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1
User
Nyland
99
2
a b
Nyland
92
3
User
Nyland
91
4
Noora Haapajarvi
Nyland
75
5
User
Nyland
71
6
User
Nyland
70
7
Ronan Doyle
Nyland
61
8
Anna Kaukkila
Nyland
61
9
User
Nyland
57
10
User
Kymenlaakso
56
11
User
Nyland
56
12
User
Kymenlaakso
55
13
User
Nyland
55
14
User
Kymenlaakso
51
15
User
Nyland
51
16
User
Nyland
51
17
User
Nyland
51
18
Marq
Nyland
49
19
JJJJ
Nyland
46
20
User
Kymenlaakso
46
21
User
Nyland
45
22
User
Kymenlaakso
44
23
Yequian
Kymenlaakso
44
24
User
Nyland
43
25
User
Birkaland
43
26
User
Nyland
42
27
User
Nyland
41
28
User
Nyland
41
29
User
Kymenlaakso
41
30
User
Nyland
41
31
User
Nyland
41
33
User
Nyland
40
34
User
Nyland
40
35
User
Nyland
40
36
User
Nyland
39
37
User
Birkaland
38
38
User
Nyland
38
39
User
Nyland
37
41
User
Nyland
37
42
User
Nyland
37
43
User
Nyland
36
44
User
Nyland
36
45
Salla P
Nyland
35
46
User
Nyland
34
48
User
Nyland
32
49
User
Nyland
32
50
Spaasi
Birkaland
32
51
User
Nyland
32
52
User
Nyland
32
53
User
Nyland
31
54
User
Keski-Suomi
31
55
User
Nyland
31
56
User
Nyland
29
57
User
Nyland
29
58
User
Nyland
29
59
User
Nyland
29
60
User
Nyland
29
61
Nikutesti
Nyland
28
62
User
Nyland
28
63
User
Etelä-Savo
27
64
User
Nyland
27
65
User
Nyland
27
66
User
Nyland
27
68
User
Nyland
27
69
User
Nyland
26
70
User
Nyland
26
71
User
Nyland
26
73
User
Nyland
25
74
User
Birkaland
25
75
User
Nyland
25
76
User
Nyland
24
77
User
Nyland
24
78
User
Lappi
24
79
User
Nyland
24
81
User
Birkaland
23
82
User
Nyland
23
83
User
Nyland
23
84
User
Nyland
23
85
User
Nyland
22
86
User
Nyland
22
91
User
Nyland
21
92
User
Birkaland
21
93
User
Birkaland
21
94
User
Nyland
21
95
User
Nyland
21
96
User
Nyland
21
97
User
Nyland
21
98
User
Nyland
21
99
User
Nyland
20
100
User
Nyland
20
101
User
Nyland
20
103
User
Österbotten
20
104
User
Nyland
20
105
User
Nyland
19
106
User
Nyland
19
107
User
Nyland
19
108
User
Nyland
19
109
User
Nyland
19
110
User
Nyland
18
111
User
Nyland
17
112
User
Österbotten
17
113
User
Nyland
17
114
User
Nyland
17
115
User
Nyland
17
116
Jarno Paavola
Etelä-Pohjanmaa
17
117
User
Nyland
17
118
User
Nyland
16
119
User
Nyland
16
120
User
Nyland
16
121
User
Nyland
16
122
User
Birkaland
16
123
User
Nyland
15
124
User
Nyland
15
125
favorable
Nyland
15
126
User
Nyland
15
127
User
Nyland
15
128
User
Nyland
15
129
User
Nyland
15
130
User
Nyland
15
131
User
Nyland
15
132
User
Österbotten
14
133
User
Nyland
14
134
User
Nyland
14
135
User
Nyland
14
137
User
Nyland
13
138
User
Birkaland
13
139
User
Nyland
13
140
User
Nyland
13
141
User
Nyland
13
142
User
Nyland
13
143
User
Nyland
12
145
User
Nyland
12
146
User
Nyland
12
147
User
Nyland
12
148
User
Nyland
12
149
User
Nyland
11
150
User
Nyland
11
151
User
