Leaderboard
Australia
Australia
Worldwide
Rank
Level
1
Callan
New South Wales
100
2
Logi SMITH
Victoria
100
3
Rio SATO
Victoria
97
4
🐧🐧🐧🐧🐧🐧🐧🐧🐧🐧🐧🐧🐧🐧🐧🐧🐧🐧🐧🐧🐧🐧🐧🐧🐧🐧🐧🐧🐧🐧🐧🐧🐧🐧🐧🐧🐧🐧🐧🐧🐧🐧🐧🐧🐧🐧🐧
Queensland
95
5
Nicolas
Queensland
95
6
Chris Weston
Queensland
93
7
Liam godly
Queensland
93
8
ryan n
Victoria
93
9
Steven Goldberg
Victoria
92
10
Wei Xi
New South Wales
92
11
Busy Bee
Victoria
88
12
Lincoln Drosse
South Australia
86
13
Terry Zhong
New South Wales
84
15
Tuusdey
South Australia
80
16
marcus dyson
Western Australia
80
19
Lyn Jin
New South Wales
78
20
Andrew Schirmer
Queensland
77
21
Luca HOSKING-HOARE
Victoria
76
22
David Guo
New South Wales
75
23
Peter Gibson
Queensland
75
24
Andy Wang
New South Wales
75
25
Kane Rex
New South Wales
75
28
Henry Yapeter
Western Australia
72
29
User
Victoria
72
30
David Malik
New South Wales
72
32
User
Victoria
71
33
Aiden Wong
New South Wales
71
34
Swiggolas
New South Wales
69
35
Mark Bergman
New South Wales
69
36
Alex LIANG
Victoria
69
37
Bob Smith
New South Wales
69
38
niber
Queensland
69
40
ID ZHAO
Victoria
67
42
EpiCuber7
New South Wales
66
43
Harrison HALL
Victoria
66
44
Nicholas MARAGELIS
Victoria
65
47
Pipi Zhao
Victoria
64
48
Joshua Peters
New South Wales
63
50
Dylan MANASSEH
Victoria
63
57
Joshua MORRISON
Victoria
61
58
User
Victoria
60
59
Crack Eggs
Victoria
60
60
Ayshani GAMPOLA
Victoria
60
61
Alberto Piva
New South Wales
60
62
Francois
Victoria
59
63
User
Victoria
59
65
Andrew
South Australia
59
66
Caroline Preston
New South Wales
59
67
Cooper FIRTH
Victoria
58
68
Troy price
Queensland
58
70
Ken Fang
New South Wales
57
71
Alexander Pham
New South Wales
57
72
(っ◔◡◔)っ 🇦🇺 AUSPRO 🇦🇺
Victoria
57
73
Madeleine Graham
Victoria
56
74
Aadi A
Tasmania
56
75
Crazynumnums
Queensland
56
78
Crispin Zou
New South Wales
55
80
rhys EAsthope
New South Wales
55
82
Jazzlyne Monzon
Victoria
55
83
Harsh Patel
Victoria
55
86
IncMan7 . .
