Leaderboard
Nevada
Rank
Level
1
User
Las Vegas
72
2
Joey
Reno
68
3
Alex
Las Vegas
59
4
Mason Sosa
Las Vegas
58
5
Traveon Parker [STUDENT]
Las Vegas
51
6
User
Las Vegas
47
7
User
Las Vegas
42
8
User
Las Vegas
41
9
Analiese Guillen [STUDENT]
Las Vegas
40
10
Grim turkey
Sparks
39
11
Andrew Smiciklas
Las Vegas
38
12
Andrew James Smiciklas
Las Vegas
37
13
User
Reno
36
14
User
Las Vegas
35
15
User
Las Vegas
35
16
User
Las Vegas
35
17
User
Las Vegas
34
18
elieven
Las Vegas
32
19
User
Las Vegas
32
20
Jacob Cox
Las Vegas
30
21
User
Las Vegas
30
22
Alex Riccio
Las Vegas
28
23
User
Las Vegas
27
24
User
Las Vegas
26
25
al s
Las Vegas
26
26
User
Las Vegas
26
27
Inayg
Las Vegas
25
28
User
Las Vegas
25
29
User
Las Vegas
25
30
User
Las Vegas
25
31
Al s
Las Vegas
25
32
Ciara Partridge
Las Vegas
24
33
User
Las Vegas
24
34
Stephanie Beecher
Las Vegas
24
35
User
Las Vegas
24
36
Abreona Browhaw
Las Vegas
23
37
Minister C.
Las Vegas
23
38
User
Las Vegas
23
39
User
Las Vegas
22
40
User
Las Vegas
22
41
User
Las Vegas
22
42
Samuel Sell
Las Vegas
22
43
User
Las Vegas
22
44
User
Las Vegas
22
45
User
Las Vegas
22
46
User
Las Vegas
21
47
User
Las Vegas
21
48
User
Las Vegas
21
49
User
Las Vegas
21
50
User
Reno
20
51
User
Las Vegas
20
52
Jean Romero [STUDENT]
Las Vegas
20
53
User
Las Vegas
20
54
User
Las Vegas
19
55
User
Las Vegas
19
56
User
Las Vegas
19
57
User
Las Vegas
19
58
User
Las Vegas
19
59
User
Las Vegas
19
60
User
Las Vegas
19
61
User
Las Vegas
19
62
User
Las Vegas
19
63
User
Las Vegas
19
64
Fabian R
Las Vegas
18
65
User
Las Vegas
18
66
User
Las Vegas
18
67
User
Las Vegas
18
68
User
Las Vegas
18
69
User
Las Vegas
18
70
User
Las Vegas
18
71
User
Las Vegas
17
72
User
Las Vegas
17
73
Nathan Kassai
Las Vegas
17
74
Da'Kyng Ruffin [STUDENT]
Las Vegas
17
75
User
Las Vegas
17
76
User
Las Vegas
17
77
Juan E. Salamanca
Las Vegas
17
78
User
Las Vegas
17
79
Giovany Alvarez Arzola [STUDENT]
Las Vegas
16
80
User
Las Vegas
16
81
User
Las Vegas
16
82
User
Las Vegas
16
83
LoopyLocokiller
Las Vegas
16
84
User
Las Vegas
16
85
User
Las Vegas
16
86
User
Las Vegas
16
87
User
Las Vegas
16
88
User
Las Vegas
16
89
User
Las Vegas
16
90
User
Las Vegas
16
91
User
Las Vegas
16
92
User
Las Vegas
15
93
User
Las Vegas
15
94
User
Las Vegas
15
95
Sebastian Escoto
Las Vegas
15
96
User
Las Vegas
15
97
User
Las Vegas
15
98
Yainek Castro [STUDENT]
Las Vegas
15
99
Aniyah Banks-Williams [STUDENT]
Las Vegas
15
100
User
Las Vegas
15
101
User
Las Vegas
15
102
User
Las Vegas
15
103
User
Las Vegas
15
104
User
Las Vegas
14
105
User
Las Vegas
14
106
User
Las Vegas
14
107
User
Las Vegas
14
108
User
Las Vegas
14
109
Nirv
