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This document is a synopsis of parts of the book
Mathemagics: How to Look Like a Genius Without Really Trying
by Benjamin, Arthur
Publisher: Lowell House Date Published: 1993 ISBN-13: 9780929923543 ISBN: 0929923545
1
A Little Give and Take:
Mental Addition and Subtraction
LEFT-TO-RIGHT ADDITION
2-DIGIT ADDITION
Add the left-most numbers
Attach the top right number to the end
Add the bottom ones digit.
67
 28
62
87
87  8  95
3-DIGIT ADDITION
Add the left-most digits
Attach the top right two digits
Add the tens digit
Add the bottom ones digit
538
 327
538
838
838  20  858
858  7  865
LEFT-RIGHT SUBTRACTION
2-DIGIT SUBTRACTION
Subtract the bottom tens digit from the top tens digit
Affix the top ones digit
Subtract the bottom ones digit
86
 25
82
66
66  5  61
3-DIGIT SUBTRACTION
Subtract the bottom hundreds digit from the top hundreds digit
Affix the right two digits
Subtract the tens digit
Subtract the ones digit
958
 417
94
558
558  10  7  541
Or:
Round up the bottom number
Subtract the bottom hundreds digit from the top hundreds digit
Affix the right two digits
Add the complement of the bottom right two digits
Note: to get the complements of a 2-digit number:
Add to the left digit to get 9
Add to the right digit to get 10
725
 468
75
225
225  32  257
2
Products of a Misspent Youth:
Basic Multiplication
2-BY-1 MULTIPLICATION PROBLEMS
Multiply the tens digit
Affix a zero
Multiply the ones digit
Add to the result
42
x7
7 x 4  28
280
7 x 2  14
280  14  294
3-BY-1 MULTIPLICATION PROBLEMS
Multiply the hundreds digit
Affix two zeroes
Multiply the tens digit
Affix one zero
Add to the previous result
Multiply the ones digit
Add to the previous result
326
x7
2100
140
2100  140  2240
42
2240  42  2282
BE THERE OR B2: SQUARING 2-DIGIT NUMBERS
Round the number up or down to the nearest hundred
Round the number in the opposite direction by the same amount
Multiply the two results
Square the round up value
Add to the previous result
132
16 x10  160
32  9
160  9  169
3
New and Improved Products:
Intermediate Multiplication
2-BY2 MULTIPLICATION PROBLEMS
The Addition Method
Multiply the tens digit by the tens digit
Attach two zeroes
Multiply the ones digit by the tens digit
Attach one zero
Add the two results
Multiply the tens digit by the ones digits
Attach a zero
Multiply the ones digits
Add the two results
Add the sum to the first sum
46
x 42
4 x 4  16
1600
4 x6  24
240
1600  240  1840
2 x4  8
80
2 x6  12
80  12  92
1840  92  1932
(Multiplying by 11)
If you have a 2-digit number whose digits add up to 9 or less:
Add the two digits together and insert the total between the original two digits
42
x 11
462
If you have a 2-digit number whose digits add a number that is greater than 9:
Increase the tens digit by one
Insert the last digit of the sum between the two numbers
76
x 11
836
The Subtraction Method
Round up the top number to the nearest hundred
Multiply the tens digits by the ones digit
Attach two zeroes
Multiply the tens digit by the ones digit
Attach a zero
Add the two results
Multiply the bottom number by the number you used to round up the top number
Subtract from the previous result
59
x 17
60
1x 6  6
600
7 x 6  42
420
600  420  1020
17 x1  17
1020  17  1003
When the subtraction component of a multiplication problem requires you to borrow a
number, use the complement.
The Factoring Method
3-DIGIT SQUARES
Use the same procedure as for 2-digit squares but round up or down to the nearest
multiple of 100.
