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TRANSCRIPT
Secrets of Mental Math
Arthur T. Benjamin Harvey Mudd College
Text
Insight Cruises October 30, 2015
Secrets of Mental Math
Arthur T. Benjamin Harvey Mudd College
Text
Multiplication and Squaring
Squaring numbers
132
10
16
3
160+ 9169
32492
48
501
2400+ 12401
552
50
60
5
3000+ 253025
982
96
1002
9600+ 49604
982
96
1002
9600+ 49604
Why does this method work?
Algebra: A2 = (A + d)(A - d) + d2
Example: 982 = (98 + 2)(98 - 2) + 22
= (100)(96) + 4
Proof by patternNumbers that add up to 20
9 + 11 = 20 8 + 12 = 20 7 + 13 = 20 6 + 14 = 20 5 + 15 = 20
10 + 10 = 20
How large can the product get?
Numbers that add up to 20
9 x 11 = 8 x 12 = 7 x 13 = 6 x 14 = 5 x 15 =
10 x 10 = 100
How large can the product get?
99 (down 1) 96 (down 4) 91 (down 9) 84 (down 16) 75 (down 25)
Numbers that add up to 20
9 x 11 = 8 x 12 = 7 x 13 = 6 x 14 = 5 x 15 =
10 x 10 = 100
How large can the product get?
99 (down 12) 96 (down 4) 91 (down 9) 84 (down 16) 75 (down 25)
Numbers that add up to 20
9 x 11 = 8 x 12 = 7 x 13 = 6 x 14 = 5 x 15 =
10 x 10 = 100
How large can the product get?
99 (down 12) 96 (down 22) 91 (down 9) 84 (down 16) 75 (down 25)
Numbers that add up to 20
9 x 11 = 8 x 12 = 7 x 13 = 6 x 14 = 5 x 15 =
10 x 10 = 100
How large can the product get?
99 (down 12) 96 (down 22) 91 (down 32) 84 (down 16) 75 (down 25)
Numbers that add up to 20
9 x 11 = 8 x 12 = 7 x 13 = 6 x 14 = 5 x 15 =
10 x 10 = 100
How large can the product get?
99 (down 12) 96 (down 22) 91 (down 32) 84 (down 42) 75 (down 25)
Numbers that add up to 20
9 x 11 = 8 x 12 = 7 x 13 = 6 x 14 = 5 x 15 =
10 x 10 = 100
How large can the product get?
99 (down 12) 96 (down 22) 91 (down 32) 84 (down 42) 75 (down 52)
Numbers that add up to 26
12 x 14 = 11 x 15 = 10 x 16 = 9 x 17 = 8 x 18 =
13 x 13 = 169
How large can the product get?
168 (down 1) 165 (down 4) 160 (down 9) 153 (down 16) 144 (down 25)
12 x 14 = 11 x 15 = 10 x 16 =
13 x 13 = 169 168 (down 12) 165 (down 22) 160 (down 32)
13 x 13 = 10 x 16 + 32
1082
100
116
8
11600+ 6411664
The most important idea for doing mental math:
Left to Right Text
Addition Example
Text
314 + 159
414 + 59
464 + 9
473
Subtraction Example
Text
314 – 159
314 – 200
Oversubtract
114 + 41
155
Add back complement
114
Subtraction Example
Text
1234 – 567
1234 – 600
Oversubtract
634 + 33
667
Add back complement
634
Multiplication and DivisionOrders of magnitude
Text
(8 digits) x (5 digits) = ?? or ?? digits
(8 digits) ÷ (5 digits) = ?? or ?? digits
Multiplication and DivisionOrders of magnitude
Text
(8 digits) x (5 digits) = 12 or 13 digits
(8 digits) ÷ (5 digits) = ?? or ?? digits
Multiplication and DivisionOrders of magnitude
Text
(8 digits) x (5 digits) = 12 or 13 digits
(8 digits) ÷ (5 digits) = 3 or 4 digits
Multiplication and DivisionOrders of magnitude
Text
(m digits) x (n digits) = m+n or m+n -1 digits
(m digits) ÷ (n digits) = m-n or m-n +1 digits
Multiplication and DivisionOrders of magnitude
Text
(m digits) x (n digits) = m+n or m+n -1 digits
Which one? Multiply leading digits
If product ≥ 10, then m+n digits If product ≤ 4, then m+n -1 digits
If 5 ≤ product ≤ 9, then look more closely
Text
(m digits) x (n digits) = m+n or m+n -1 digits
Which one? Multiply leading digits
If product ≥ 10, then m+n digits If product ≤ 4, then m+n -1 digits
If 5 ≤ product ≤ 9, then look more closely
What’s more probable? m+n or m+n-1?
If digits are chosen at random (uniformly) then m+n digits is much more probable.
Mathematical Aside
What’s more probable? m+n or m+n-1?
If digits are chosen at random (uniformly) then m+n digits is much more probable.
