
MarqueE
Nyland
68
11
User
Keski-Suomi
65
12
Ronan Doyle
Nyland
61
13
Anna Kaukkila
Nyland
61
14
User
Nyland
57
15
User
Kymenlaakso
56
16
User
Nyland
56
17
User
Kymenlaakso
55
18
User
Nyland
55
19
User
Birkaland
53
21
User
Nyland
52
22
User
Nyland
51
23
User
Kymenlaakso
51
24
User
Nyland
51
25
User
Nyland
51
26
User
Nyland
51
27
User
Nyland
49
28
Marq
Nyland
49
29
User
Keski-Suomi
48
30
User
Nyland
48
31
User
Nyland
48
32
User
Nyland
47
34
User
Keski-Suomi
46
35
User
Kymenlaakso
46
36
User
Nyland
45
37
User
Kymenlaakso
44
38
Yequian
Kymenlaakso
44
39
User
Nyland
43
40
User
Birkaland
43
41
User
Nyland
42
42
User
Nyland
41
43
User
Nyland
41
44
User
Kymenlaakso
41
45
User
Nyland
41
47
User
Nyland
40
48
User
Nyland
40
49
User
Nyland
40
50
User
Keski-Suomi
39
51
User
Nyland
39
52
User
Nyland
39
53
User
Nyland
38
54
User
Etelä-Savo
37
55
User
Nyland
37
57
User
Nyland
37
58
User
Keski-Suomi
36
59
User
Nyland
36
60
User
Nyland
36
61
User
Nyland
36
62
Salla P
Nyland
35
63
Nikutesti
Nyland
34
64
Elina Hamalainen
Egentliga Finland
34
65
User
Nyland
34
66
Eva Stina
Norra Österbotten
33
68
User
Nyland
32
69
User
Nyland
32
70
User
Nyland
32
71
Spaasi
Birkaland
32
72
User
Nyland
32
73
User
Nyland
32
74
User
Nyland
31
75
User
Keski-Suomi
31
76
User
Nyland
31
77
User
Nyland
30
78
User
Nyland
29
79
User
Nyland
29
80
User
Nyland
29
81
User
Nyland
29
82
User
Nyland
29
83
User
Nyland
28
84
User
Etelä-Savo
27
85
User
Nyland
27
86
User
Nyland
27
87
User
Nyland
27
89
User
Nyland
27
90
User
Nyland
26
91
User
Nyland
26
92
User
Nyland
26
93
User
Nyland
26
95
User
Nyland
25
96
User
Nyland
25
97
User
Birkaland
25
98
User
Nyland
25
99
User
Nyland
24
100
User
Nyland
24
101
User
Lappi
24
102
User
Nyland
24
104
User
Birkaland
23
105
User
Nyland
23
106
User
Nyland
23
107
User
Nyland
23
108
User
Nyland
22
109
User
Nyland
22
113
Jari
Norra Karelen
21
114
User
Nyland
21
115
User
Birkaland
21
116
User
Birkaland
21
117
User
Nyland
21
118
User
Nyland
21
119
User
Nyland
21
120
User
Nyland
21
121
User
Nyland
21
122
User
Nyland
20
123
User
Nyland
20
124
User
Nyland
20
126
User
Österbotten
20
127
User
Nyland
20
128
User
Nyland
19
129
User
Nyland
19
130
User
Nyland
19
131
User
Nyland
19
132
User
Nyland
19
133
User
Nyland
18
134
User
Nyland
17
135
User
Österbotten
17
136
User
Nyland
17
137
User
Nyland
17
138
User
Nyland
17
139
Jarno Paavola
Etelä-Pohjanmaa
17
140
User
Nyland
17
141
User
Nyland
16
142
User
Nyland
16
143
User
Nyland
16
144
User
Nyland
16
145
User
Birkaland
16
146
User
Nyland
15
147
User
Nyland
15
148
User
Nyland
15
149
favorable
Nyland
15
150
User
Nyland
15
151
User
Nyland
15
152
User
Nyland
15
153
User
Nyland
15
154
User
Nyland
15
155
User
Nyland
15
156
User
Österbotten
14
157
User
Nyland
14
158
User
Nyland
14
159
User
Nyland
14
161
User
Nyland
13
163
User
Nyland
13
164
User
Birkaland
13
165
User
Nyland
13
166
User
Nyland
13
167
User
Nyland
13
168
User
Nyland
13
169
User
Birkaland
12
170
User
Nyland
12
172
User
Nyland
12
173
User
Nyland
12
174
User
Nyland
12
175
User
Nyland
12
176
User
Nyland
11
177
User
Nyland
11
178
User
Nyland
11
179
User
Nyland
11
180
User
Nyland
11
181
User
Nyland
11
182
User
Nyland
11
183
User
Nyland
11
184
User
Nyland
11
185
User
Nyland
10
186
User
Nyland
10
190
User
Nyland
10
191
Nico Nyberg
Birkaland
10
192
User
Nyland
10