Western Australia
55
87
Mumud Santun
Victoria
54
88
User
Victoria
54
90
User
Victoria
54
91
Shikamaru Nara
New South Wales
54
92
Garrythehobo Interesting
New South Wales
54
93
User
Victoria
53
95
Emil TANSEY
Victoria
53
96
Thomas TEMPLETON
Victoria
53
99
User
Queensland
53
100
Zawad Aziz
Queensland
52
101
Deepak RJ
Western Australia
52
102
Charlie
Victoria
52
103
Shyam Maharaj
New South Wales
52
104
Jordan Craw
New South Wales
52
107
Jarryd Beck
New South Wales
52
108
M P
Victoria
52
109
User
Queensland
52
112
xenxa
New South Wales
51
113
Michelle Barret
New South Wales
51
114
User
Victoria
51
115
User
Victoria
51
117
Rick
Queensland
51
118
User
Victoria
51
120
User
Queensland
50
121
User
Queensland
50
122
User
Victoria
50
123
Tyson Field
Victoria
50
124
FrostByte -
New South Wales
50
125
Helena SEREVETAS
Victoria
50
127
User
Victoria
49
128
Ethan Nguyen
New South Wales
49
129
Nirav Keshava
New South Wales
49
130
simon Woo
New South Wales
49
132
Trung Le
Victoria
48
133
bobothy bobson
New South Wales
48
134
james
Victoria
47
135
User
Queensland
47
136
Woody
Victoria
47
138
MikDaBozz
Victoria
47
139
User
Victoria
47
140
User
Victoria
47
141
User
Queensland
47
142
Jess Vereker
Victoria
46
146
User
Victoria
46
147
User
Queensland
46
148
User
Victoria
46
149
User
Victoria
46
152
Sam Sepial
New South Wales
46
153
Namitha Jacob
Queensland
45
155
User
Queensland
45
156
Taeyeong Kim
Victoria
45
157
Soren Stonebank
New South Wales
45
158
Kyle Montgomery
Queensland
45
159
Sam DORMER
Victoria
45
161
Thomas Kammermann
Western Australia
45
163
User
Victoria
44
165
User
Victoria
44
166
Ryhan Khan
Victoria
44
168
Aurelia HARVEY
Victoria
44
169
User
Queensland
44
170
Ari MOULIERIS
Victoria
44
171
User
Victoria
44
172
User
Queensland
44
177
User
Queensland
43
178
User
Victoria
43
180
User
Victoria
43
185
Shahar Eyal
Western Australia
43
186
Rezkiller
New South Wales
42
189
User
Victoria
42
190
Sofija CRNCEVIC
Victoria
42
191
Astrid Streeter-Smith
Queensland
42
192
User
Victoria
42
197
Tasmania
41
198
User
Queensland
41
199
Jamiele Cabato
Victoria
41
200
siccc
New South Wales
41
202
Angie Jiffry
Victoria
41
205
User
Victoria
41
206
Gus ALEKNA
Victoria
41
210
User
Queensland
40
211
User
Victoria
40
213
AlsoDaCsgoGod
Victoria
40
215
It’s me Pad
New South Wales
40
218
Lily Ypinazar
Victoria
40
220
User
Queensland
40
221
User
Victoria
40
222
marcus dyson
Western Australia
40
223
Anthony Xerri
Victoria
40
224
User
Victoria
40
225
User
Victoria
40
226
Nugget
New South Wales
40
229
User
Victoria
40
230
User
Victoria
40
231
User
Victoria
39
232
User
Victoria
39
233
User
Victoria
39
236
Harry STEWART
Victoria
39
237
Nicholas Wardle
New South Wales
39
238
Jack Kennedy
Western Australia
39
239
Prijwal
Queensland
39
240
Arham Zakir
New South Wales
39
241
User
Queensland
39
244
Callum
New South Wales
39
246
User
Victoria
39
247
User
Queensland
39
248
Hugo DOOLEY
Victoria
39
249
User
Victoria
38
250
User
Victoria
38
251
User
Victoria
38
252
Isuzu
Victoria
38
254
Mars Ultor
Queensland
38
255
Peter A
Victoria
38
256
User
Victoria
38
260
Big Numb
Tasmania
38
261
Wilson Jiang