Aside
But real data isn’t always uniformThink of your home address
How many have leading digit 1, 2, or 3?
How many have leading digit 7, 8, or 9?
Many data sets (populations, addresses, stock prices, lengths of rivers, etc.) follow
Benford’s Law
P(Leading digit = d) = log(d+1) – log(d)
Prob
abilit
y
0
10
20
30
40
Leading Digit1 2 3 4 5 6 7 8 9
Benford’s Law
P(Leading digit = d) = log(d+1) – log(d)
1 2 3 4 5 6 7 8 9
30.1 17.6 12.5 9.7 7.9 6.7 5.8 5.1 4.6
p1 + p2 + p3 = 60.2%; p7 + p8 + p9 = 15.5%
Text
(m digits) x (n digits) = m+n or m+n -1 digits
Which one? Multiply leading digits
If product ≥ 10, then m+n digits If product ≤ 4, then m+n -1 digits
If 5 ≤ product ≤ 9, then look more closely
What’s more probable? m+n or m+n-1?
If digits are chosen by Benford’s Law, then the probability of m+n digits is ??????????
Mathematical Aside
Text
(m digits) x (n digits) = m+n or m+n -1 digits
Which one? Multiply leading digits
If product ≥ 10, then m+n digits If product ≤ 4, then m+n -1 digits
If 5 ≤ product ≤ 9, then look more closely
What’s more probable? m+n or m+n-1?
If digits are chosen by Benford’s Law, then the probability of m+n digits is exactly 1/2.
Mathematical Aside
Text
Fundamentals of multiplication
42 x 7
7 x 40 = 2807 x 2 = +14
294
78 x 8
8 x 70 = 5608 x 8 = +64
624
2-by-1 multiplication
Text
Fundamentals of multiplication
624 x 6
6 x 600 = 36006 x 20 = +120
3720
3-by-1 multiplication
6 x 4 = + 243744
2-by-2 multiplication
✦ Addition Method ✦ Subtraction Method ✦ Factoring Method ✦ Close Together Method
Addition Method
47 x 31
30 x 47 = 1 x 47 = + 47
1457
31 = 30 + 1
1410
Subtraction Method
47 x 78
80 x 47 = 3760 -2 x 47 = – 94
3666
78 = 80 - 2
Subtraction Method
47 x 78
50 x 78 = 3900 -3 x 78 = –234
3666
47 = 50 - 3
Factoring Method
48 x 78
8 x 78 = 624 x 6
3744
48 = 8 x 6
Close Together Method
Multiplying numbers near 100
107 x 111
(7) (11)
118 77
Multiplying numbers near 100
103 x 106
(3) (6)
109 18Cool! Why?
Algebra!
107 x 111
(7) (11)
118 77(z+a)(z+b) = z2 + za + zb + ab
= z(z + a + b) + abExample: z = 100, a = 7, b =11 (107)(100+11) = 100(107 + 11) + (7)(11)107 x 111 = 100 x 118 + 77
z + a = 107
Multiplying numbers near 100
96 x 97
(-4) (-3)
93 12
Multiplying numbers near 100
107 x 97
(7) (-3)
104 -21
Multiplying numbers near 100
107 x 97
(7) (-3)
104 -21
Multiplying numbers near 100
107 x 97
(7) (-3)
104-2100
103 79
Multiplying numbers near 40
43 x 48
Addition Method
43 x 48
40 x 48 = 3 x 48 = +144
2064
43 = 40 + 3
1920
Subtraction Method
43 x 48
50 x 43 = 2150 -2 x 43 = – 86
2064
48 = 50 - 2
Factoring Method
43 x 48
43 x 6 = 258 x 8
2064
48 = 6 X 8
Close Together Method
43 x 48
(3) (8)
40 x 51 = 2040 3 x 8 = + 24
2064
Arthur Benjamin Harvey Mudd College [email protected]
Mental Math (2 DVDs)
Games & Puzzles (3 DVDs)
Joy of Math (4 DVDs)
Discrete Math (4 DVDs)
Online price 200
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Total: 930
Mental Math (2 DVDs)
Games & Puzzles (3 DVDs)
Joy of Math (4 DVDs)
Discrete Math (4 DVDs)
My price 40
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3 items: $10 off All 4: $190All forms of payment accepted
My college
Harvey Mudd College
3142
300
328
14
98400+ 19698596
3-Digit Square
7532
706
80047
3-Digit Square
7532
706
80047
564800+ 2209
567009
3-Digit Square Calendar Calculations
Sunday
Moon-day
Saturn-day
Mardi/Martes
Jeudi/Jueves
Mercredi/Miércoles
Vendredi/Viernes
Sunday
Moon-day
Saturn-dayWoden
Thor
TiwFreya
Sunday
Moon-day
Saturn-dayWoden’s day
Thor’s day
Tiw’s dayFreya’s day
Calendar Calculations2015
4M T W Th F Sat Sun1 2 3 4 5 6 7or 0
Jan Apr Jul Oct
Feb May Aug Nov
Mar Jun Sep Dec
Calendar Calculations2015
4M T W Th F Sat Sun1 2 3 4 5 6 7or 0
Jan 6* Apr 5 Jul 5 Oct 6
Feb 2* May 0 Aug 1 Nov 2
Mar 2 Jun 3 Sep 4 Dec 4
* For leap years, Jan = 5 and Feb = 1
Mnemonics for Month CodesJan = 6: W-I-N-T-E-R
Feb = 2: 2nd month
Mar = 2: March 2 the beat
Apr = 5: F-O-O-L-S
May = 0: Hold the May0
June = 3: B-U-G
July = 5: 5erworks!
