Lightning Calculation

Transcription

1 Lightning Calculation Fast mental techniques for arithmetic operations and elementary functions A 0 Calendar

2 Notes

3 Introduction The art of lightning calculation reveals the remarkable potential of the mind. Individuals with preternatural abilities to calculate arithmetic results without pen, paper or other instruments, and to do so at astonishing speed, are the stuff of mathematical and psychological lore. These lightning calculators were sometimes of limited mental ability, sometimes illiterate but of average intelligence, and often exceptionally bright, this despite the popular notion of the idiot savant. The techniques used by these people are not generally well known. The history of lightning calculators is interesting from a human standpoint, but it s perhaps more intriguing because the methods they learned or developed are uniquely suited for fast mental calculation. These methods are different from the ones taught in school for pencil-and-paper solution, and therefore most people are quite surprised when they find out that other algorithms such as these exist. Arthur F. Griffith (880-9) The types of calculations performed by lightning calculators were historically quite limited, notable mainly for the size of the numbers and the speed at which they were manipulated. But remember that the questioner had to verify every calculation by hand, making higher powers and roots (particularly inexact roots) much less feasible. The dawn of calculators and computers propelled some of these tasks into hitherto uncharted territories such as th or rd roots, deep roots of inexact powers, and so forth, much of it supported by more sophisticated mathematics. In this calendar we will explore the methods of calculation used in the past, many of them not commonly known, as well as other techniques that are relatively new. Each month in this calendar is devoted to a different type of mental calculation. Exercises are provided within the dates displayed for each month, as described in the Legend for that month. The answers are contained within the boxes connected with the dates involved, but small enough that they are not visible from a distance. Every day starting in February also poses a day-date calculation, a very popular pastime of current and historical mental calculators that is discussed below. The answers here are coded in a simple manner to avoid spoiling your fun! The calendar format naturally encourages daily exercise, but I encourage you to skip ahead to other days and months. After all, the topic of a particular month may not appeal to you, and in any event if you work ahead you will be able to breeze through the exercises when you get there. My philosophy on mental calculation: It is important to realize that lightning calculators were highly individual in how they approached these tasks, and most calculators have such a vast knowledge of number facts that answers are often obtained immediately from memory or following only slight adjustment. As one example, Wim Klein learned through experience the multiplication table through 00x00 and used it to great advantage doing cross-multiplication in -digit by -digit groups. He also knew squares of integers up to 000, cubes up to 00, and roughly all primes below 0,000. He also knew logarithms to digits for integers up to 0. Sometimes calculators used a mnemonic scheme, often of their own design, to help remember these number facts. Use whatever knowledge you can muster!. Numbers are not just collections of digits; they have properties that can be exploited. It helps greatly to work from left-to-right.. Try to use convenient numbers near the actual ones and correct the result at the end.. Have confidence! Don t limit yourself by saying it s too hard. Don t worry about speed at first just completing a calculation is often a real achievement, and speed comes naturally with practice. Have a happy 0! Ron Doerfler (doerfpub@myreckonings.com)

4 Using the Calendar There are 98 mental calculation exercises in this calendar. Thought you would want to know that. This calendar is packed with challenging mental calculations! We will explore a different field of mental calculation each month.the upper half of each calendar page describes methods that can be used to solve problems of that month, and the lower half offers opportunities to test yourself and practice these techniques daily using the actual dates shown in the calendar. Of course, you should feel free to try all the dates at once, or bounce around between months, or do whatever you want. After all, this is all for fun! In nearly all cases the answers are provided right in the calendar itself. That s where the legend comes in. Every month includes a legend that describes how the dates are used in the calculations. The calculations in some months use - or -digit numbers, and in that case the legend describes how to form those numbers from the -digit dates. Two-Date Single- Date The boxes contain products of the connected dates. Blue End: Double the date Red End: Triple the date Example for the box at the bottom:. Left date: Blue end x 8 =. Right date: Blue end x =. x = 8 (answer in box) 8^ = Add : 8 + = 9 9^ = 0 Add : 8 + = 79 79^ = Day of the Week Code: Add to the date, sum the digits, check day-code table: = 7, + 7 = 9 Jan 8, 9, was a Sunday 8^ 9^ 79^ 8 Jan 8 9 Legend ^ ^ 7^ Nov = = Nov, 799, was a Tuesday Mon Tue Wed Thu Fri Sat 9 Sun 89 9 Feb 99 The answers for calculations that involve pairs of dates appear in boxes that connect the two dates, or connect a date on the edge of the calendar to a date printed next to it. These answers are printed in very small font to avoid revealing them too readily. Since it would be too easy to accidentally detect the answer for the day of the week in the calendar calculations (on months after January ), a simple code based on the date is used. This is consistent throughout the year but is described in the legend for every month. 0 8 May 8 0 So take a minute to look over the legend for a typical month shown above along with the corresponding date blocks on the left. The colored end of each box is a key to the operation to perform first on the date at that end of the box. Here we are told to double the date if a tip is blue and triple it if it is red, then multiply it by the result of the operation on the date at the other end of the box. Sometimes the colored ends add values to the date rather than double or triple them, so please look at the legend for before starting a month. Here we also square the date, (date + ), and (date + ), and those answers are shown right above the date. It s all much easier than you might think at first.

5 Day-Date Calendar Calculations Finding the Day of the Week is a very common task. Zeller s Congruence: A straightforward calculation. Mechanical Jan / Jul Feb 8/9 Aug 8 Mar 7 Sep Apr Oct 0 May 9 Nov 7 Jun Dec Pericles Diamandi (88-?) visualized a table with a rotating grille for day-date calculations. Doomsday Method: Find a nearby doomsday and adjust it. Days are numbered 0- for Sunday-Saturday. Each century has a Doomsday for its first year: 700,00,00, Sunday (0) 800,00,00, Friday () 900,00,700, Wednesday () 000,00,800, Tuesday () For y = last two digits of the year you are calculating, where [ ] means to round down to the integer and R is the remainder of [y/]. Subtract multiples of 7 to get a number less than 7 to find the Doomsday for your year. Doomsday is Monday for 0. Every day after January contains a date between the adoption of the Gregorian calendar in Britain and its colonies (7) and the year 099. What day of the week does that date fall on? The answer is found from a simple code described in the Legend for each month. Shown here are two common methods used by lightning calculators for day-date calculations. Then remember the table of dates that fall on the Doomsday and calculate the difference for the date you are calculating, or use this: For M = the number of the month, Doomsday is The Doomsday method was invented by John Conway in 98 - The last day of February and January (where Jan = Feb ) - M for an even month - M+ for an odd month with days - M- for an odd month with 0 days - Work from 9 to at the 7- for Sep, May 9, July and Nov 7 May 7, 889: Anchor = [89/] = 7 with remainder R= [/] = and forget the remainder 7+++ = 8, subtract to get, so the Doomsday for 889 is Thursday So May 9 is a Thursday, and May 7 is 8 days later, or a Friday. Tip: Every years the Doomsday repeats, so we can simplify: y 89 8 = d = day of week numbered 0- for Saturday-Friday n = day of the month m = month (-, where Jan/Feb are / of the previous year) y = last two digits of the year (or previous year if m = or ) C = first two digits of the year where [ ] means to round down to the integer and mod 7 indicates the remainder when divided by 7. To simplify the addition we can find this remainder for each term as we go. May 7, 889: d = (since 7 mod 7 = ) + (since x/0 mod 7 = ) + (since 89 mod 7 = ) + (since 89/ mod 7 = ) + (since 8/ mod 7 = ) (since x8 mod 7 = ) = mod 7 =, or a Friday You probably know the day of the week for many personal, family and historical events. Calculate from one of these if it is convenient! Some Historical Date Exercises in this Calendar Jul, 77 Feb, 789 Jul, 789 Dec, 80 Sep, 80 Nov 9, 8 Apr, 8 Jul, 87 Jan, 90 Dec 7, 90 Apr, 9 Nov 7, 97 Nov, 9 May, 97 Mar, 90 May, 97 Dec 7, 9 Jun, 9 May 8, 9 Aug, 9 Aug, 97 May, 98 May 9, 9 Feb, 9 Oct, 97 Aug 8, 9 Nov, 9 Apr, 98 Jul, 99 Apr, 98 Jun, 989 Nov 9, 989 Oct, 990 Feb, 00 Dec, 0 Apr, 09 Jan 9, 08 United States Independence George Washington elected Storming of the Bastille Napoleon declares himself emperor Mexico Cry of Independence Gettysburg Address Abraham Lincoln assassinated Canada Independence Australia Independence Wright brother's first powered flight Titanic sinks Bolshevik Revolution Howard Carter discovers King Tut's tomb Charles Lindbergh lands in Paris Gandhi begins Dandi Salt March Hindenburg disaster Pearl Harbor D-Day Victory in Europe Hiroshima India Independence Israel Independence Edmund Hillary / Tenzing Norgay reach Mt. Everest summit Elizabeth II becomes queen Sputnik launched March on Washington / ML King "I Have a Dream" speech John F. Kennedy assassinated Martin Luther King assassinated First man on the moon Chernobyl disaster Tiananmen Square massacre Fall of Berlin Wall Reunification of Germany Challenger space shuttle disaster Mayan calendar ends Asteroid Apophos near-miss of Earth Unix time ends