Las Vegas
14
110
User
Las Vegas
14
111
User
Las Vegas
14
112
Vince Chongdee [STUDENT]
Las Vegas
14
113
User
Las Vegas
14
114
User
Las Vegas
14
115
User
Las Vegas
14
116
User
Las Vegas
14
117
User
Las Vegas
14
118
david lopez
Las Vegas
14
119
User
Las Vegas
14
120
User
Las Vegas
14
121
User
Las Vegas
14
122
User
Las Vegas
14
123
User
Las Vegas
13
124
User
Las Vegas
13
125
User
Las Vegas
13
126
User
Las Vegas
13
127
User
Las Vegas
13
128
User
Las Vegas
13
129
Allen
Las Vegas
13
130
User
Las Vegas
13
131
User
Las Vegas
13
132
User
Las Vegas
13
133
User
Las Vegas
13
134
User
Las Vegas
13
135
User
Las Vegas
13
136
Justin Kavanagh
Reno
13
137
User
Las Vegas
13
138
User
Las Vegas
13
139
User
Las Vegas
13
140
User
Las Vegas
12
141
User
Las Vegas
12
142
Natalia Mankova
Las Vegas
12
143
User
Las Vegas
12
144
User
Las Vegas
12
145
User
Las Vegas
12
146
User
Las Vegas
12
147
User
Las Vegas
12
148
User
Las Vegas
12
149
User
Las Vegas
12
150
User
Las Vegas
12
151
User
Las Vegas
12
152
Alexis Carson
Las Vegas
12
153
User
Las Vegas
12
154
Jeraldyne Diaz Blakeley [STUDENT]
Las Vegas
12
155
User
Las Vegas
11
156
User
Las Vegas
11
157
User
Las Vegas
11
158
User
Las Vegas
11
159
User
Las Vegas
11
160
User
Las Vegas
11
161
User
Las Vegas
11
162
User
Las Vegas
11
163
User
Las Vegas
11
164
User
Las Vegas
11
165
User
Las Vegas
11
166
User
Las Vegas
11
167
User
Las Vegas
11
168
User
Las Vegas
11
169
Adan Quintana [STUDENT]
Las Vegas
10
170
User
Las Vegas
10
171
User
Las Vegas
10
172
User
Las Vegas
10
173
User
Las Vegas
10
174
User
Las Vegas
10
175
User
Las Vegas
10
176
User
Las Vegas
10
177
User
Las Vegas
10
178
User
Las Vegas
10
179
User
Las Vegas
10
180
User
Las Vegas
10
181
User
Las Vegas
10
182
User
Las Vegas
10
183
User
Las Vegas
10
184
User
Las Vegas
10
185
User
Las Vegas
10
186
User
Las Vegas
10
187
User
Las Vegas
9
188
User
Las Vegas
9
189
User
Las Vegas
9
190
User
Las Vegas
9
191
User
Las Vegas
9
192
User
Las Vegas
9
193
User
Las Vegas
9
194
User
Las Vegas
9
195
User
Las Vegas
9
196
User
Las Vegas
9
197
User
Las Vegas
9
198
User
Las Vegas
9
199
User
Las Vegas
9
200
User
Las Vegas
9
201
User
Las Vegas
9
202
User
Las Vegas
9
203
User
Las Vegas
9
204
User
Las Vegas
8
205
User
Las Vegas
8
206
User
Las Vegas
8
207
User
Las Vegas
8
208
User
Las Vegas
8
209
User
Las Vegas
8
210
User
Las Vegas
8
211
Kimberly Carrasco-Ruiz [STUDENT]
Las Vegas
8
212
User
Las Vegas
8
213
User
Las Vegas
8
214
User
Las Vegas
8
215
User
Las Vegas
8
216
User
Las Vegas
8
217
User
Las Vegas
8
218
User
Las Vegas
8
219
User
Las Vegas
8
220
User
Las Vegas
8
221
User
Las Vegas
8
222
Eme Fazili [STUDENT]
