Leaderboard
New Zealand
New Zealand
Worldwide
Rank
Level
1
Chris Stockley
Auckland
85
2
Taylor Taylor Yi
Auckland
71
3
Eamon Sun
Otago
62
4
Richard Gottlieb
Canterbury
60
5
User
Auckland
58
6
Matthew Inglis
Auckland
56
7
Luke Willis
Canterbury
53
8
User
Auckland
51
9
User
Auckland
48
10
Cameron Forster
Northland
48
11
Calum McKay
Canterbury
42
12
Amy Lee
Auckland
40
13
User
Auckland
39
15
User
Hawke\'s Bay
39
16
Olivia Landau
Manawatu-Wanganui
39
17
a b
Canterbury
38
18
User
Auckland
38
19
User
Hawke\'s Bay
37
20
User
Hawke\'s Bay
36
21
User
Hawke\'s Bay
36
22
User
Hawke\'s Bay
36
23
Abdurrahman Obaidullah
Otago
35
24
User
Hawke\'s Bay
35
25
User
Hawke\'s Bay
35
26
User
Hawke\'s Bay
34
27
User
Auckland
33
28
Alyssa Leddie
Hawke\'s Bay
33
29
NGAIO
Hawke\'s Bay
33
30
User
Wellington
33
31
tupu
Hawke\'s Bay
32
32
User
Wellington
32
33
User
Hawke\'s Bay
32
34
User
Canterbury
32
35
User
Hawke\'s Bay
31
37
User
Taranaki
30
38
westen
Hawke\'s Bay
30
39
Zac L
Otago
30
40
User
Auckland
29
41
Anu
Canterbury
29
42
User
Hawke\'s Bay
28
43
User
Hawke\'s Bay
28
44
Mandiv Sujan
Auckland
28
45
User
Auckland
27
46
User
Otago
27
47
Danielle McKay
Canterbury
26
48
User
Auckland
25
49
User
West Coast
25
50
User
Northland
25
51
User
Otago
25
52
User
Wellington
25
53
User
Hawke\'s Bay
25
54
User
Auckland
24
55
James
Hawke\'s Bay
24
56
User
Hawke\'s Bay
24
57
User
Hawke\'s Bay
24
58
User
Hawke\'s Bay
24
59
User
Auckland
24
60
User
Auckland
24
62
User
Auckland
23
63
User
Otago
23
64
User
Auckland
23
65
User
Auckland
22
66
time zone
Canterbury
22
67
User
Auckland
22
68
Shadman Jahin
Bay of Plenty
21
69
User
Wellington
21
71
User
Auckland
21
72
User
Auckland
20
73
User
Auckland
20
74
h toast
Bay of Plenty
20
75
Dion
Auckland
20
76
User
Wellington
20
77
User
Auckland
20
78
tupu
Hawke\'s Bay
19
79
Aaron Kramer
Canterbury
19
80
User
Auckland
19
81
wryao64
Auckland
19
82
User
Auckland
19
83
Jasmine Eyley
Manawatu-Wanganui
19
84
User
Auckland
19
85
User
Auckland
19
86
User
Auckland
18
87
User
Wellington
18
88
Winston Zhao
Auckland
18
89
User
Hawke\'s Bay
18
90
User
Wellington
17
92
Pablo 3620
Otago
17
93
User
Wellington
17
94
User
Wellington
17
95
User
Auckland
17
96
User
Auckland
16
97
timothy m
Canterbury
16
98
User
Otago
16
99
User
Hawke\'s Bay
16
100
Noah Bradley
Hawke\'s Bay
16
102
User
Auckland
16
103
cheryl
Otago
16
104
User
Canterbury
16
105
Alyssa Leddie
Hawke\'s Bay
16
106
User
Wellington
15
107
User
Auckland
15
108
User
Hawke\'s Bay
15
109
User
Auckland
15
110
User
Auckland
15
111
Roan Parker
Auckland
14
112
User
Northland
14
113
User
Auckland
14
114
User
Auckland
14
115
User
Auckland
14
116
User
Otago
14
117
User
Wellington
14
118
User
Auckland
14
119
User
Auckland
14
120
User
Auckland
14
121
User
Auckland
14
122
User
Bay of Plenty
14
123
User
Auckland
14
124
User
Auckland
13
125
Poppy
Hawke\'s Bay
13
126
User
Auckland
13
127
User
Auckland
13
128
User
Auckland
13
129
User
Wellington
13
130
User
Hawke\'s Bay
13
131
User
Auckland
13
133
Adda Meow
Wellington
13
135
User
Canterbury
13
136
User
Auckland
13
137
User
Auckland
13
138
User
Canterbury
13
139
User
Auckland
13
140
User
Canterbury
13
141
User
Hawke\'s Bay
13
142
User
Auckland
13
143
User
Auckland
13
144
jahseh
Auckland
12
145
User
Hawke\'s Bay