6 -Digit Multiplication Multiplication is the most common and useful type of lightning calculation. There are a variety of methods you can choose among to instantly find the product of any pair of two-digit numbers. In one performance, Maurice Dagbert (9-99) extracted a fifth root (answer: ) in seconds; a seventh root (answer: ) in seconds; a cube root (answer: 78,7) in minutes seconds; a fifth root (answer: 89) in minutes seconds; and raised 87 to its cube in seconds. Mechanical Methods Partial Products are the combinations of the individual digit multiplications. They are added from left to right to find the product: x 8 = 0x0 + 0x8 + x0 + x8 = = 8 The terms are added as they are calculated, so when 0x8 is calculated, it is added to 000 to get 0, then x0 is added to get 0, and finally x8 is added to yield 8. Fast, with only one running total to remember! Good Neighbor Method: Is one of the numbers near a very round number? Multiply by the round number instead and adjust for the difference at the end: 9x = 0x = 98 x = 0x + x = To find 0x here, we would multiply from left to right: 0x0 + 0x = 00. Then subtract 0 and add to subtract. In the second example, 0x = 0x0 + 0x = 0, and add x =. Is a number a multiple of 9 or? Use nearby multiple of 0 and subtract or add /0: x: Find 0x = 080, then = 88 x: Find 0x = 080, then = 87 (subtract 00, then 8 more!) x = 0x + x = x = 0x + x = 8 97x9 = 00x89 + x8 = 89 7x = 0x - x = Easily scaled to larger numbers, but the answer is found from right to left so it must be reversed to recite it. 8 8 Becoming Neighbors: Bringing two multipliers nearer can sometimes allow use of other methods. For example:. Subtract one number from a very round number (or add it to a very round number) to bring it closer to the other number: x7 = (00 ) = 00 x = 00 (0x + x). Divide or multiply one number by a low integer and add a correction: x 7 = xx + = (0x + x) + 8 x = 0x + x = 88 This is much easier to use than it might appear! Visualize anchoring one multiplier at the round number, and then literally stringing out the differences (plus or minus) of the original numbers from this anchor to find the other multiplier. If the differences have the same sign, the correction at the end will be added, and if one is positive and the other is negative the correction will be subtracted. x 8: : x8 = 8, or 8 with a carry of : x8 + x + =, or with a carry of : x + = Answer: 8. Break one number into two convenient parts: x7 = (0+7) = 00/ + x7 = TIPS The two most common techniques used by lightning calculators Legend connected dates. Blue End: Double the date 8 x = 0x x = The boxes contain products of the Red End: Triple the date 9 x 8 x 7 x 7 x 8 x 8 x Anchor Method: Anchor one multiplier at a nearby round number a, and then string out the differences (plus or minus) of the original numbers from this anchor to find the other multiplier. Then add the product of the differences. Algebraically this is represented as Cross Multiplication adds single-digit products that contribute to each digit of the result, including carries: Example for the rightmost box here:. Bottom date: Blue end x 9 = 8. Top date: Blue end x = more multiplication methods in February s Calendar---take a look!. 8 x = 9 (answer in box) 9

7 Sunday Monday Tuesday Wednesday Thursday Friday Saturday December 00 February 0 JANUARY

8 Squares and -Digit Multiplication Squares and general multiplication share a strange and productive relationship. Squares can be used to greatly ease general multiplication, while general multiplication can be used to greatly ease squaring. Midpoint Method: One of the most powerful tools in mental calculation is converting the multiplication of two different numbers into the square of the average minus the square of the distance to the average. This is the Midpoint Method, an algebraic identity: Very useful! where a is the average of the two numbers, (a+c) is one of the numbers, and (a c) is the other number. This is equivalent to the Anchor Method when the anchor is midway between the two multipliers. For example, Jedidiah Buxton (707 77) was an illiterate lightning calculator who once calculated the product of a farthing doubled 9 times., which expressed in pounds has thirty-nine digits. No Midpoint Method: We might have the case where there is no midpoint of the two multipliers here we can adjust one of the multipliers by one, do the calculation, and then provide a correction to account for the original adjustment, as for 8x = 8x + 8 = but in this particular case it may be easier to use the Anchor Method (see January) to get 8x = 0x x 8x = 0 x78 = x 8 = The vertical bar separates two-digit groups. If we end up with a -digit result in a grouping, its most significant digit would be added to the group to its left. A 0. in a group adds 0 to the group to its right. An anchor of 00 is very common, say, 8 = 00x8 +. With an anchor of 00, we can find the value 8 simply by doubling 8 and using the last two digits of the resulting 8 rather than by finding the difference between 00 and 8 and subtracting this again from 8. Binomial Expansion for Squares: We can express the number to be squared as the sum of two other numbers that are more easily squared: Very useful for larger numbers = (0+) = 0 + x0x + = 9 = (70 ) = 70 x70x + = 7 Special Neighbors: To find the square of a number near 0, add the difference from 0 to, multiply by 00, and add the difference squared. If the number is within 0 of 0, we can add the difference to and simply append the distance squared rather than adding it. In this notation, (+b) = ( + b/) ( + b ) (0+b) = ( + b) b (7+b) = ( + b + b/) ( + b ) 7 = ( + ) ( + ) = 79 = ( + ) = 70 8 = ( ) = = 78 = ( + +.) ( + ) = 0. = 08 Reverse Midpoint Method: To calculate a square, we can split it into the product of two numbers equidistant from the original number, and add the square of that distance (one scenario of the Anchor Method). For example, we can find = 0x70 + = = 0x + = 9 It is helpful to remember that the average squared will always be larger than the spread numbers multiplied, so when spreading a square to the product of two numbers you add the correction, and when collapsing two multipliers to a square you subtract the correction. These follow from the binomial expansions, such as (0+b) = b + b Neighbors of Squares: Since (a+) = a + a + (a+), we can find = = 9. Similarly, 9 = = 8. For other neighboring numbers we can find the square of the convenient number, then add or subtract the original number, the final number, and twice each number in between, so = x + = 0. A shortcut for squaring a number ending in is a = a (xa) where means to limit the middle value to one digit by merging any upper digit to the left, as in = = 9 or = = (+) = 7. Recognize the trick for squaring numbers ending in? Multiply the number left of the units digit by that number plus one, and then append, as in = x7 =. Two-Date Single- Date The boxes contain products of the connected dates. Blue End: Double the date Red End: Triple the date Example for the box at the bottom:. Left date: Blue end x 8 =. Right date: Blue end x =. x = 8 (answer in box) 8^ = Add : 8 + = 9 9^ = 0 Add : 8 + = 79 79^ = Day of the Week Code: Add to the date, sum the digits, check day-code table: = 7, + 7 = 9 Jan 8, 9, was a Sunday Cool Facts: For squares of two-digit numbers ending in 7, 8 or 9: (0a+7) = 00a(a+) + 0(a+) + 9 (0a+8) = 00a(a+) + 0(a+) + (0a+9) = 00a(a+) + 80(a+) + where the red digits make up the squares of the units digits. 8^ 9^ 79^ 8 Jan = = 87 = = 79 Legend ^ ^ 7^ Nov = = Nov, 799, was a Tuesday Mon Tue Wed Thu Fri Sat 9 Sun