Las Vegas
8
223
User
Las Vegas
8
224
User
Las Vegas
8
225
User
Las Vegas
8
226
User
Las Vegas
8
227
User
Las Vegas
8
228
User
Las Vegas
8
229
Rachel Andersen
Las Vegas
8
230
User
Las Vegas
7
231
User
Las Vegas
7
232
Allison Su
Kingsbury
7
233
User
Las Vegas
7
234
User
Las Vegas
7
235
User
Las Vegas
7
236
User
Las Vegas
7
237
User
Las Vegas
7
238
User
Las Vegas
7
239
User
Las Vegas
7
240
User
Las Vegas
7
241
Keyan Winn Calbert [STUDENT]
Las Vegas
7
242
User
Las Vegas
7
243
User
Las Vegas
7
244
User
Las Vegas
7
245
User
Las Vegas
7
246
User
Las Vegas
7
247
User
Las Vegas
7
248
User
Las Vegas
7
249
User
Las Vegas
7
250
User
Las Vegas
7
251
Xhalyhanna McFadden [STUDENT]
Las Vegas
7
252
User
Las Vegas
7
253
User
Las Vegas
7
254
User
Las Vegas
7
255
User
Las Vegas
7
256
User
Las Vegas
7
257
User
Las Vegas
7
258
User
Las Vegas
7
259
User
Las Vegas
7
260
User
Las Vegas
7
261
User
Las Vegas
7
262
User
Las Vegas
6
263
User
Las Vegas
6
264
User
Las Vegas
6
265
User
Las Vegas
6
266
User
Las Vegas
6
267
User
Las Vegas
6
268
User
Las Vegas
6
269
User
Las Vegas
6
270
User
Las Vegas
6
271
User
Reno
6
272
User
Las Vegas
6
273
User
Las Vegas
6
274
Luis Moller
Las Vegas
6
275
User
Las Vegas
6
276
User
Las Vegas
6
277
User
Reno
6
278
User
Henderson
6
279
User
Las Vegas
6
280
User
Las Vegas
6
281
User
Las Vegas
6
282
User
Las Vegas
6
283
User
Las Vegas
6
284
User
Las Vegas
6
285
User
Las Vegas
6
286
User
Las Vegas
6
287
User
Las Vegas
6
288
User
Las Vegas
6
289
User
Las Vegas
6
290
User
Las Vegas
6
291
User
Las Vegas
6
292
User
Las Vegas
6
293
User
Las Vegas
6
294
User
Las Vegas
6
295
User
Las Vegas
5
296
User
Las Vegas
5
297
User
Las Vegas
5
298
Torintino Garrett [STUDENT]
Las Vegas
5
299
User
Las Vegas
5
300
User
Las Vegas
5
301
User
Las Vegas
5
302
User
Las Vegas
5
303
User
Las Vegas
5
304
User
Las Vegas
5
305
User
Las Vegas
5
306
Natalie Mackey [STUDENT]
Las Vegas
5
307
User
Las Vegas
5
308
User
Las Vegas
5
309
User
Las Vegas
5
310
User
Las Vegas
5
311
User
Reno
5
312
User
Las Vegas
5
313
User
Las Vegas
5
314
User
Las Vegas
5
315
User
Las Vegas
5
316
User
Las Vegas
5
317
User
Las Vegas
5
318
User
Las Vegas
5
319
User
Las Vegas
5
320
nm
Las Vegas
5
321
User
Las Vegas
5
322
User
Reno
5
323
User
Las Vegas
5
324
User
Las Vegas
5
325
User
Las Vegas
5
326
User
Las Vegas
5
327
User
Las Vegas
5
328
User
Las Vegas
5
329
User
Las Vegas
5
330
User
Las Vegas
5
331
User
Las Vegas
4
332
User
Las Vegas
4
333
User
Las Vegas
4
334
User
Las Vegas
4
335
User
Las Vegas
4
336
User
Las Vegas
4
337
User
Las Vegas
4
338
User
Las Vegas
4
339
User
Las Vegas
4
340
User
Las Vegas
4
341
User
Las Vegas
