Leaderboard
New Zealand
New Zealand
Worldwide
Rank
Level
1
Taylor Taylor Yi
Auckland
97
2
User
Auckland
82
3
User
Auckland
80
4
Richard Gottlieb
Canterbury
72
5
Luke Willis
Canterbury
66
6
User
Auckland
66
7
Mandiv Sujan
Auckland
66
8
User
Auckland
65
9
Eamon Sun
Otago
61
11
Calum McKay
Canterbury
56
12
User
Auckland
55
13
a b
Canterbury
53
14
User
Auckland
53
15
User
Auckland
53
16
Olivia Landau
Manawatu-Wanganui
52
17
User
Auckland
52
18
User
Hawke\'s Bay
50
19
User
Hawke\'s Bay
50
20
User
Hawke\'s Bay
50
21
User
Hawke\'s Bay
49
22
User
Hawke\'s Bay
47
23
User
Wellington
47
24
User
Hawke\'s Bay
46
25
User
Wellington
46
26
User
Auckland
46
27
Abdurrahman Obaidullah
Otago
45
28
User
Auckland
45
29
User
Taranaki
44
30
User
Auckland
44
31
User
Auckland
43
32
User
Hawke\'s Bay
40
33
User
Wellington
40
34
User
Hawke\'s Bay
39
35
Zac L
Otago
38
36
User
Auckland
38
37
User
Auckland
38
38
User
Hawke\'s Bay
37
39
NGAIO
Hawke\'s Bay
37
40
User
Auckland
37
41
User
Hawke\'s Bay
37
42
User
Northland
37
43
User
Otago
37
44
User
Wellington
37
45
User
Auckland
36
46
User
Wellington
36
47
User
Canterbury
35
49
User
Hawke\'s Bay
34
50
User
Auckland
34
51
User
Auckland
34
52
Christine
Auckland
34
53
User
Hawke\'s Bay
33
54
User
Auckland
33
55
Shadman Jahin
Bay of Plenty
33
56
User
Auckland
33
57
User
Auckland
32
58
Dion
Auckland
32
59
User
Auckland
32
60
Winston Zhao
Auckland
32
62
User
Wellington
31
63
Danielle McKay
Manawatu-Wanganui
31
64
User
Auckland
31
65
wryao64
Auckland
31
66
User
Auckland
31
67
User
Auckland
31
68
User
Auckland
30
69
User
Auckland
29
70
User
Auckland
29
71
User
Taranaki
29
72
User
Canterbury
29
73
User
Auckland
29
74
User
Auckland
28
75
User
Auckland
28
76
Joseph Edwards
Wellington
28
77
User
Auckland
28
78
James
Hawke\'s Bay
27
79
User
Hawke\'s Bay
27
80
User
Hawke\'s Bay
27
81
User
Auckland
27
82
User
Auckland
27
83
User
Auckland
27
84
User
Auckland
27
85
jahseh
Auckland
27
86
User
Hawke\'s Bay
26
87
timothy m
Canterbury
26
88
Condog
Auckland
26
89
User
Auckland
26
90
User
Auckland
25
91
User
Otago
25
92
User
Auckland
25
93
User
Auckland
25
94
Oliver Wilkinson
Canterbury
25
95
Jasmine Eyley
Manawatu-Wanganui
24
96
User
Auckland
24
97
User
Auckland
24
98
User
Auckland
24
99
tupu
Hawke\'s Bay
23
100
cheryl
Otago
23
101
User
Auckland
23
102
User
Canterbury
23
103
User
Canterbury
23
104
User
Hawke\'s Bay
23
105
User
Auckland
23
106
Pablo 3620
Otago
22
107
User
Canterbury
22
108
User
Auckland
22
109
User
Wellington
22
110
User
Auckland
22
111
User
Nelson City
22
112
User
Wellington
22
113
User
Auckland
22
114
User
Auckland
22
115
User
Auckland
22
116
User
Otago
21
117
User
Otago
21
118
User
Auckland
21
119
User
Auckland
21
120
Amy Lauridsen
Wellington
21
121
Best Moonsan
Auckland
21
122
User
Canterbury
21
123
User
Auckland
21
124
User
Auckland
21
125
Adda Meow
Wellington
20
126
User
Otago
20
127
User
Auckland
20
128
User
Wellington
20
129
User
Wellington
20
130
User
Wellington
20
131
User
Auckland
20
132
Kay Paterson
Canterbury
20
133
User
Auckland
19
134
User
Auckland
19
135
User
Northland
19
136
User
Auckland
19
137
User
Wellington
19
138
User
Auckland
19
139
User
Otago
19
140
User
Hawke\'s Bay
18
142
User
Canterbury
18
143
User
Wellington
18
144
User
Auckland
18
145
User
Wellington
18
146
User
Northland
18
147
Jason van Schaeck Mathon