9 Sunday Monday Tuesday Wednesday Thursday Friday Saturday FEBRUARY Jan 8 (0) Feb Dec (80) Apr July 7 90 () May Jul 0 () June June 9 () 7 Mar Oct 800 (0) Mar Dec 80 () Sept 9 0 () Aug 78 (0) Nov Nov 0 () Apr 0 98 () Jan 7 97 (0) Feb 8 () Feb Dec 789 (0) May Jan 97 January Aug 9 () 89 9 Feb Aug Nov 8 (0) March

10 Reciprocals and Division We can usually manipulate divisions to -digit or -digit divisors depending on the accuracy needed. Division is an important skill. Salo Finkelstein (89/7-?) was notable for his mental addition and number memorization abilities, and was one of the many lightning calculators who used cross multiplication to calculate products of large numbers. Simplify the Denominator: Try to reduce the denominator to an integer of one or two digits and then do short division.. Shift decimal point:.7 / 0.07 =.7 / 7 =.. Divide the numerator and denominator by low common factors: 0.0 / 7. =.0 / 7 = 0.0 / 8 = 0.0. Divide by low factors of the denominator: 0.0/8 = (0.0/) / 9 = 0.0 / 9 = 0.0 / = (/) / =.8 / = 0.97 Adjust the Denominator: Adjust the denominator to a round number and then adjust the numerator by the same percentage, or roughly so for an approximation: 7 / 9: Adjusting 9 up to 0 is a change of about /0, so we adjust 7 by : 9 / 0 =.9 / =.070 Actual Value:.07 For more accuracy, notice that 7 is twice 9 plus ~0%, so adjust 7 by.:.9 / = / : Adjusting down to 00 is a change of just less than %, so we adjust 9 down by 9 for a first estimate: 90 / 00 = 9.0 /. = 8. / 7 =.889 Since % of 00 = rather than, an error of /00 /0000, we should adjust 90 down by more to get a more accurate answer: 90 / 00 = 9.0 /. = 8.0 / 7 =.900 Actual Value:.90 Friendly Neighbor Exact: Divide by nearby round number and adjust the remainder in each step of Short division. If rounding up, add (adjustment x quotient digit), otherwise subtract. 7 / 78: Adjusting 78 up to 80 will add x quotient digit to each remainder : / 80 = Remainder = + x = / 80 = Remainder = + x = 7 77 / 80 = 8 Remainder = 77 + x8 = 9 But 9>80, so we change the quotient 8 to 9: 77 / 80 = 9 Remainder = - + x8 = / 80 = Remainder = 7 + x = 7 70 / 80 = 9 Remainder = 0 + x9 = 8, etc. Placing the decimal point, we have / 0: Adjusting 0 down to 00 will subtract x quotient digit from each remainder : 70 / 00 = Remainder = 0 x = 8 80 / 00 = Remainder = 80 x = 0 / 00 = Remainder = 0 x = 8 80 / 00 = Remainder = 80 x = 7 70 / 00 = 9 Remainder = 0 x9 = 8, etc. Placing the decimal point, we have / 9: Multiply top and bottom by 8 / 98 will provide single-digit divisor. Adjusting 98 up to 7000 will add x quotient digit to each remainder : 8 / 7000 = Remainder = + x = 0 00 / 7000 = Remainder = 00 + x = 0 / 7000 = Remainder = 0 + x = 0 / 7000 = Remainder = 0 + x = 0 / 7000 = 9 Remainder = 0 + x9 = 0, etc. Placing the decimal point, we have.9 Friendly Neighbor Approximation: For small b, /(+b) b, so Two-Date Single- Date The boxes contain quotients of the connected dates. Blue End: Add to date Red End: Add to date Example for the box at the bottom:. Left date: Blue end 7 + = 8. Right date: Blue end + =. 8 / =.87 (answer in box) /7 = Add : 7 + = 8 / 8 = Add : 7 + = 78 /78 = Day of the Week Code: Add to the date, sum the digits, check day-code table: =, + = Jan 7, 98, was a Thursday and 7 / 0 (7 / 00) ( /00) = 0.7 ( /00) = = 0. Actual Value: 0.9 / ( / 00) ( /00). ( 0.00) = =.8 Actual Value:. /7 /8 /78 7 Jan 7 98 Legend / / /7 Nov 798 () + = = 9 Nov, 798, was a Sunday Mon Tue Wed Thu Fri Sat 9 Sun

11 Sunday Monday Tuesday Wednesday Thursday Friday Saturday MARCH Sep Nov July Dec 7 87 () Feb 9 () Sep Oct 8 () Jan 8 97 (0) July May May 0 (0) June 9 0 () June 78 (7) Dec 7 () Oct 0 78 (7) Mar Apr Mar Aug (8) Aug Feb Mar 890 Mar Mar Jan 9 08 (0) Dec (00) (8) (0) Feb May Oct () Aug 0 89 (0) July February April