4
Divide and Conquer:
Mental Division
1-DIGIT DIVISION
THE RULE OF THUMB
2-DIGIT DIVISION
SIMPLIFYING DIVISION PROBLEMS
Divide both numbers by a common number
MATCHING WITS WITH A CALCULATOR:
LEARNING DECIMALIZATION
1
2
 .50
1
3
1
 .25
4
 .571428
 .333
2
3
 .666
2 1
3
  .50
 .75
4
4 2
4
1
2
3
4
 .20
 .40
 .60
 .80
5
5
5
5
1
2 1
3 1
 .1666
  .333
  .50
6
6 3
6 2
4 2
5
  .666
 .8333
6 3
6
1
2 1
 .125
  .25
8
8 4
3
1
4 1
 .375 (3   3  .125  .375)
  .50
8
8
8 2
5
1
6 3
 .625 (5   5  .125  .625)
  .75
8
8
8 4
7
1
 .875 (7   7  .125  .875)
8
8
1
2
3
4
 .111
 .222
 .333
 .444
9
9
9
9
5
6
7
8
 .555
 .666
 .777
 .888
9
9
9
9
1
2
3
 .10
 .20
 .30
10
10
10
4
5
6
 .40
 .50
 .60
10
10
10
7
8
9
 .70
 .80
 .90
10
10
10
1
2
 .0909
 .1818 (2  .0909)
11
11
3
4
5
 .2727 (3  .0909)
 .3636
 .4545
11
11
11
6
7
8
 .5454
 .6363
 .7272
11
11
11
9
10
 .8181
 .9090
11
11
1
2
3
 .142857
 .285714
 .428571
7
7
7
7
5
7
 .714285
6
7
 .857142
Divide the denominator into the numerator. The first digit of the quotient is the first
number of the sequence.
TESTING FOR DIVISIBILITY
By 2: The last digit is even (0, 2, 4, 6, 8)
By 4: The last two digits are divisible by 4
By 8: The last three digits are divisible by 8 (If not divisible by 4, not divisible by 8)
By 3: The sum of its digits is divisible by 3
By 9: The sum of its digits is a multiple of 9
By 6: The number is divisible by 2 and 3
By 5: The number ends in 5 or 0
By 11: Alternately subtract and add its digits. If the result ends in 0 or a multiple of 11
By 7: Subtract or add a multiple of 7 to the number to get a zero on the end. Throw the
zero away and continue the steps until you reach a multiple of 7
Any odd number not ending in 5: Same as for 7
5
THE ART OF “GUESSTIMATION”
Addition Guesstimation
Guesstimating at the Supermarket
Subtraction Guesstimation
Division Guesstimation
Multiplication Guesstimation
SUARE ROOT ESTIMATION:
DIVIDE AND AVERAGE
6
Math for the Board:
Pencil-And-Paper Mathematics
7
A Memorable Chapter
THE PHONETIC CODE
1 is the “t” or “d” sound.
2 is the “n” sound.
3 is the “m” sound.
4 is the “r” sound.
5 is the “l” sound.
6 is the “j,” “ch,” ”or “sh” sound.
7 is the “k” or hard “g” sound.
8 is the “f” or “v” sound.
9 is the “p” or “b” sound.
0 is the “z” or “s” sound.
8
The Tough Stuff Made Easy:
Advanced Multiplication
9
Mathematical Magic
QUICK CUBE ROOTS
13 
1
2 
8
3
3  27
3
43  64
53  125
63  216
7 3  343
83  512
93  729
103  1000
The thousands number is between n3 and m3 and the first digit of the cube root is n.
To determine the last digit of the cube root, find a cube where the last digit is equals the
last digit of the number. The base is the last digit of the cube root.
314,432
314 lies between 6  216 and 7 3  343
3
The first digit of the cube root is 6
8  512
3
The last digit of the cube root is 8
68
A DAY FOR ANY DATE
Divide the last two digits of the year by 4.
Ignore the remainder.
Add the quotient to the last two digits of the year.
Add the number corresponding to the month from Figure 1.
Add the day of the month.
Divide by 7.
The remainder corresponds to the day of the week in Figure 2.
Figure 1
Month
Add
January
1
February
4
March
4
April
0
May
2
June
5
July
0
August
3
September
6
October
1
November
4
December
6
Leap Years
0
3
Figure 2
Remainder
Day
1
Sunday
2
Monday
3
Tuesday
4
Wednesday
5
Thursday
6
Friday
Mnemonic
First (1)
cold
lion/lamb
fool
2 on a maypole
bride
boom
hot
school
broom stick (1)
thankful 4
season
0
Saturday
December 20, 1935
35  4  8
35  8  43
43  6  49
49  20  69
69  7  9 R 6
Friday
June 19, 1943
43  4  10
43  10  53
53  5  58
58  19  77
77  7  11R0
Saturday
For 1800’s, add 2 to the remainder. (5+2=0; 6+1=0: 6+2=1)
June 4, 1899
99  4  24
99  24  123
123  5  128
128  4  132
132  7  18 R 6
6 2 1
Sunday
For 200’s, subtract 1 from the remainder. (0-1=6)
June 3, 2005
05  4  1
05  1  6
6  5  11
11  3  14
14  7  2 R0
0 1  6
Friday