Wellington
18
148
User
Auckland
18
149
Andrew Sunde
Auckland
18
150
User
Wellington
18
151
User
Hawke\'s Bay
17
152
Noah Bradley
Hawke\'s Bay
17
153
User
Hawke\'s Bay
17
154
User
Auckland
17
155
User
Auckland
17
156
User
Auckland
17
157
User
Waikato
17
158
User
Auckland
17
159
User
Canterbury
17
160
User
Hawke\'s Bay
16
161
User
Hawke\'s Bay
16
162
User
Waikato
16
163
User
Auckland
16
164
User
Waikato
16
165
User
Auckland
16
166
User
Hawke\'s Bay
15
167
Poppy
Hawke\'s Bay
15
168
User
Wellington
15
169
User
Hawke\'s Bay
15
170
User
Auckland
15
171
User
Auckland
15
172
sai panat
Auckland
15
173
User
Wellington
15
174
User
Auckland
15
175
User
Auckland
15
176
User
Auckland
15
177
User
Auckland
15
178
User
Auckland
15
179
User
Auckland
15
180
User
Auckland
14
181
User
Auckland
14
182
User
Auckland
14
183
User
Auckland
14
184
User
Auckland
14
185
User
Canterbury
14
186
User
Auckland
14
187
User
Wellington
14
188
User
Auckland
14
189
User
Auckland
14
190
User
Bay of Plenty
14
191
User
Bay of Plenty
14
192
User
Wellington
14
193
User
Auckland
14
194
User
Auckland
14
195
User
Auckland
14
196
User
Bay of Plenty
14
197
User
Wellington
14
198
User
Wellington
14
199
User
Canterbury
13
200
User
Hawke\'s Bay
13
201
User
Auckland
13
202
User
Hawke\'s Bay
13
203
Owen Follington
Auckland
13
204
User
Auckland
13
205
User
Auckland
13
206
User
Auckland
13
207
User
Auckland
13
208
User
Auckland
13
209
User
Wellington
13
210
User
Hawke\'s Bay
12
211
User
Hawke\'s Bay
12
212
User
Canterbury
12
213
josie
Hawke\'s Bay
12
214
Sarah Y
Auckland
12
215
User
Auckland
12
216
User
Wellington
12
217
User
Auckland
12
218
Kupa
Waikato
12
219
User
Wellington
12
220
User
Auckland
12
221
User
Auckland
12
222
User
Auckland
12
223
User
Auckland
12
224
User
Wellington
12
225
imissuhard lol
Auckland
12
226
User
Auckland
12
227
User
Auckland
12
228
h toast
Bay of Plenty
11
229
User
Auckland
11
230
User
Auckland
11
231
User
Auckland
11
232
User
Hawke\'s Bay
11
233
User
Wellington
11
234
User
Auckland
11
235
User
Auckland
11
236
User
Auckland
11
237
User
Northland
11
238
User
Auckland
11
239
User
Canterbury
11
240
User
Auckland
11
241
User
Wellington
11
242
User
Auckland
11
243
User
Wellington
11
244
User
Auckland
11
245
User
Auckland
11
246
User
Auckland
11
247
User
Bay of Plenty
11
248
User
Wellington
11
249
User
Wellington
11
250
User
Auckland
11
251
Max
Hawke\'s Bay
10
252
User
Canterbury
10
253
User
Hawke\'s Bay
10
254
User
Auckland
10
255
User
Canterbury
10
256
User
Auckland
10
257
User
Auckland
10
258
User
Auckland
10
259
User
Auckland
10
260
User
Taranaki
10
262
User
Wellington
10
263
User
Auckland
10
264
User
Auckland
10
265
User
Hawke\'s Bay
9
266
Sophie Munro
Hawke\'s Bay
9
267
User
Otago
9
268
User
Auckland
9
269
User
Auckland
9
271
User
Hawke\'s Bay
9
272
Maree
Canterbury
9
273
tupu
Auckland
9
274
Joy Heather
Auckland
9
275
User
Auckland
9
276
User
Auckland
9
277
User
Waikato
9
278
User
Auckland
9
279
User
Auckland
9
280
User
Auckland
9
281
User
Waikato
9
282
User
Wellington
9
283
User
Canterbury
9
284
User
Hawke\'s Bay
8
285
User
Canterbury
8
286
User
Auckland
8
287
User
Auckland
8
288
User
Otago
8
289
User
Canterbury
8
290
User
Auckland
8
291
User
Auckland
8
292
User
Auckland
8
293
Ngaio
Auckland
8
294
Leila Tansley
Auckland
8
295
User
Otago
8
296
User