12 Square roots are often encountered in technical work, and the ability to quickly calculate them is easily acquired. Square Roots One-Digit Endings Is N a Perfect Square? Power. If the last two digits of N are one less than a multiple of, N cannot be a square.. Perfect squares end in 00, e, e,, d or e9 for d = odd digit and e = even digit.. A square ending in must end in, or.. A square ending in e or e9 with e divisible by must have an odd thousands digit: is not a square, may be a square 789 is not a square, 89 may be a square Square Root Averaging: A good method to approximate a non-integer square root. For an estimate r, a better estimate is the average of r and N/r: : Estimate 7 Better estimate is (7 + /7)/ = 00/ = 0/7 = 7.9 For an estimate of a fraction s/t, this average can be re-written as: : Estimate s/t = 0/7 Better estimate is (00 + x9)/ x0x7 = 999/700 = 7. Digit-by-Digit Extraction: Step : Move the decimal point in N left two places at a time until only one or two digits are left of the decimal point. Find the closest digit A whose square will be less than this new N. Find the remainder R0 = N-A using only the integer portion of N. Square root of : N., A = 7, R0 = 7 = Quartering the Difference: A more accurate approximation but more difficult. For an estimate r, a better estimate is the weighted average of r and d = N r: Estimate: 7 R0= Step : Find 0 x R0 / and add ½ the next digit of N if it exists. Divide by A to get the next digit B and remainder R. The current square root is now A.B B = (0x/ + /) / 7 = remainder : Estimate 7 Better estimate is 7 x (9 + ¾ x ) / (9 + ¼ x ) = 7 x 0./9. = 7 x 0/99 = 707/99 = 7. which is better than averaging once but poorer than averaging twice Estimate: 7. R= Step : Find 0 x R and add ½ the next digit of N if it exists. Subtract B/. If the result is negative, reduce B by and add A to R and try again. Otherwise, divide by A to get the next positive digit C and remainder R. The current square root is now A.BC C = (0x + / /) / 7 = 8 remainder The boxes contain square roots of the connected dates separated by a decimal point. Blue End: Double the date Two-Date F = (0x + 0/ x 8x) / 7 = 0 remainder Estimate: 7.80 Final Step: Move the decimal point of the answer to the right once for each time it was moved left in Step. Red End: Triple the date Example for the box at the bottom: 9/ 0/ 70/ 09. Left date: Blue end x = 0. Right date: Blue end x 09 = 8 Nov (0.8)/ =.98 (answer in box) (0) / =.8798 Estimate: 7.8 R=0 E = (0x0 + 0/ x 8/) / 7 is negative, so previous estimate 7.8 with R = 0+7 = 7 then E = (0x7 + 0/ x 8/) / 7 = remainder Estimate: 7.8 R= Legend Estimate: 7.8 R= Later Steps: Repeat for more digits. In each step, subtract multiplications of digits pairing them inward to the middle, then if there is a digit left over we subtract half its square. See the pattern: A.B subtract B/ A.BC subtract BxC A.BCD subtract (BxD + C/) A.BCDE subtract (BxE + CxD) A.BCDEF subtract (BxF + CxE + D/) If a negative value occurs, back up and reduce the previous digit by, add A to the previous R, and start again from there. D = (0x + 0/ x8) / 7 = remainder 0 John Wallis (-70) wrote that one night in 9 he mentally extracted the square root of x0^ to digits and two months later did the same for a -digit number to 7 digits. SingleDate Add : + = / = Add : + = 7 7/ = Day of the Week Code: Add to the date, sum the digits, check day-code table: + =, + = Jan, 9, was a Thursday / / 7/ = += Nov 9, 799, was a Saturday Mon Tue Wed Thu Jan 9 Fri Sat () 9 Sun

13 Sunday Monday APRIL M T W T Wednesday Thursday May 0 March 0 S Tuesday F S S M T W T F S Friday Saturday Feb 00 Sep 80 () Mar 77 Nov 9 Sep 08 May 97 Apr 7 07 June 8 9 Jan 9 78 (7) () () () (0) Mar 0 07 Sep 800 Oct 78 Aug 99 Nov 88 Feb 799 Mar 0 (0) () () (0) Apr 7 88 Dec 8 08 Oct 9 8 Apr 0 99 Feb 88 Oct 78 Apr 9 () Nov 90 Oct 79 Mar 89 Jan 7 7 Sep July 9 88 Nov 0 9 () (7) (0)

14 x-digit Multiplication Multiplication of larger numbers can be performed using the same methods we use for multiplying smaller numbers. Remember Partial Products? x = 0x00 + 0x0 + 0x + x00 + x0 + x = 808 Remember Cross Multiplication? 7x: : x =, or with a carry of 0 : 7x + x + 0 =, or with a carry of : 7x + x + = 9, or 9 with a carry of : 7x + = Answer: 9 The calculating prodigy Truman Henry Safford (8-90) grew to become the director of the Hopkins Observatory at Williams College. Remember the Good Neighbor Method? Multiply by a nearby round number and adjust for the difference: 9x = 0x = 998 x = 0x + x = 80 To find 0x here, we would multiply from left to right: 0x00 + 0x0 + 0x = 00 Then to subtract we can subtract 0 and add 8. Remember the Midpoint Method? This technique converts the multiplication of two different numbers into the square of the average minus the square of the distance to the average Very useful! where a is the average of the two numbers, (a+c) is one of the numbers, and (a c) is the other number. Of course, this applies to larger numbers. Here the midpoint of and 8 is at a distance of 7 from each number: x7 = x80 x = ( 7 )x0 9 = 90 Here we can find as 0x + using the Midpoint Method again, or or (x) = 8 by the Neighbors of Squares Method (see February). Then we subtract 7 by subtracting 0 and adding, attach a 0 to the end, and subtract 9 by subtracting 00 and adding. We are not yet squaring -digit numbers, just multiplying a -digit number by a -digit number. The August calendar covers x-digit multiplication, and there we will extend our previous rules for -digit squares to -digit squares. If you can t wait, do a time leap to August now! Remember the Anchor Method? Anchor one multiplier at the round number a, and then string out the differences (plus or minus) of the original numbers from this anchor to find the other multiplier. If the differences have the same sign, the correction at the end will be added, and if one is positive and the other is negative the correction will be subtracted. For a x-digit multiplication, the numbers are generally far apart, but the -digit number might be broken into a multiple of 0 plus an offset: x = x0 + x = (0x + x)x0 + x = = 77 87x98 = 87x x = (00x8 + x)x0 + 87x = / = 89 x = x0 + x = (0x 7x)x0 + x = (70 8)x0 + (x0 + x) = 8 (where we subtract 8 by adding 00 and subtracting ) Two-Date Single- Date The boxes contain products of the connected dates. Blue End: Double the date Red End: Triple the date Then attach the original last digit of the leftmost date to the start of the rightmost date and multiply. Example for the box at the bottom:. Left date: Blue end x =. Right date: Blue end x 0 =. Attach the last digit from the leftmost date to the front. x = 088 (answer in box) x = Add : + = x = 79 Add : + = 7 7 x 7 = 79 Day of the Week Code: Add to the date, sum the digits, check day-code table: + =, + = Jan, 98, was a Saturday x x 7 x 7 Jan 98 Legend + 0 = = Nov, 798, was a Tuesday 0 x 0 7 x 77 7 x 77 0 Nov 798 Mon Tue Wed Thu Fri Sat 9 Sun

15 Sunday Monday Tuesday Wednesday Thursday Friday Saturday MAY July 87 (00) Dec Jan Nov 907 () Feb May 07 (0) Apr 8 () June Apr 7 9 () May 788 () July 98 () Aug 880 (9) Jan Nov 8 80 (0) Feb Nov Sep 80 (0) Nov (0) May 7 0 () 79 Dec 79 0 Mar 80 () Dec 8 Oct May Oct 78 July Oct Dec 8 79 (7) (0) Mar 9 08 (0) May 0 8 (0) 0 8 Aug April June