12
146
User
Hawke\'s Bay
12
147
Condog
Auckland
12
148
User
Otago
12
149
User
Hawke\'s Bay
12
150
User
Auckland
12
151
User
Auckland
12
152
User
Auckland
12
153
User
Wellington
12
154
User
Canterbury
12
155
User
Hawke\'s Bay
12
156
User
Hawke\'s Bay
12
157
User
Hawke\'s Bay
12
158
User
Auckland
12
159
jongin
Otago
12
160
User
Otago
12
161
User
Waikato
11
162
User
Auckland
11
163
User
Auckland
11
164
User
Auckland
11
165
User
Waikato
11
166
User
Auckland
11
167
User
Wellington
11
168
sai panat
Auckland
11
169
User
Waikato
11
170
User
Canterbury
11
171
User
Auckland
11
172
Nathan Overall
Wellington
11
173
User
Canterbury
11
174
User
Auckland
11
175
jeff barnes
Auckland
11
176
User
Waikato
11
177
User
Auckland
11
178
User
Auckland
11
179
User
Canterbury
11
180
User
Wellington
11
181
User
Auckland
11
182
User
Auckland
11
183
User
Waikato
11
184
User
Auckland
11
185
User
Northland
10
186
User
Nelson City
10
187
User
Hawke\'s Bay
10
188
User
Canterbury
10
189
User
Auckland
10
190
User
Auckland
10
191
josie
Hawke\'s Bay
10
192
User
Canterbury
10
193
Sarah Y
Auckland
10
194
User
Wellington
10
195
User
Wellington
10
196
User
Auckland
10
197
User
Auckland
10
198
Max
Hawke\'s Bay
10
199
User
Wellington
10
200
Owen Follington
Auckland
10
201
User
Auckland
10
202
Amy Lauridsen
Wellington
10
203
User
Auckland
10
204
User
Auckland
10
205
User
Hawke\'s Bay
9
206
User
Wellington
9
207
User
Wellington
9
208
User
Auckland
9
209
User
Auckland
9
210
User
Auckland
9
211
User
Auckland
9
212
User
Auckland
9
213
User
Auckland
9
214
User
Northland
9
215
User
Auckland
9
216
User
Taranaki
9
217
User
Auckland
9
218
User
Hawke\'s Bay
9
219
Sophie Munro
Hawke\'s Bay
9
220
User
Auckland
9
221
User
Wellington
9
222
User
Auckland
9
223
User
Wellington
9
224
User
Hawke\'s Bay
9
225
User
Auckland
9
226
User
Auckland
9
227
User
Canterbury
9
228
Joseph Edwards
Wellington
9
229
User
Auckland
9
230
User
Auckland
9
231
User
Auckland
8
232
User
Auckland
8
233
User
Wellington
8
234
User
Wellington
8
235
User
Auckland
8
236
User
Auckland
8
237
User
Auckland
8
238
User
Taranaki
8
239
Jason van Schaeck Mathon
Wellington
8
240
User
Auckland
8
241
User
Auckland
8
243
User
Auckland
8
244
User
Auckland
8
245
Best Moonsan
Auckland
8
246
User
Canterbury
8
247
User
Auckland
8
248
User
Auckland
8
249
User
Hawke\'s Bay
8
250
User
Canterbury
8
251
Kupa
Waikato
8
252
User
Auckland
8
253
User
Otago
8
254
User
Auckland
8
255
User
Auckland
8
256
User
Otago
8
257
User
Auckland
7
258
User
Northland
7
259
User
Wellington
7
260
User
Auckland
7
261
User
Waikato
7
262
User
Auckland
7
263
User
Hawke\'s Bay
7
264
User
Auckland
7
265
User
Auckland
7
266
User
Canterbury
7
269
User
Canterbury
7
270
User
Auckland
7
271
User
Hawke\'s Bay
7
272
User
Auckland
7
273
User
Auckland
7
274
User
Hawke\'s Bay
7
276
User
Wellington
7
277
User
Auckland
7
278
User
Auckland
7
279
User
Wellington
7
280
Maree
Canterbury
7
281
User
Hawke\'s Bay
7
282
User
Canterbury
7
283
User
Auckland
7
284
User
Auckland
7
285
User
Auckland
7
286
User
Auckland
7
287
User
Auckland
7
288
User
Auckland
7
289
User
Auckland
7
290
User
Auckland
7
291
User
Hawke\'s Bay
7
292
User
Canterbury
7
293
User
Auckland
7
294
User
Hawke\'s Bay
7
295
User
Canterbury
7
296
User
Auckland
7
297
User