Auckland
8
297
User
Auckland
8
298
User
Wellington
8
299
User
Auckland
8
300
User
Auckland
8
301
User
Auckland
8
302
User
Auckland
8
303
User
Wellington
8
304
User
Northland
8
305
User
Wellington
8
306
User
Hawke\'s Bay
7
307
User
Auckland
7
308
User
Auckland
7
309
User
Hawke\'s Bay
7
310
User
Hawke\'s Bay
7
311
User
Nelson City
7
312
User
Auckland
7
313
Danisha
Hawke\'s Bay
7
314
User
Hawke\'s Bay
7
315
User
Auckland
7
316
User
Canterbury
7
317
User
Hawke\'s Bay
7
318
User
Auckland
7
319
User
Auckland
7
320
User
Canterbury
7
321
Nik Voykovich
Auckland
7
323
User
Auckland
7
324
User
Waikato
7
325
User
Auckland
7
326
User
Wellington
7
327
User
Auckland
7
328
User
Wellington
7
329
User
Wellington
7
330
User
Auckland
7
332
tyler p
Auckland
6
333
User
Canterbury
6
334
User
Auckland
6
335
User
Auckland
6
336
Nathan Overall
Waikato
6
337
User
Auckland
6
338
User
Hawke\'s Bay
6
339
User
Auckland
6
340
User
Auckland
6
341
User
Hawke\'s Bay
6
342
User
Canterbury
6
343
User
Hawke\'s Bay
6
344
User
Auckland
6
345
User
Auckland
6
346
User
Canterbury
6
347
User
Auckland
6
348
User
Auckland
6
349
Maxwell Fysh
Auckland
6
350
User
Auckland
6
351
User
Canterbury
6
352
User
Otago
6
353
User
Auckland
6
354
User
Auckland
6
355
User
Auckland
6
357
User
Auckland
6
358
User
Wellington
6
359
User
Auckland
6
360
User
Wellington
6
361
User
Auckland
6
362
bobknight42
Wellington
6
363
User
Auckland
6
364
Georgia
Hawke\'s Bay
5
365
User
Auckland
5
366
User
Auckland
5
367
User
Auckland
5
368
User
Auckland
5
369
MERSAKK
Bay of Plenty
5
371
User
Auckland
5
372
User
Auckland
5
373
User
Canterbury
5
374
User
Auckland
5
375
User
Auckland
5
376
User
Waikato
5
377
User
Auckland
5
378
User
Canterbury
5
379
Kimberlyn Vaka
Hawke\'s Bay
5
380
User
Hawke\'s Bay
5
381
Pirate X
Northland
5
383
Jane
Canterbury
5
384
User
Auckland
5
385
Tom Bell
Wellington
5
386
User
Auckland
5
387
User
Auckland
5
388
User
Canterbury
5
389
User
Auckland
5
390
Alyssa Leddie
Auckland
5
391
Vicki Crozier
Hawke\'s Bay
5
392
User
Auckland
5
393
User
Wellington
5
394
User
Auckland
5
395
User
Auckland
5
397
User
Auckland
5
398
User
Auckland
5
399
User
Auckland
5
400
User
Waikato
5
401
Hayden Rabel
Canterbury
5
402
User
Hawke\'s Bay
4
403
User
Auckland
4
404
User
Auckland
4
405
User
Auckland
4
406
User
Auckland
4
407
User
Auckland
4
408
User
Auckland
4
409
User
Auckland
4
410
User
Auckland
4
411
User
Auckland
4
412
User
Auckland
4
413
User
Hawke\'s Bay
4
414
User
Auckland
4
415
User
Auckland
4
416
User
Auckland
4
417
User
Auckland
4
418
User
Canterbury
4
419
User
Auckland
4
420
User
Canterbury
4
421
User
Canterbury
4
422
User
Auckland
4
423
User
Auckland
4
424
User
Auckland
4
425
User
Auckland
4
427
Lybeck
Canterbury
4
428
User
Auckland
4
429
User
Auckland
4
430
User
Auckland
4
431
luca
Auckland
4
432
User
Auckland
4
433
User
Auckland
4
434
User
Canterbury
4
435
User
Auckland
4
436
Tommy
Waikato
4
437
User
Waikato
4
438
User
Waikato
4
439
User
Auckland
4
440
User
Auckland
4
441
User
Auckland
4
444
User
Auckland
4
445
User
Auckland
4
446
User
Auckland
4
447
User
Auckland
4
448
User
Wellington
4
449
User
Auckland
4
450
User
Auckland
4
451
User
Wellington
4
452
User
Auckland
4
453
User
Auckland
4
454
User
Auckland
4
455
User
Auckland
4
456
User
Canterbury
4
457
User
Waikato
4
459
User
Canterbury
4
460