16 Cubes and Cube Roots Cubes are often asked of lightning calculators, but non-integer cube roots are very rarely encountered. A.C. Aitken (89-97) excelled in noninteger root extraction. The Thirding method here was pioneered by Aitken. Square and Multiply: Square, then multiply again using Partial Products or Cross Multiplication: = so = x = = 90 or x : : x =, or with a carry of : x + x + = 0, or 0 with a carry of : x + x + =, or with a carry of : x + x + = 9, or 9 with a carry of 0 : x + 0 = Answer: 90 Weighted Averaging: A good method to approximate a non-integer cube root. For an estimate r, a better estimate is the weighted average of r and N/r: (9) / : Estimate Better estimate is (x + 9/)/ = (0 + 7/00)/ =.7/ =.9 For an estimate expressed as a fraction s/t, this average can be re-written as: Binomial Expansion for Cubes: As we did for squares, we can express the number to be cubed as the sum of two other numbers that are more easily cubed: Very useful for larger numbers = (0+) = x0xx = x = 70 + (00 0) = 90 9 = (70 ) = 70 x70xx9 = 000 x70x9 = 999 x70 + x70 = 809 Thirding the Difference: Our most accurate approximation but more difficult. For an estimate r, a better estimate is the weighted average of r and d = N r : (9) / : Estimate Better estimate is x [ + (/)(-)] / [ + (/)(-)] = x / = 0/ =.9899 Actual Value:.988 Legend (9) / : Estimate s/t = 9/0 Better estimate is (00 + 9x000)/ x0x0 = 00/700./7.0 =./[7(+0.0/7)] (./7)x( /00) =.988 Actual Value:.988 Two-Step Approximation: An improvement to one weighted average. For an initial estimate r, (9) / : Estimate add b = (9 ) / 7 = to get.9 subtract c = 0.08 / = to get.987 Actual Value:.988 Our answer of.987 is better than weighted averaging once Two-Date Single- Date The boxes contain cube roots of the connected dates separated by a decimal point. Blue End: Double the date Red End: Triple the date Example for the box at the bottom:. Left date: Blue end x = 0. Right date: Blue end x 09 = 8. (0.8) / =.8 (answer in box) ^ = 7 Add : + = ^ = 97 Add : + = 7 7^ = 897 Day of the Week Code: Add to the date, sum the digits, check day-code table: + =, + = Jan, 9, was a Thursday ^ ^ 7^ Jan 9 () 9^ 0^ 70^ 09 Nov (0) = + = Nov 9, 799, was a Saturday Mon Tue Wed Thu Fri Sat 9 Sun

17 Sunday Monday Tuesday Wednesday Thursday Friday Saturday JUNE Nov 87 (7) Dec 77 () May July Mar 79 () Mar Aug June May Aug Oct 8 97 (0) Aug 9 79 May Sep June 787 () Feb 0 87 () July 7 98 () 87 7 Apr 98 (7) June 988 () Jan 8 8 (0) May 9 99 Mar Dec July 8 Dec Aug June 9 9 (0) (0) May Aug 7 88 (0) June 8 99 (0) Apr July

18 Factoring and finding roots of perfect powers is a fun and challenging pursuit. Factoring and Integer Roots As a child, Zerah Colburn(8089) factored into x 7007 Integer Roots: For a -digit integer cube root, fifth root or seventh root, as in the exercises in this month s calendar: Seventh root of N =,998,98,9,79,7:. The one s digit is unique for all these roots. It is the same as in the power for a root of order of the form k+ like the fifth root. The third and seventh root differ only in the blue rows of the table here. For a seventh power ending in, the root is nn7. Is the year 0 prime? Difference/Sum of Powers: Is N one of these forms? N = (a b) has factors (a+b) and (a-b) N = (a b) has a factor (a-b) N = (a + b) has a factor (a+b). The hundreds digit is found from the ranges of the powers. Here 007 = 8x0 and 007 = 87x0 so the fifth root here would be in the 00s. The root is n7. 0 is not! 9 = 0 = x 8 97 = 0 has factor 7 00 = 0 + has factor. The tens digit is found by matching the -Remainders for cube or seventh roots, or the -Remainders for fifth or seventh roots, since these produce unique results. Sieve: Eliminate small prime factors of N, since 8% of random -digit numbers (such as this month s exercises) have prime factors <=.. Factor of if N is even; if divides last two digits; 8 if 8 divides last three digits 0 not even, does not divide, and 8 does not divide 0. Factor of if N ends in -Remainders (Remainders after division by ) Subtract even digits from odd digits in N, repeat until <, adding s if needed until >0. Check table for the matching result for the root. Then do the -Remainder on the root and deduce n: -Remainder on,998,98,9,79,7 = - 8, so -Remainder on n7 =, so n must equal and so the root = 7. One-Digit Endings 0 does not end in. Factors of or 9 if they divide the 9-Remainder (Add all digits, repeat until < 0) = is not a multiple of or 9. Factors of if divides the -Remainder (odd-place digits minus even-place digits, repeat until less than, add s if needed to get result >0) = - so add to get 9, which is not a multiple of Or use -Remainders: We can reduce N using Other Divisor Tests or simply divide the entire N by to get a remainder of. From the -test table, the -Remainder on n7 =, and by trial we find root = 7. Other Divisor Tests: Simplify very large numbers before testing some primes by mental division. Repeat until N is too small for these to be useful. Remove m = last digits of N (not useful for -digit N in this calendar unless m is small) Find (N-m) for 7,, 0 = -9 not divisible by 7,, Find (N+m) for = not divisible by 7 Find (N-m) for, 9 0 = -0 not divisible by Find (N+m) for, = 90 not divisible by, Remove k = last digits of N (useful for most -digit N in this calendar ) Find (N+k) for = not divisible by 9 Find (N+8k) for 7, = 08 not divisible by 7,7 Find (N+k) for = 9 not divisible by Find (N-k) for 7 0 = - not divisible by 7 Find (N-8k) for = - not divisible by 89 Find (N-9k) for 0 99 = -77 not divisible by Cube Root Fifth Root Seventh Root s Test Fifth Seventh Root Root Legend The boxes contain prime factors of the connected dates Blue End: Add to date Red End: Add to date x y z7 Example for the box at the bottom: Two-Date. Left date: Blue end + =. Right date: Blue end 09 + = 0. 0 = xxxxxx9 or ^xx9 (answer in box) If the letter P appears anywhere in the factor list the number is actually prime! 09 Nov (0) We don t need to test for primes above 0 = We ve tested all primes < 00 (the range for the -digit numbers in this calendar) except 9, 9,, 7, 7, 79, 8, and 97. The only one less than is 9, and 0/9 is not an integer, so. The year 0 is prime! Power s Test Power Cube Seventh Root Root No answers are provided for these (use the Calculator on your PC) Find x = integer cube root Find y = integer fifth root Find z = integer seventh root Day of the Week Code: Add to the date, sum the digits, check day-code table: + =, + = Jan, 9, was a Thursday x y z = += Nov 9, 799, was a Saturday Mon Tue Wed Thu Jan 9 Fri Sat () 9 Sun

19 Sunday Monday Tuesday Wednesday Thursday Friday Saturday JULY June Oct 990 (8) Sep (00) August July 77 (8) Oct June 87 (8) May Feb 9 () Jan 889 () Feb Aug 90 () Mar 89 () Sep 8 80 () Oct 9 08 (0) Nov June 9 () Apr Dec Apr (8) Nov Feb 0 7 (0) Jan 9 () Sep 89 Mar 00 (0) Mar 8 Sep Apr Nov Aug 8 9 (0) Dec 9 8 Jan (0) Mar