4
342
User
Las Vegas
4
343
User
Las Vegas
4
344
User
Las Vegas
4
345
Rastafarian 38 Mcallister
Las Vegas
4
346
User
Reno
4
347
User
Las Vegas
4
348
User
Las Vegas
4
349
User
Las Vegas
4
350
User
Las Vegas
4
351
User
Las Vegas
4
352
User
Las Vegas
4
353
User
Las Vegas
4
354
User
Winnemucca
4
355
User
Las Vegas
4
356
Dee Brown
Las Vegas
4
357
User
Las Vegas
4
358
User
Las Vegas
4
359
User
Las Vegas
4
360
User
Las Vegas
4
361
User
Las Vegas
4
362
User
Las Vegas
4
363
User
Reno
4
364
User
Reno
4
365
User
Las Vegas
4
366
User
Las Vegas
4
367
User
Las Vegas
4
368
User
Las Vegas
4
369
User
Las Vegas
4
370
User
Las Vegas
4
371
User
Las Vegas
3
372
User
Las Vegas
3
373
User
Las Vegas
3
374
User
Las Vegas
3
375
User
Las Vegas
3
376
User
Las Vegas
3
377
Marcos Munoz
Las Vegas
3
378
User
Las Vegas
3
379
User
Las Vegas
3
380
User
Las Vegas
3
381
User
Las Vegas
3
382
User
Las Vegas
3
383
User
Las Vegas
3
384
User
Las Vegas
3
385
User
Las Vegas
3
386
User
Las Vegas
3
387
User
Las Vegas
3
388
User
Las Vegas
3
389
User
Las Vegas
3
390
User
Las Vegas
3
391
Carlos Ortigoza [STUDENT]
Las Vegas
3
392
User
Las Vegas
3
393
User
Las Vegas
3
394
User
Las Vegas
3
395
User
Las Vegas
3
396
User
Las Vegas
3
397
User
Las Vegas
3
398
User
Las Vegas
3
399
User
Las Vegas
3
400
User
Las Vegas
3
401
User
Las Vegas
3
402
User
Las Vegas
3
403
User
Las Vegas
3
404
User
Las Vegas
3
405
User
Las Vegas
3
406
User
Las Vegas
3
407
User
Las Vegas
3
408
User
Las Vegas
3
409
User
Las Vegas
3
410
User
Las Vegas
3
411
User
Las Vegas
2
412
User
Las Vegas
2
413
User
Las Vegas
2
414
User
Las Vegas
2
415
User
Las Vegas
2
416
User
Las Vegas
2
417
User
Las Vegas
2
418
User
Las Vegas
2
419
User
Las Vegas
2
420
User
Las Vegas
2
421
User
Las Vegas
2
422
User
Las Vegas
2
423
User
Las Vegas
2
424
User
Las Vegas
2
425
User
Las Vegas
2
426
User
Las Vegas
2
427
User
Las Vegas
2
428
User
Las Vegas
2
429
User
Las Vegas
2
430
User
Las Vegas
2
431
User
Las Vegas
2
432
User
Las Vegas
2
433
User
Las Vegas
2
434
User
Las Vegas
2
435
User
Las Vegas
2
436
User
Las Vegas
2
437
Daiyan Galvez Hodelin [STUDENT]
Las Vegas
2
438
User
Las Vegas
2
439
User
Las Vegas
2
440
User
Las Vegas
2
441
User
Las Vegas
2
442
User
Las Vegas
2
443
User
Las Vegas
2
444
User
Las Vegas
2
445
User
Las Vegas
2
446
User
Las Vegas
2
447
User
Las Vegas
2
448
User
Las Vegas
2
449
User
Las Vegas
2
450
User
Las Vegas
2
451
User
Las Vegas
2
452
User
Las Vegas
2
453
User
Las Vegas
2
454
User
Las Vegas
2
455
User
Las Vegas
2
456
User
Las Vegas
2
457
User
Las Vegas
2
458
User
Las Vegas
2
459
User
Las Vegas
2
460
User
Las Vegas
2
461
Bryce Galuz
Las Vegas
2