Emily ( ͡° ͜ʖ ͡°)
Auckland
3
461
User
Auckland
3
462
User
Auckland
3
463
User
Auckland
3
464
User
Hawke\'s Bay
3
465
User
Auckland
3
466
User
Auckland
3
467
User
Auckland
3
468
User
Auckland
3
469
User
Canterbury
3
470
User
Auckland
3
471
User
Auckland
3
472
User
Auckland
3
473
User
Auckland
3
474
User
Auckland
3
475
User
Auckland
3
476
User
Auckland
3
477
User
Auckland
3
478
User
Hawke\'s Bay
3
479
User
Canterbury
3
480
User
Region
3
481
User
Auckland
3
482
User
Auckland
3
483
User
Auckland
3
484
User
Auckland
3
485
User
Auckland
3
486
User
Canterbury
3
487
User
Auckland
3
488
User
Auckland
3
490
NOAH
Auckland
3
491
User
Wellington
3
492
User
Auckland
3
493
User
Auckland
3
494
User
Otago
3
495
gmoney
Auckland
3
496
User
Auckland
3
497
User
Auckland
3
498
User
Auckland
3
499
User
Auckland
3
500
Mafi-Haapai Valentine Manufekai
Auckland
3
501
User
Auckland
3
502
User
Auckland
3
503
User
Auckland
3
504
User
Auckland
3
505
User
Auckland
3
506
User
Auckland
3
507
User
Auckland
3
508
User
Waikato
3
509
User
Auckland
3
510
User
Auckland
3
511
User
Wellington
3
512
User
Auckland
3
513
User
Wellington
3
514
User
Wellington
3
515
User
Auckland
3
516
User
Auckland
3
517
User
Canterbury
3
518
User
Waikato
3
519
User
Auckland
3
520
User
Auckland
3
521
User
Otago
3
522
User
Auckland
3
523
User
Canterbury
3
524
User
Wellington
3
525
User
Auckland
3
526
User
Auckland
3
527
User
Auckland
3
528
User
Auckland
3
529
User
Auckland
2
530
User
Auckland
2
531
User
Auckland
2
532
User
Auckland
2
533
User
Hawke\'s Bay
2
534
User
Auckland
2
535
User
Auckland
2
536
User
Auckland
2
537
User
Auckland
2
538
User
Auckland
2
539
User
Auckland
2
540
User
Auckland
2
541
User
Auckland
2
542
User
Auckland
2
543
User
Auckland
2
544
User
Auckland
2
545
User
Auckland
2
546
User
Auckland
2
547
User
Auckland
2
548
User
Wellington
2
549
User
Auckland
2
550
User
Auckland
2
551
User
Auckland
2
552
User
Hawke\'s Bay
2
553
User
Auckland
2
554
User
Auckland
2
555
Iraia Wensor
Auckland
2
556
User
Canterbury
2
557
User
Auckland
2
558
User
Auckland
2
559
User
Auckland
2
560
User
Auckland
2
561
User
Nelson City
2
562
User
Southland
2
563
User
Canterbury
2
564
User
Canterbury
2
565
User
Auckland
2
566
User
Auckland
2
567
User
Auckland
2
568
User
Otago
2
569
User
Auckland
2
570
User
Auckland
2
571
User
Auckland
2
572
User
Auckland
2
573
User
Auckland
2
574
User
Auckland
2
575
User
Auckland
2
576
User
Canterbury
2
577
User
Waikato
2
578
User
Auckland
2
579
User
Auckland
2
580
sophie
Auckland
2
581
User
Auckland
2
582
User
Auckland
2
583
User
Auckland
2
584
User
Auckland
2
585
User
Auckland
2
586
User
Wellington
2
587
User
Auckland
2
588
User
Auckland
2
589
Rahuls
Auckland
2
590
User
Wellington
2
591
User
Canterbury
2
592
User
Auckland
2
593
User
Taranaki
2
594
User
Auckland
2
595
User
Canterbury
2
596
User
Auckland
2
597
User
Wellington
2
598
User
Wellington
2
599
User
Wellington
2
600
User
Auckland
2
601
User
Auckland
2
602
User
Auckland
2
603
User
Wellington
2
604
User
Auckland
2
605
User
Auckland
2
606
User
Auckland
2
607
viksingh
Waikato
2
608
Jay Singh
Auckland
2
609
User
Auckland
2
610
User
Auckland
2
611
User
Auckland
2
612
User
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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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