20 -Digit Multiplication Partial Products: The abilities of Jacques Inaudi (87-90) were investigated and written about by Alfred Benet and others. Special Neighbors: (the comma indicates a -digit group, so a fourth digit is added to the group on the left) (0+a) = ( + a/), 00+a (00+a) = (0 + a), a (70+a) = ( + a/), 00+a 8 = ( + 9), 00+8 = 7,8 = (0 + ), =,9 9 = (0-7), 7 = 79, 0 = 8,0 77 = ( + /), 00+7 = 7., 9 = 7, 09 = 7,09 Advanced: Can you separate out a product that is easy to multiply by the rest? 7 x = 0 x = 99 9 x = 99 7 x 7 = x 9 = 80 x 7 = 90 7 x 7 = x x = 00 x 9 x = 00 x x = x 7 = (x)x70 + x9 = x = 78 7x = 00x x0 + 00x + 0x00 + 0x0 + 0x + 7x00 + 7x0 + 7x = 87 Multiplication of x-digit numbers extends our earlier techniques (January/February/May) Cross Multiplication: Midpoint Method: Square the average and subtract the square of the distance to that average, or algebraically, where a is the average of the two numbers, (a+c) is one of the numbers, and (a c) is the other number. Here the midpoint of and 8 is, a distance of 9 from each number. Then we usethe Reverse Midpoint Method to simplify : x8 = 9 = (00x + ) 9 Anchor Method: Anchor one multiplier at a nearby round number a, and then string out the differences (plus or minus) of the original numbers from this anchor to find the other multiplier. Then add the product of the differences. Algebraically this is represented as If the differences have the same sign, the correction at the end will be added, and if one is positive and the other is negative the correction will be subtracted. x = 00x + x = (x00 + x0 + x)x00 + x0 + x = 8 9x98 = 000x x = x0 + ½(78x0) = Cool Facts: For numbers ending in or 7, (n) = (n + n/) 0 (n7) = (n + n + n/) = ( +.) 0 = 0. 0 = 0 7 = ( + + ) = x = 00x99 7x = 99000/ (0x + x) = 900 = 88 (where we might subtract by subtracting 00 and adding 8) 87x: : x =, or with a carry of 0 : 7x + x + 0 =, or with a carry of : 8x + 7x + x + =, or with a carry of : 8x + 7x + = 8, or 8 with a carry of : 8x + = 9 Answer: 98 Two-Date Single- Date Good Neighbor Method: Multiply by a nearby round number and adjust for the difference: 9x7 = 00x7 x7 = 98 x = 00x + x = 7 To find 00x, we can multiply from left to right as (x00 + x0 + x)x00 = 000. Then x7 = 9, so we subtract 000 and add 8. The boxes contain products using digits of the connected dates. Blue End: Double the date Red End: Triple the date. Attach the original first digit of the rightmost date (without doubling or tripling) to the end of the leftmost date. Attach the original last digit of the leftmost date (without doubling or tripling) to the start of the rightmost date. Multiply the two numbers Example for the second box from the bottom:. Left date: Blue end x = Attach 0 from 0 to end 0. Right date: Red end x 0 = 8 Attach from to start 8. 0 x 8 = 0 (answer in box) ^ = 9 Add : + = ^ = 79 Add : + = 7 7^ = 99 Day of the Week Code: Add to the date, sum the digits, check day-code table: + =, + = Jan, 98, was a Saturday Reverse Midpoint Method: Split the square into the product of two numbers equidistant from the original number, and add the square of that distance (a type of Anchor Method): 7 = 00x + 7 = (x00 + x0 + x)x00 + (x0 + ) = 8089 When spreading a square to the product of two numbers you add the correction, and when collapsing two multipliers to a square you subtract the correction. Binomial Expansion for Squares: Separate the number into the sum of two other numbers that are more easily squared: = (00+) = 00 + x00x + = 9 8 = (00-) = 00 x00x + = 7 ^ ^ 7^ Jan = = Nov, 798, was a Tuesday Legend 0^ 77^ 77^ 0 Nov 798 Mon Tue Wed Thu Fri Sat 9 Sun

21 Sunday Monday Tuesday Wednesday Thursday Friday Saturday AUGUST Oct 7 80 (9) Sep Aug Dec May Dec Nov July 89 () July Jan Jan Jan Oct 97 (9) 0 Jan () Apr 0 () 7 8 Nov 7 9 (0) Sep 8 7 (0) June Feb Aug 9 () Nov July 9 87 () June June Dec May 0 0 (0) Sep (0) (0) (0) (0) (0) Feb Aug 9 80 () Apr Oct July September

22 -Digit Multiplication Multiplication of x-digit numbers can benefit from our earlier techniques for special cases, but cross-multiplication is typically the most realistic option. We also introduce a new group technique here. Group Notation: It is convenient to treat hundreds groupings as separate blocks for multiplication. The notation " n" represents a two-digit number string. If more than two digits exist in n, they are merged (or added) to the digits to the left of the " " sign. The hundreds groups can carry or borrow as needed from neighboring groups to make each group positive and less than 00. = 0 9 = 9 = 9 - = (00-) = 7 Do you see that -8 =? Cross Multiplication: 87x: : x =, or with a carry of 0 : 7x + x + 0 =, or with a carry of : 8x + 7x + x + =, or with a carry of : x + 8x + 7x + x + = 8, or with a carry of 8 : x + 8x + 7x + 8 = 9, or with a carry of 9 : x + 8x + 9 = 7, or with a carry of 7 7: x + 7 = 7 Answer: 7 x Group Multiplication: x 89: Represent this as x 8 9 or A B x C D. Find AxC = x8 = 708. Find BxD = x9 = = 7 77 It helps if you are skilled at x multiplications! x Cross Multiplication: 87x: : 7x = 0, or with a carry of 0 : 8x + 7x + 0 = 70, or with a carry of 70 : 8x + 70 = 7 Answer: 7 Johann Dase (8-8) once multiplied 798 by in seconds; two 0-digit numbers in minutes; two 0- digit numbers in 0 minutes; and two 00-digit numbers in 8 hours minutes.. Add the last two numbers, +77, and subtract (A-B) x (C-D) (-)x(8-9) = 0 + x8 = 99. Previous Results: Answer: 7 (7 + ) Merge terms left to right as we go to get x:. AxC = 8x = BxD = 7x = 0 0 = 8. Add the last two numbers, 8+, and subtract (A-B) x (C-D) (8-7)x(-) = 0 + x =. Answer: ( + 8) = 7. Requires only three x digit multiplications. Naturally produces a leftto-right answer. Mechanical process is simple when learned. Ideal when A and B or C and D are close in value! Two-Date Single- Date The boxes contain products using digits of the connected dates. Blue End: Double the date Red End: Triple the date. Form a number from the leftmost date followed by the rightmost date. Double or triple if blue or red end on left date.. Form a number from the rightmost date followed by the leftmost date. Double or triple if blue or red end on right date.. Multiply the two numbers Example for the second box from the bottom:. Left date: Blue end x 09 = 08. Right date: Red end x 09 = x 7 = 880 (answer in box) Add reversed date to front and square ^ = Add : + = ^ = 09 Add : + = 7 77^ = 97 Day of the Week Code: Add to the date, sum the digits, check day-code table: + =, + = Jan, 9, was a Thursday ^ ^ 77^ Jan 9 () Legend 9009^ 00^ 0770^ 09 Nov (0) = + = Nov 9, 799, was a Saturday Mon Tue Wed Thu Fri Sat 9 Sun

