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Secrets of Mental Math

Arthur T. Benjamin Harvey Mudd College

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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

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(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

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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 benjamin@hmc.edu

Mental Math (2 DVDs)

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Joy of Math (4 DVDs)

Discrete Math (4 DVDs)

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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

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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!