Nyland
11
152
User
Nyland
11
153
User
Nyland
11
154
User
Nyland
11
155
User
Nyland
11
156
User
Nyland
11
157
User
Nyland
10
158
Nico Nyberg
Birkaland
10
159
User
Nyland
10
160
User
Nyland
10
161
User
Nyland
10
162
User
Nyland
10
163
User
Nyland
10
164
User
Nyland
10
165
User
Nyland
10
166
User
Nyland
10
167
User
Nyland
10
168
User
Birkaland
10
169
User
Nyland
10
170
Elina Hamalainen
Nyland
9
171
User
Nyland
9
172
Jaakko
Nyland
9
173
User
Nyland
9
174
User
Nyland
9
175
User
Nyland
9
176
User
Nyland
9
177
User
Nyland
9
178
User
Nyland
9
179
User
Nyland
9
181
User
Lappi
8
182
User
Nyland
8
183
User
Nyland
8
184
User
Nyland
8
185
User
Nyland
8
186
User
Birkaland
8
187
User
Nyland
8
188
User
Nyland
8
189
User
Nyland
8
190
User
Nyland
8
191
User
Nyland
8
192
User
Birkaland
8
194
User
Nyland
7
196
User
Nyland
7
197
User
Nyland
7
198
User
Nyland
7
199
User
Nyland
7
200
User
Nyland
7
201
User
Nyland
7
203
User
Nyland
6
205
User
Nyland
6
206
User
Kymenlaakso
6
207
User
Nyland
6
208
User
Nyland
6
209
User
Nyland
6
210
User
Nyland
6
211
User
Nyland
6
212
User
Nyland
6
214
User
Birkaland
6
215
User
Nyland
6
217
User
Nyland
5
218
User
Birkaland
5
219
User
Birkaland
5
220
Matthew Maunula
Egentliga Finland
5
221
User
Nyland
5
222
User
Nyland
5
223
User
Nyland
5
224
User
Nyland
5
225
User
Nyland
5
227
User
Birkaland
4
228
User
Nyland
4
229
User
Nyland
4
230
User
Nyland
4
231
Topi
Nyland
4
232
User
Nyland
4
233
User
Nyland
4
234
User
Nyland
4
235
User
Nyland
4
236
User
Nyland
4
237
User
Keski-Suomi
4
238
User
Keski-Suomi
4
239
User
Nyland
4
240
User
Keski-Suomi
4
241
User
Lappi
3
242
User
Nyland
3
245
User
Österbotten
3
246
User
Nyland
3
247
User
Nyland
3
248
User
Nyland
3
249
User
Nyland
3
250
User
Keski-Suomi
3
251
User
Nyland
3
253
User
Nyland
3
254
User
Nyland
3
255
User
Nyland
3
256
User
Nyland
3
257
User
Birkaland
3
258
User
Keski-Suomi
3
259
User
Nyland
3
260
User
Nyland
3
261
User
Nyland
3
262
User
Nyland
3
263
User
Nyland
3
264
User
Nyland
3
266
User
Nyland
2
268
Nico Nyberg
Birkaland
2
270
User
Nyland
2
271
User
Nyland
2
272
User
Nyland
2
273
User
Nyland
2
274
User
Nyland
2
275
User
Nyland
2
277
User
Birkaland
2
278
User
Nyland
2
279
User
Nyland
2
280
User
Nyland
2
281
User
Birkaland
2
282
User
Nyland
2
283
User
Nyland
2
284
User
Nyland
2
285
User
Nyland
2
286
User
Nyland
2
287
User
Nyland
2
288
User
Nyland
2
289
User
Nyland
2
291
User
Nyland
2
292
User
Nyland
2
293
User
Nyland
2
294
User
Nyland
2
295
User
Etelä-Savo
2
296
User
Birkaland
2
297
User
Nyland
1
298
User
Nyland
1
299
User
Nyland
1
300
User
Nyland
1
303
Kim Malinen
Nyland
1
304
User
Nyland
1
305
User
Birkaland
1
306
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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