23 Sunday Monday Tuesday Wednesday Thursday Friday Saturday SEPTEMBER Jun Jan 7 (0) Mar 8 88 (0) July 8 August October Dec July Dec 9 77 (7) Feb July Apr 09 (9) Sep 0 8 (0) June 7 7 (8) Mar July 789 (9) May July Nov 980 () Aug 8 8 (9) Apr 9 (8) Aug 788 (0) May Apr 8 () Mar Feb 8 () June Sep 0 97 (70) Sep June Jan 88 (9) May 0 0 (00) (0)

24 Trigonometric Functions Tangent with Memorization (~ digit accuracy):. If d > (or 0.78), replace d with (90 d) or (.7 d).. Find the angle or sum or difference of a pair of angles in Table that is nearest to d. For a single angle, the fraction in column is our first estimate. If a pair, use the equation below to add or subtract their fractions, where N and D are the numerator and denominator of each fraction. Swap signs to subtract a fraction.. If the remaining difference is < ±0. (or.00 radians), skip to Step. Otherwise, convert it to radians if in degrees (multiply by the simpler ratio 7/00). Find a simple fraction that approximates it and use the equation in Step to add this fraction to the result of Step.. Flip over the fraction if d was originally > (or 0.78). Divide fraction to digits.. Convert the remaining angle difference b to radians if still in degrees, again using 7/00. Then calculate c below to digits, where tan a is our current estimate from Step. Subtract c if d was originally >, otherwise add c. Degrees/Radians Conversion: d (degrees) 0/7 x d (radians) d (radians) 7/0 x d (degrees) (7/00 /000) x d (degrees) Cosine without Memorization: For an angle in degrees d, cos 8. : 000 cos d x9. / 7 = x. = cos d = Actual Value: Cosine with Memorization (~ digit accuracy): For an angle in degrees d, split it into two parts d = a + b, where cos a has been memorized: tan 8. : Doubling.0 in the table gives Add N/D = ¼ and N/D = ¼ using our formula: cos 8. : 000 cos d 000 cos 0 (-.)( ) / 7 = x8.= 879. cos d = Actual Value: For angle d, calculate the sine: cos d sin (90 d) (x + x) / (x x) = 8/ Since b = = 0. is just over 0., we might skip Step and divide 8/ to get 0.. Convert b to radians and add c: 0. x 7/00 = 0.08 x 0.07 = 0.00 c = 0.00 ( + 0.) 0.00 ( + 0. ) = = 0.0 Actual Value: 0.07 Sine without Memorization: For an angle in degrees d, If we do not skip Step, approximate the difference 0. with /7 = 0.8 to simplify the radian conversion: /7 x 7/00 = /00. Then (8x00 + x) / (x00 8x) = /99 = 0.98 sin 8. : 000 sin d (8./0)[7. 8.x9./0] = 7. sin d = 0.7 Actual Value: 0.7 Sine with Memorization (~ digit accuracy): For an angle in degrees d, split it into two parts d = a + b, where sin a has been memorized: sin 8. : 000 sin d 000 sin 0 + (-./0)[7 0x8./0] = (7 0x0.7) = 7. sin d = 0.7 Actual Value: 0.7 Final difference 0.0 x 7/00 = c = ( ) = , and = 0.0 For angle d, calculate the cosine: sin d cos (90 d) The boxes contain sines and cosines of the connected dates separated by a decimal point, in degrees Blue End: Double the date Two-Date (concatenated answer in box is.08.97) For angle d, calculate as: tan d / tan (90 d) SingleDate sin = cos = tan = 0. Day of the Week Code: Add to the date, sum the digits, check day-code table: + =, + = Jan, 98, was a Saturday Legend sin cos tan 0. Left date: Blue end x =. sin. =.08 and cos. =.97 tan 8. = 7 / 879 8/9000 = 0.0 Actual Value: 0.07 tan 8. : 000 tan d 000 tan 0 + (-./0)(7 + 0x8./0) =. tan d = 0. Actual Value: 0.07 Red End: Triple the date Example for the box at the bottom:. Right date: Blue end x 0 = Tangent by Definition: Tangent with Memorization (~ digit accuracy): For angle d, split it into two parts d = a + b, where tan a has been memorized The mathematician Johann Carl Friedrich Gauss (7778) possessed enormous skill in lightning calculation. Nov 798 sin cos tan + 0 = 0 +0= Nov, 798, was a Tuesday Mon Tue Wed Thu Jan 98 Fri Sat 9 Sun

25 Sunday Monday Tuesday Wednesday Thursday Friday Saturday OCTOBER September Oct 9 (9) Nov () July May 7 November May 8 (8) Nov Mar (0) Sep Aug Feb 0 () Oct 8 7 () Aug 07 () Oct 00 (7) Dec Mar 9 00 () Sep 98 (0) Aug 9 (0) June 7 (9) Jan 0 8 (00) May Nov Apr July 99 (79) Mar 8 9 (7) Jan 90 (9) July 8 07 (9) May Apr June Dec May

26 Logarithms Logarithms appear quite often in mathematics and engineering and calculating them is an impressive skill. When combined with exponentials, they can be used to calculate high-order roots. We will be working here in common logarithms, or logarithms to base 0. To calculate natural logarithms we can use the relation ln N =.0 log N = (log N) / 0., or for easier calculation, Integers with Low Factors: Memorize the logarithms of low primes in the table to the right. Notice how the other logarithms in the table can be calculated as simple combinations of these---you can combine these to find logarithms of many numbers according to these rules: Use whichever is easier for a number N Example: log 7 = 0.80 ln 7 (0.80/0.) x ( 0.0) = =.9 where Actual Value:.99 Series Approximation: Factor out a nearby number whose logarithm is easier to find. Then add to the first term or two in the following series expansion for ln (+a) (multiplied by 0. for log N): = 0 x ( + /0), so ln = ln 0 + ln ( + /0): log 0 = log (0 x ) = + log =.0879 log ( + /0) 0. x /0 = first term only so log =.08 Actual Value:.08 George Parker Bidder Method: Factor out a nearby number whose logarithm is easy to find, then use the Following expression, where m is the number of places that a needs to be shifted to lie between and 0: log = log 00 + ( + /00): log 00 = + log (x) = + log + log = x /00 = /. which lies between and 0, so m = so log ( + /00) /. log.00 = /. x = so log = =.08 Actual Value:.08 Note how far we were able to go from to the very convenient 00 and still get a very accurate answer for log! To use this method we need to learn log (+0 m ) except for m=, in which case a closer convenient number is needed for accuracy. These values approach 0.x0 m as m increases: log.0 = 0.00 log.00 = log.000 = Start off with lower accuracy before attempting -digit solutions! log ab = log a + log b log = log + log + log 7 log (a/b) = log a log b log. = log (/) = log log log a b = b log a log 9 = log 7 = log 7 log a /b = (/b) log a log / = ½ log = ½ (log 9 + log 7) log (0 b N) = b + log N log 0 = + log = + log. log (0 -b N) = log N b log 0.0 = log = log. - Series Modification: Modify the first term in the series expansion, where the 0.88 is omitted if calculating ln (N+a). where 0.88 = x0. and may be calculated as 0.8 ( + 0.0) if desired (add final result shifted right by two places). If a = this will be accurate to decimal places for any N>0. log log (/) = / (9 x 9) =.08 Actual Value:.08 log log (/) for N = and a = - log = log = (0.00) = (/) = ( + 0.0)x0.8/ = ( + 0.0)x0.0 = 0.07 so log =.799 Actual Value:.799 Two-Date Single- Date The boxes contain common (base 0) logarithms of the connected dates separated by a decimal point. Blue End: Double the date Red End: Triple the date Example for the box at the bottom:. Left date: Blue end x =. Right date: Blue end x 0 = 0. log.0 =.078 (answer in box) log =.0 Add : + = 7 log 7 =.70 Add : + = 77 log 77 =.889 Day of the Week Code: Add to the date, sum the digits, check day-code table: + 07 =, + = Jan, 9, was a Friday log log 7 log 77 Jan 9 Among many other facts, Wim Klein (9-98) drew on his knowledge of logarithms to five places of the first 0 integers N Legend log 0 log log 7 0 Nov (7) = = 9 Nov 0, 799, was a Sunday log N.00 = x log.9897 = log.778 = log + log = x log 9.9 = x log Mon Tue Wed Thu Fri Sat 9 Sun