Otago
7
298
User
Auckland
7
299
User
Auckland
7
300
User
Wellington
7
301
imissuhard lol
Auckland
7
302
User
Otago
7
303
User
Auckland
7
304
User
Auckland
7
305
User
Otago
6
306
User
Wellington
6
307
User
Auckland
6
308
User
Auckland
6
309
Joy Heather
Auckland
6
310
User
Auckland
6
311
User
Otago
6
312
User
Auckland
6
313
User
Auckland
6
314
User
Taranaki
6
315
Kay Paterson
Canterbury
6
316
User
Auckland
6
317
User
Hawke\'s Bay
6
318
Zachariah Ryan
Otago
6
319
User
Wellington
6
320
User
Auckland
6
321
User
Auckland
6
322
User
Auckland
6
323
User
Hawke\'s Bay
6
324
User
Auckland
6
325
User
Auckland
6
326
User
Auckland
6
327
User
Auckland
6
328
User
Auckland
6
329
User
Auckland
6
330
User
Nelson City
6
331
User
Auckland
6
332
User
Auckland
6
333
User
Waikato
6
334
User
Region
6
335
User
Auckland
6
336
Danisha
Hawke\'s Bay
6
337
Andrew Sunde
Auckland
6
338
User
Hawke\'s Bay
6
339
User
Auckland
6
340
Ngaio
Auckland
5
341
User
Auckland
5
342
User
Auckland
5
343
User
Canterbury
5
344
Leila Tansley
Auckland
5
345
User
Hawke\'s Bay
5
346
User
Waikato
5
347
User
Otago
5
348
User
Canterbury
5
349
User
Auckland
5
350
User
Auckland
5
352
User
Wellington
5
353
User
Otago
5
354
User
Wellington
5
355
User
Northland
5
356
User
Auckland
5
357
Logan
Bay of Plenty
5
358
User
Auckland
5
359
tyler p
Auckland
5
360
User
Canterbury
5
361
User
Waikato
5
362
User
Auckland
5
363
User
Auckland
5
364
User
Hawke\'s Bay
5
365
User
Waikato
5
366
User
Auckland
5
367
User
Auckland
5
368
User
Auckland
5
369
User
Auckland
5
370
User
Auckland
5
371
User
Canterbury
5
372
User
Auckland
5
373
User
Hawke\'s Bay
5
374
User
Auckland
5
375
User
Auckland
5
376
User
Wellington
5
378
User
Auckland
5
379
User
Hawke\'s Bay
5
380
User
Auckland
5
381
Nik Voykovich
Auckland
5
382
User
Canterbury
5
383
User
Auckland
5
384
User
Hawke\'s Bay
5
385
User
Auckland
5
386
User
Auckland
5
387
User
Auckland
5
388
User
Canterbury
4
389
User
Auckland
4
390
User
Auckland
4
391
Georgia
Hawke\'s Bay
4
392
User
Auckland
4
393
User
Auckland
4
394
User
Wellington
4
395
User
Auckland
4
396
User
Auckland
4
397
User
Auckland
4
398
Maxwell Fysh
Auckland
4
400
User
Auckland
4
401
User
Waikato
4
402
User
Waikato
4
403
User
Auckland
4
404
User
Auckland
4
405
User
Auckland
4
406
User
Canterbury
4
407
User
Auckland
4
408
User
Waikato
4
409
User
Auckland
4
410
User
Waikato
4
411
User
Wellington
4
412
User
Auckland
4
413
User
Auckland
4
414
User
Wellington
4
415
User
Wellington
4
416
User
Auckland
4
417
Emily ( ͡° ͜ʖ ͡°)
Auckland
4
419
MERSAKK
Bay of Plenty
4
420
User
Auckland
4
421
User
Canterbury
4
422
User
Auckland
4
423
User
Auckland
4
424
User
Wellington
4
425
User
Wellington
4
426
User
Auckland
4
427
User
Auckland
4
428
User
Wellington
4
429
User
Otago
4
430
User
Waikato
4
431
User
Auckland
4
432
User
Auckland
4
433
User
Auckland
4
434
User
Auckland
4
435
User
Canterbury
4
436
User
Auckland
4
438
User
Hawke\'s Bay
4
439
Kimberlyn Vaka
Hawke\'s Bay
4
440
Pirate X
Northland
4
441
User
Auckland
4
442
User
Auckland
4
443
User
Auckland
4
444
User
Canterbury
4
445
User
Auckland
3
446
User
Auckland
3
447
User
Auckland
3
448
User
Canterbury
3
450
User
Auckland
3
452
User
Auckland
3
453
josie
Hawke\'s Bay
3
454
User
Auckland
3
455
User
Auckland
3
456
User
Wellington