Aug = 1: begins with A
Sep = 4: F-A-L-L
Oct = 6: T-R-I-C-K-S
Nov = 2: 2rkey!
Dec = 4: L-A-S-T
20154
M T W Th F Sat Sun1 2 3 4 5 6 7or 0
Jan 6* Apr 5 Jul 5 Oct 6
Feb 2* May 0 Aug 1 Nov 2
Mar 2 Jun 3 Sep 4 Dec 4
* For leap years, Jan = 5 and Feb = 1
October 30, 20156 + 30+ 4 = 40– 35 = 5 (mod 7)
= Friday
subtract 7, 14, 21, 28, 35…
20154
M T W Th F Sat Sun1 2 3 4 5 6 7or 0
Jan 6* Apr 5 Jul 5 Oct 6
Feb 2* May 0 Aug 1 Nov 2
Mar 2 Jun 3 Sep 4 Dec 4
* For leap years, Jan = 5 and Feb = 1
December 25, 20154 + 25+ 4 = 33– 28 = 5 (mod 7)
= Friday
subtract 7, 14, 21, 28,…
20154
Jan 6* Apr 5 Jul 5 Oct 6
Feb 2* May 0 Aug 1 Nov 2
Mar 2 Jun 3 Sep 4 Dec 4
* For leap years, Jan = 5 and Feb = 1
What about next year?
2016*6
20177 ≡ 0
20181
20192
2020*4
20154
Deriving the year code
2016*6
20177 ≡ 0
20181
20192
2020*4
Given: 2000* has year code 0. In 2015 the calendar has shifted 15 times, including 3 shifts for leap years (2004*, 2008*, 2012*) so the year code should be:
15 + 3 ≡ 18 – 14 = 4.
Other useful or interesting facts1600*
01700
51800
31900
12000*
0Gregorian calendar cycles every 400 years.
Between 1901 and 2099, it cycles every 28 years.
January 1, 2001 was a Monday.
Shakespeare and Cervantes both died on April 23, 1616 — yet they died 10 days apart!
Deriving month codes
Jan 6* Apr 5 Jul 5 Oct 6
Feb 2* May 0 Aug 1 Nov 2
Mar 2 Jun 3 Sep 4 Dec 4
* For leap years, Jan = 5 and Feb = 1
In a non-leap year, February has 28 days, so March will have the same month code.
Since March has 31 = 28 + 3 days, April’s month code will be 3 days later.
Large calculations require mnemonics
Phonetic code (major system)1 = t or d
2 = n
3 = m
4 = r
5 = L
6 = ch, sh, or j
7 = k or (hard) g
8 = f or v
9 = p or b
0 = s or z
Digits of Pi
π ≈ 3.14159265358979323846264…m t r
motormetermeteormetromatterMudder
Digits of Pi
3.1415 926 5358 979 32 384 6264…m t r
My turtle
t l pnch l mlv pkp mn mvr jnjr
Pancho will, my love,pickup my new mover Ginger!
π
Digits of Pi
π ≈ 3.1415 9265358 979 32 3846264m t r t l pnj l mlv pkp mn mvr jnjr
My movie monkey plays in a favorite bucket! 3 38 327 950 2 8841 971m mv mnk pls n fvrt bkt
69 3 99 375 1 05820 97494…shp m pp mkl t slvns bkrbr
Ship my puppy Michael to Sullivan’s back rubber!
100 Digits of Piπ ≈ 3.1415 9265358 979 32 3846264
3 38 327 950 2 8841 971 69 3 99 375 1 05820 97494
45 92 307 81 640 62 8 620
8 99 86 28 0 3482 5 3421 1 70 67…A really open music video cheers Jenny F. Jones.
Have a baby fish-knife, so Marvin will marinate the goose-chick!
Large calculations require mnemonics 23582
2000
2716
358
5,432,000
4-Digit Square
“5 million…”
r mn“Roman”
3582
316
40042
126,400
3-Digit Square
+ 1,764128,164
tch r“teacher”
23582
2000
2716
358
5,432,000
4-Digit Square
r mn“Roman”
+ 128,560,
“teacher”164
5-Digit Square1 = t or d
2 = n
3 = m
4 = r
5 = L
6 = ch, sh, or j
7 = k or (hard) g
8 = f or v
9 = p or b
0 = s or z
Thank you!