New South Wales
38
263
User
Tasmania
38
268
Fabian Wise
Tasmania
37
270
User
Queensland
37
274
User
Victoria
37
277
LKT
Victoria
37
279
User
Victoria
37
281
User
Victoria
37
283
Laura Scott
New South Wales
37
284
Eyu Chyu
Victoria
37
287
Ben Murrie
New South Wales
36
288
Terry Roy
New South Wales
36
290
User
Victoria
36
291
User
Queensland
36
292
Joshua HUTIUK
Victoria
36
294
User
Queensland
36
295
Boyd Hillman
Victoria
36
296
User
Victoria
36
297
User
Victoria
36
298
User
Queensland
36
300
User
Victoria
36
301
Chrysostomos KONTOS
Victoria
36
302
FOFY
Victoria
36
303
User
Queensland
36
304
Brendan Fulton
New South Wales
36
305
User
Victoria
36
307
User
Victoria
35
310
User
Queensland
35
311
Jerry Fung
Victoria
35
313
Jackson Hawkins
Victoria
35
315
User
Queensland
35
317
Andrew Cain
New South Wales
35
318
Ericsuperman
Victoria
35
319
User
Victoria
35
320
ShadowRange
New South Wales
35
321
User
Queensland
35
322
User
Victoria
35
323
Random Person
Victoria
35
324
Frankie Fleming
Victoria
35
325
Kristian Rados
Victoria
35
326
User
Queensland
35
327
Jakmac
Victoria
35
328
George
Tasmania
35
335
User
Victoria
34
336
User
Victoria
34
340
User
Queensland
34
341
User
Victoria
34
343
User
Victoria
34
350
HGFlame
Victoria
34
351
User
Victoria
33
352
User
Victoria
33
353
User
Victoria
33
355
Vincent Phoummathep
New South Wales
33
356
User
Tasmania
33
357
The Table
Tasmania
33
359
User
Victoria
33
361
Eric Sciberras
Victoria
33
363
User
Queensland
33
366
User
Victoria
33
368
Yuta Masuda
South Australia
32
374
User
Victoria
32
375
User
Victoria
32
378
User
Tasmania
32
380
User
Queensland
32
385
User
Victoria
32
386
AGC
Victoria
31
388
User
Victoria
31
390
Annie PIDGEON
Victoria
31
391
Zhennan WANG
Victoria
31
392
User
Victoria
31
393
User
Victoria
31
394
Natasha SIM
Victoria
31
396
Liya Joseph
South Australia
31
397
User
Queensland
31
398
Chloe Lin
New South Wales
31
403
User
Victoria
31
404
Patrick WOODHOUSE
Victoria
31
405
LKT
Victoria
31
407
User
Queensland
31
408
prasidha
New South Wales
31
411
Ava YPINAZAR
Victoria
31
412
nik
Victoria
31
414
Harrison
Victoria
30
415
ChiniousVids
Tasmania
30
416
User
Victoria
30
419
Jack Wall
New South Wales
30
420
Jim Vella
Queensland
30
422
G Davies
Victoria
30
423
User
Victoria
30
425
Sue tucker
New South Wales
30
426
Posaunekaiser
Victoria
30
427
Noah A Kool Kid
New South Wales
30
428
User
Queensland
30
430
User
Victoria
30
431
User
Victoria
30
432
im noah
New South Wales
30
433
User
Queensland
30
434
User
Victoria
30
435
User
Tasmania
30
437
Seth Thuraisingham
South Australia
30
439
Jake Demasi
South Australia
29
441
User
Victoria
29
443
User
Queensland
29
444
User
Victoria
29
445
User
Victoria
29
446
BigManOnion
Victoria
29
447
User
Victoria
29
452
User
Victoria
29
453
User
Victoria
29
457
User
Victoria
29
458
Nola
Queensland
29
459
User
Victoria
29
461
User
Victoria
29
463
User
Victoria
29
464
User
Queensland
29
465
Jasmin Clarke
South Australia
29
469
Kristy-Lee
New South Wales
28
473
User
Victoria
28
475
User
Victoria
28
477
User
Queensland
28
478
User
Victoria
28
479
Jess MILDENHALL
Victoria
28
483
BC