193
User
Nyland
10
194
User
Nyland
10
195
User
Nyland
10
196
User
Nyland
10
197
User
Nyland
10
198
User
Nyland
10
199
User
Nyland
10
200
User
Nyland
10
201
User
Birkaland
10
202
User
Nyland
10
203
User
Birkaland
9
204
User
Nyland
9
205
Jaakko
Nyland
9
206
User
Nyland
9
207
User
Nyland
9
208
User
Nyland
9
209
User
Nyland
9
210
User
Nyland
9
211
User
Nyland
9
212
User
Nyland
9
214
User
Nyland
8
215
User
Nyland
8
216
User
Nyland
8
217
User
Nyland
8
218
User
Lappi
8
219
User
Nyland
8
220
User
Nyland
8
221
User
Nyland
8
222
User
Nyland
8
223
User
Birkaland
8
224
User
Nyland
8
225
User
Nyland
8
226
User
Nyland
8
227
User
Nyland
8
228
User
Nyland
8
229
User
Birkaland
8
230
User
Nyland
7
231
User
Nyland
7
233
User
Nyland
7
235
User
Nyland
7
236
User
Nyland
7
237
User
Nyland
7
238
User
Nyland
7
239
User
Nyland
7
240
User
Nyland
7
241
User
Nyland
6
242
User
Nyland
6
243
User
Nyland
6
244
User
Nyland
6
246
User
Nyland
6
248
User
Nyland
6
249
User
Kymenlaakso
6
250
User
Nyland
6
251
User
Nyland
6
252
User
Nyland
6
253
User
Nyland
6
254
User
Nyland
6
256
User
Birkaland
6
257
User
Nyland
6
258
User
Nyland
5
259
User
Nyland
5
260
User
Nyland
5
261
User
Nyland
5
262
User
Nyland
5
264
User
Nyland
5
265
User
Birkaland
5
266
User
Birkaland
5
267
Matthew Maunula
Egentliga Finland
5
268
User
Nyland
5
269
User
Nyland
5
270
User
Nyland
5
271
User
Nyland
5
272
User
Nyland
5
274
User
Nyland
4
275
User
Nyland
4
276
User
Kymenlaakso
4
278
User
Nyland
4
279
User
Birkaland
4
280
User
Nyland
4
281
User
Nyland
4
282
User
Nyland
4
283
Topi
Nyland
4
284
User
Nyland
4
285
User
Nyland
4
286
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Nyland
4
287
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Nyland
4
288
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Nyland
4
289
User
Keski-Suomi
4
290
User
Keski-Suomi
4
291
User
Nyland
4
292
User
Keski-Suomi
4
293
User
Nyland
3
295
User
Nyland
3
298
User
Lappi
3
299
User
Nyland
3
302
User
Österbotten
3
303
User
Nyland
3
304
User
Nyland
3
305
User
Nyland
3
306
User
Nyland
3
307
User
Keski-Suomi
3
308
User
Nyland
3
310
User
Nyland
3
311
User
Nyland
3
312
User
Nyland
3
313
User
Nyland
3
314
User
Birkaland
3
315
User
Keski-Suomi
3
316
User
Nyland
3
317
User
Nyland
3
318
User
Nyland
3
319
User
Nyland
3
320
User
Nyland
3
321
User
Nyland
3
325
User
Nyland
2
326
User
Nyland
2
327
User
Keski-Suomi
2
328
User
Nyland
2
329
User
Nyland
2
331
User
Nyland
2
333
Nico Nyberg
Birkaland
2
335
User
Nyland
2
336
User
Nyland
2
337
User
Nyland
2
338
User
Nyland
2
339
User
Nyland
2
340
User
Nyland
2
342
User
Birkaland
2
343
User
Nyland
2
344
User
Nyland
2
345
User
Nyland
2
346
User
Birkaland
2
347
User
Nyland
2
348
User
Nyland
2
349
User
Nyland
2
350
User
Nyland
2
351
User
Nyland
2
352
User
Nyland
2
353
User
Nyland
2
354
User
Nyland
2
356
User
Nyland
2
357
User
Nyland
2
358
User
Nyland
2
359
User
Nyland
2
360
User
Etelä-Savo
2
361
User
Birkaland
2
363
User
Nyland
1
365
User
Nyland
1
366
User
Österbotten
1
368
User
Nyland
1
369
User
Nyland
1
370
User
Nyland
1
371
User
Nyland
1
374
Kim Malinen
Nyland
1
375
User
Nyland
1
376
User
Birkaland
1
377
Hassan Abdi
Nyland
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.

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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