27 Sunday Monday Tuesday Wednesday Thursday Friday Saturday NOVEMBER Dec 79 () May 79 (9) Aug 0 8 (0) Dec 7 9 () Jan 7 90 () Oct 90 (0) June 9 (0) Apr Sep June Sep 8 () Feb 78 (9) Nov () Mar () Apr 9 89 () Nov July 0 () Oct 0 87 () July Aug 0 79 (00) Feb 7 8 (0) Dec Oct July 98 (9) June Nov 0 (0) Jan Sep Aug 97 October () Feb 009 December

28 Exponentials Calculating exponentials involves raising a number to a power, such as 0.8 = 7.09 The value e N is often a solution to mathematical or physical equations, but this is a simple extension of finding 0 N since e N = 0 (0. ) N. George Parker Bidder (80-878) was a calculating prodigy who in 8 described his method of mentally calculating compound interest. Step : Add or subtract up to two copies of log = 0.77 to make the number close to a multiple of 0.. You can determine the right number of copies by looking at the second and third digits of the number, where adding log is like subtracting in those digits, and vice versa. log M M.09 /0.087 /9.80 /8.99 /7. /. /.9 / log M M.0 0/9.0 /.0 /.0 /7.07 7/0. 89/.0 8/.7 /9.00. /. /.77 Step : Extract powers of 0 from N (by subtracting or adding integers) to leave the smallest difference from zero: x In the calendar, all exponents lie between 0 and, so 0 is extracted if greater than 0.. Good at memorization? You can extract logarithms of conveniently multiplied and divided fractions from these tables. Bemer Method Step : Add or subtract memorized logarithms of low numbers or convenient fractions from the remaining exponent to reduce it to near zero (within ±0.09 if -digit accuracy is desired). Subtracting a logarithm will flip the multiplying number or fraction. Use creativity! We may know that log 7 = 0.80, so x and we are already at Step! We can find a higher-order root r of a number N as 0 log(n)/r. So to find the th root of, we find log =.70, divide by to get 0.7, and find =.70 Here you only need to know log and log! McIntosh- Doerfler Method = 7/ x since = = 7/ x /0 x since = Using the fractions in the /n table often cancels terms, as here: = 7/0 x Step : Calculate the following for our remainder b, where digits are sufficient for the last term (omit it completely for less accuracy): +.b ( ) + (.b) / +.(0.007)( ) + (.x0.007) / = + ( ) =.07 Step : Multiply the results: x 7/0 x.07 = 0 x 7 x.07 = 7.07 Actual Value: 7.09 Two-Date Single- Date log = where 7 is nearly x Step : Add or subtract up to five copies of log =.00 to make the decimal part close to zero, and separate out the integer part. Notice how easy it is to multiply by log since log = ! log = since (0.00) =.0 Step : Calculate the following for our remainder b, where digits are sufficient for the last term (omit it completely for less accuracy): The boxes contain exponentials (in powers of 0) for the connected dates following a decimal point. Blue End: Double the date Red End: Triple the date Example for the box at the bottom:. Left date: Blue end x =. Right date: Blue end x 0 = 0. 0^(0.0) =.0989 (answer in box) 0^(0.) =. Add.: = 0.7 0^( 0.7) =.9 Add.: = ^(0.77) =.888 Day of the Week Code: Add to the date, sum the digits, check day-code table: + 07 =, + = Jan, 9, was a Friday +.b ( ) + (.b) / +.(0.0087)( ) + (.x0.0) / = =.0 Step : Undo Steps and. Divide if we added, multiply if we subtracted. Perform divisions last. We added log, so we divide by. We added log so we divide by =. Then shift the decimal point based on the integers we extracted along the way: x.0 / (9x) = 0 x = 7.09 Actual Value: ^(0.) 0^(0.7) 0^(0.77) Jan 9 Legend 0^(0.0) 0^(0.) 0^(0.7) 0 Nov (7) = = 9 Nov 0, 799, was a Sunday Mon Tue Wed Thu Fri Sat 9 Sun

29 Sunday Monday Tuesday Wednesday Thursday Friday Saturday DECEMBER Sep 8 (8) Nov 900 () Feb (8) Dec 800 (0) November May 0 () Aug 8 (9) June Jan 08 (9) January Oct 777 (0) Oct Oct 0 0 () Dec 7 9 (9) Mar 008 (8) Apr Feb 8 80 (9) Aug Jan Sep July 7 89 (0) Apr Jan 8 77 () Sep 9 90 (0) Jan 890 (9) May 9 0 () Dec 8 (0) Mar 0 8 () Aug July 0 88 (0) Oct 7 00 () July Dec

30 Lightning Calculators Lightning calculators possess startling abilities to mentally compute products, quotients, powers, roots, and sometimes functions such as logarithms and exponentials. This calendar presents methods used by these individuals, along with daily exercises for fun and practice. The study of lightning calculators of the past and present is fascinating from more than a mathematical aspect. Many presentations, particularly in the popular media, ascribe abilities in these areas to mysterious machinations in the minds of remote geniuses, which makes for a good story but can be discouraging. In fact, these individuals through talent and training acquired a knack for racing headlong through calculations that can often seem mysterious to the uninformed. Other than rough estimation, techniques of mental calculation are not being taught in our schools today. Yet presentations on even the most basic methods of mental calculation are met with incredible interest among people. This calendar attempts to address that need. Mental calculation can be a highly creative and satisfying endeavor offering a variety of interesting strategies, many more than most people realize. It is a skill that engages both children and adults, and one that naturally leads to a real familiarity with the properties and relationships of numbers. It provides a useful and fun approach for developing a number sense and generating a true appreciation for the elegance of elementary mathematics. It is an art as fundamentally important as other areas of mathematics. For information on obtaining this calendar, including a free PDF download to create it on your own printer, visit For additional details on advanced topics of mental calculation, please see Lightning Calculators I - III essays at Benjamin, Arthur. Secrets of Mental Math: The Mathemagician's Guide to Lightning Calculation and Amazing Math Tricks (00). Doerfler, Ronald. Dead Reckoning: Calculating Without Instruments (99). Lane, George. Mind Games: Amazing Mental Arithmetic Tricks Made Easy (00). 00 Ron Doerfler All Rights Reserved Lightning front cover photo by John A. Cobb Lightning back cover photo by Erica Burrell