Victoria
28
486
User
Victoria
28
489
User
Queensland
28
490
Otto LANGTRY-KIERCE
Victoria
28
491
Matthew
Victoria
28
495
Alexander Etges
Queensland
28
496
User
Queensland
28
498
User
Queensland
28
499
Tait Dwyer
Tasmania
28
500
User
Queensland
27
501
User
Victoria
27
502
User
Victoria
27
503
User
Queensland
27
508
User
Victoria
27
509
User
Victoria
27
510
Kline Nazareth
Victoria
27
511
Adem
Victoria
27
512
User
Victoria
27
514
User
Queensland
27
515
Sophia BOWDEN
Victoria
27
519
Maddy McBrearty
Victoria
27
522
Emmanuel Bosco
Tasmania
27
525
User
Victoria
27
527
User
Queensland
27
530
Viola LINDEWALD
Victoria
27
534
User
Queensland
26
536
User
Victoria
26
537
User
Queensland
26
538
User
Victoria
26
542
Dragoness
Queensland
26
543
Samuel LAWSON
Victoria
26
547
Emilia PERRY
Victoria
26
549
JJChicken
New South Wales
26
550
User
Queensland
26
554
Cody Bellgrove
Victoria
26
555
Martijn Muis
Western Australia
26
559
Josie COLE
Victoria
26
560
Lauren M
Victoria
26
562
User
Queensland
25
563
Luke Wanless
New South Wales
25
564
Kevin Trang
New South Wales
25
565
User
Victoria
25
566
User
Victoria
25
567
Lucy GUSTIN
Victoria
25
568
Raif KARRAR
Victoria
25
570
Mahya
New South Wales
25
572
Lerrrrm
Queensland
25
578
User
Victoria
25
579
User
Victoria
25
581
B.T123
New South Wales
25
582
Ryan HARTANTO
Victoria
25
583
User
Victoria
25
584
Shane
Queensland
25
585
Monkey Bong
New South Wales
25
587
User
Victoria
25
589
User
Victoria
25
592
Ella ROSS
Victoria
25
593
Emil Kiroff
New South Wales
25
594
Matthew Ward
South Australia
25
595
ąŋɬɧཞơƖƖɛ ƈཞơơʂ
Victoria
25
596
User
Victoria
24
599
User
Victoria
24
600
User
Queensland
24
601
User
Queensland
24
602
Zaid Karim
New South Wales
24
603
User
Victoria
24
604
User
Victoria
24
605
James Lavis
Queensland
24
606
User
Victoria
24
607
User
Victoria
24
610
Spencer GREENBERG
Victoria
24
612
Jorjia
New South Wales
24
613
OMG
Victoria
24
615
User
Victoria
24
617
im the best and humble
Queensland
24
619
User
Victoria
24
622
User
Queensland
24
623
Joelle Thrush
Queensland
24
624
User
Victoria
24
627
User
Queensland
24
629
SquareVoyage 753
Queensland
24
631
Athiya Azeem
New South Wales
23
632
User
Queensland
23
633
User
Queensland
23
634
User
Victoria
23
635
User
Queensland
23
636
User
Victoria
23
637
User
Victoria
23
639
Saksham
South Australia
23
640
User
Victoria
23
641
Maximilian PAPASPIROPOULOS
Victoria
23
644
User
Victoria
23
645
User
Victoria
23
649
Wang Ziyuan
New South Wales
23
652
User
Queensland
23
657
User
Victoria
23
658
User
Queensland
23
659
Shane C
New South Wales
23
661
Gidon Kessler
New South Wales
23
663
User
Victoria
23
664
User
Victoria
23
665
Jeffrey bob two
Victoria
23
666
User
Queensland
22
669
Asad Khan
South Australia
22
670
Arhum Moin
Victoria
22
672
User
Victoria
22
675
User
Victoria
22
676
Ilona DIGHT
Victoria
22
678
User
Victoria
22
679
User
Victoria
22
683
User
Victoria
22
684
User
Victoria
22
687
User
Tasmania
22
689
User
Victoria
22
690
User
Victoria
22
692
User
Victoria
22
693
User
Queensland
22
696
Will Udovenya
Victoria
22
699
User
Queensland
22
700
User
Queensland
22
701
User
Queensland