3
457
User
Auckland
3
458
User
Wellington
3
459
User
Hawke\'s Bay
3
460
User
Wellington
3
461
User
Auckland
3
462
User
Wellington
3
463
User
Auckland
3
464
User
Auckland
3
465
Jane
Canterbury
3
466
ngaio
Hawke\'s Bay
3
467
Tom Bell
Wellington
3
468
User
Auckland
3
469
User
Auckland
3
470
User
Auckland
3
471
User
Auckland
3
472
User
Auckland
3
473
User
Auckland
3
474
User
Hawke\'s Bay
3
475
User
Auckland
3
476
User
Auckland
3
477
User
Auckland
3
478
User
Otago
3
479
User
Canterbury
3
480
User
Auckland
3
481
User
Auckland
3
482
User
Northland
3
483
User
Auckland
3
484
User
Canterbury
3
485
User
Auckland
3
486
User
Auckland
3
487
User
Canterbury
3
488
User
Auckland
3
489
User
Auckland
3
490
User
Canterbury
3
491
Vicki Crozier
Hawke\'s Bay
3
492
User
Auckland
3
493
User
Wellington
3
494
User
Auckland
3
495
User
Auckland
3
496
User
Auckland
3
497
User
Auckland
3
498
User
Auckland
3
499
User
Auckland
3
500
User
Auckland
3
502
User
Auckland
3
503
User
Auckland
3
506
Lybeck
Canterbury
3
508
User
Hawke\'s Bay
3
509
User
Auckland
3
510
User
Auckland
3
511
User
Auckland
3
512
User
Canterbury
3
513
User
Auckland
3
514
User
Auckland
2
515
User
Auckland
2
516
User
Auckland
2
517
luca
Auckland
2
518
User
Auckland
2
519
User
Auckland
2
520
User
Auckland
2
521
User
Auckland
2
522
User
Hawke\'s Bay
2
523
User
Northland
2
524
User
Canterbury
2
525
User
Auckland
2
526
User
Auckland
2
527
Tommy
Waikato
2
528
User
Waikato
2
529
User
Auckland
2
530
User
Waikato
2
531
User
Auckland
2
532
User
Canterbury
2
533
User
Auckland
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
Hawke\'s Bay
2
546
User
Canterbury
2
547
User
Auckland
2
548
User
Auckland
2
549
User
Auckland
2
550
User
Region
2
552
User
Auckland
2
553
User
Auckland
2
554
User
Auckland
2
555
User
Auckland
2
556
User
Auckland
2
557
User
Canterbury
2
558
User
Auckland
2
559
User
Auckland
2
561
User
Waikato
2
562
User
Auckland
2
563
User
Auckland
2
564
User
Auckland
2
565
User
Auckland
2
566
User
Auckland
2
567
User
Auckland
2
568
User
Auckland
2
569
User
Wellington
2
570
User
Auckland
2
571
User
Auckland
2
572
User
Auckland
2
573
User
Wellington
2
574
User
Auckland
2
576
User
Auckland
2
577
NOAH
Auckland
2
578
User
Wellington
2
579
User
Wellington
2
580
User
Auckland
2
581
User
Auckland
2
582
User
Auckland
2
583
User
Otago
2
584
gmoney
Auckland
2
585
User
Auckland
2
586
User
Auckland
2
587
User
Auckland
2
588
User
Auckland
2
589
User
Wellington
2
590
Mafi-Haapai Valentine Manufekai
Auckland
2
591
User
Auckland
2
592
User
Auckland
2
593
User
Auckland
2
594
User
Auckland
2
595
Iraia Wensor
Auckland
2
596
User
Otago
2
597
User
Auckland
2
598
User
Auckland
2
599
User
Auckland
2
600
User
Auckland
2
601
User
Auckland
2
602
User
Auckland
2
603
User
Auckland
2
604
User
Auckland
2
606
User
Auckland
2
607
User
Auckland
2
608
User
Auckland
2
609
User
Auckland
2
610
User
Auckland
2
611
User
Hawke\'s Bay
2
612
User
Auckland
2
614
User
Auckland
2
616
User
Auckland
2
617
User
Auckland
2
618
User
Auckland
2
619
User
Auckland
2
620
User
Auckland
2
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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 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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