462
User
Las Vegas
2
463
User
Las Vegas
2
464
User
Las Vegas
2
465
User
Las Vegas
2
466
User
Las Vegas
2
467
User
Las Vegas
2
468
User
Las Vegas
2
469
User
Las Vegas
2
470
User
Las Vegas
2
471
User
Las Vegas
1
472
User
Las Vegas
1
473
User
Las Vegas
1
474
User
Las Vegas
1
475
User
Las Vegas
1
476
User
Las Vegas
1
477
User
Las Vegas
1
478
User
Las Vegas
1
479
User
Las Vegas
1
480
User
Las Vegas
1
481
User
Las Vegas
1
482
User
Las Vegas
1
483
User
Las Vegas
1
484
User
Las Vegas
1
485
User
Las Vegas
1
486
User
Las Vegas
1
487
User
Las Vegas
1
488
User
Las Vegas
1
489
User
Las Vegas
1
490
User
Las Vegas
1
491
User
Las Vegas
1
492
User
Las Vegas
1
493
User
Las Vegas
1
494
Gionna Anderson
Las Vegas
1
495
User
Las Vegas
1
496
User
Las Vegas
1
497
User
Las Vegas
1
498
User
Las Vegas
1
499
User
Las Vegas
1
500
User
Las Vegas
1
501
User
Las Vegas
1
502
User
Las Vegas
1
503
User
Las Vegas
1
504
User
Las Vegas
1
505
User
Las Vegas
1
506
User
Las Vegas
1
507
User
Las Vegas
1
508
User
Las Vegas
1
509
User
Las Vegas
1
510
User
Las Vegas
1
511
User
Las Vegas
1
512
User
Las Vegas
1
513
User
Las Vegas
1
514
User
Las Vegas
1
515
User
Las Vegas
1
516
User
Las Vegas
1
517
User
Las Vegas
1
518
User
Las Vegas
1
519
User
Las Vegas
1
520
User
Las Vegas
1
521
User
Las Vegas
1
522
User
Las Vegas
1
523
User
Las Vegas
1
524
User
Las Vegas
1
525
User
Las Vegas
1
526
User
Las Vegas
1
527
User
Las Vegas
1
528
User
Las Vegas
1
529
User
Las Vegas
1
530
User
Las Vegas
1
531
User
Las Vegas
1
532
Kimberly Dicken
Las Vegas
1
533
User
Carson City
1
534
User
Reno
1
535
User
Las Vegas
1
536
User
Las Vegas
1
537
User
Las Vegas
1
538
User
Sparks
1
539
User
Las Vegas
1
540
User
Las Vegas
1
541
User
Las Vegas
1
542
User
Las Vegas
1
543
User
Las Vegas
1
544
User
Las Vegas
1
545
User
Las Vegas
1
546
User
Las Vegas
1
547
User
Las Vegas
1
548
User
Las Vegas
1
549
User
Sparks
1
550
User
Las Vegas
1
551
User
Las Vegas
1
552
User
Reno
1
553
User
Las Vegas
1
554
User
Las Vegas
1
555
User
Las Vegas
1
556
Heyzel Funes [STUDENT]
Las Vegas
1
557
User
Las Vegas
1
558
User
Las Vegas
1
559
User
Las Vegas
1
560
User
Fallon
1
561
User
Las Vegas
1
562
User
Las Vegas
1
563
User
Las Vegas
1
564
User
Las Vegas
1
565
User
Las Vegas
1
566
User
Las Vegas
1
567
Rashid Ishimwe [STUDENT]
Las Vegas
1
568
Christinatclara tosambe
Las Vegas
1
569
Nelson Lopez Coreas [STUDENT]
Las Vegas
1
570
Rashid 2k19
Las Vegas
1

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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.

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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