22
705
User
Victoria
21
706
User
Victoria
21
708
Waiting Foramate
Queensland
21
709
Yash
Victoria
21
710
Nathan Trinh
New South Wales
21
711
fin bob
New South Wales
21
712
User
Victoria
21
713
User
Victoria
21
714
User
Victoria
21
715
DaCsgoGod
Victoria
21
716
User
Victoria
21
717
Tharani Murugananthan
Queensland
21
718
Harrison Jin
New South Wales
21
719
User
Victoria
21
720
Tomthy
New South Wales
21
722
Amal
Queensland
21
723
User
Victoria
21
724
User
Victoria
21
726
User
Victoria
21
728
User
Victoria
21
729
User
Victoria
21
732
GARRY Giltendez
Queensland
21
733
User
Victoria
21
734
User
Queensland
21
736
User
Victoria
21
737
User
Victoria
21
740
User
Victoria
21
741
User
Queensland
21
743
User
Queensland
21
744
Lily SIMPSON
Victoria
21
745
chin is thot
Tasmania
21
748
User
Victoria
20
750
User
Victoria
20
754
User
Victoria
20
757
User
Victoria
20
758
User
Victoria
20
759
Rita KIM
Victoria
20
761
Zee Gee
New South Wales
20
764
Sahil JOSHI
Victoria
20
765
User
Queensland
20
769
User
Victoria
20
770
User
Victoria
20
771
User
Victoria
20
773
David Houghton
New South Wales
20
776
User
Queensland
20
777
User
Queensland
20
778
Patrick
Queensland
20
779
User
Queensland
20
780
User
Tasmania
20
783
User
Victoria
20
784
Lazy Cow
Victoria
20
785
User
Victoria
20
786
Kelvin Meng
New South Wales
20
788
User
Queensland
20
789
Lavra
Victoria
19
790
User
Queensland
19
792
Lachlan Taylor
New South Wales
19
793
User
Victoria
19
795
Polly
Queensland
19
796
Jake Kerr
New South Wales
19
799
User
Queensland
19
800
Isabelle LEONDAS
Victoria
19
801
John Kam
Victoria
19
802
User
Victoria
19
804
User
Victoria
19
805
User
Victoria
19
807
User
Queensland
19
808
User
Queensland
19
812
User
Queensland
19
814
User
Queensland
19
815
User
Queensland
19
816
User
Queensland
19
819
User
Queensland
19
821
Maria VAKOUSSEVITCH
Victoria
19
823
Brendan D
Victoria
19
824
Alan Perry
Victoria
19
828
User
Queensland
19
829
Juuso
New South Wales
19
831
User
Queensland
19
835
User
Victoria
19
837
User
Victoria
19
838
User
Victoria
19
839
User
Victoria
19
840
User
Victoria
19
842
User
Victoria
19
843
Oliver Hoang
New South Wales
19
844
User
Victoria
19
846
User
Victoria
19
847
Chloe SALIARIS
Victoria
19
848
Isaac HALL
Victoria
19
850
Richard Huang
Victoria
18
853
User
Queensland
18
857
Semyon Furman
Victoria
18
858
User
Queensland
18
859
User
Victoria
18
860
User
Queensland
18
862
Grigory
Victoria
18
863
User
Tasmania
18
864
Amelie BRYANT
Victoria
18
865
Shakira ANDERSON
Victoria
18
866
Colleen McCasker
Queensland
18
869
User
Victoria
18
872
User
Victoria
18
879
User
Queensland
18
880
Renee Nysten
South Australia
18
881
sanaz mozhdehifard
New South Wales
18
883
User
Victoria
18
884
User
Victoria
18
885
Kenneth Huang
New South Wales
18
887
Tylor White
Western Australia
18
888
Crack Eggs Some More
Victoria
18
889
User
Victoria
18
891
Charlie MOTION
Victoria
18
892
User
Victoria
18
893
User
Queensland
18
894
User
Queensland
18
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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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