1000 logs book

1,000 Logarithms

by Jim HallCopyright November 28, 2009

Introduction

use with a soroban. A soroban is a ja

methods involve more than the most basic math. All of the tables are laid out for legibility and ease of use by the average person.

make copies for others. If you would like to translate this work, please do so.

not be sold or otherwise altered, in whole or in part.

Sincerely,

Jim Hall

November, 2009

Table of Contents

Chapter one

Squaring on a soroban, p.7Napier’s bones, p.8

Chapter twoFinding square roots

Chapter threeFinding Logarithms

Method #1, p.15Method #2, p.16Method #3, p.19Method #4, p.22Method #5, p.23Method #6, p.25

Chapter fourTables

Table #1, 1,000 logarithms, p.29Table #2, small logarithms, p.40Perfect squares, p.41Table #3, multiplication, p.41Table #4, powers of ½, p.42

About Me - p.43

1,000 Logarithms 2 [email protected]

46

Squaring on a soroban, p7Napier’s bones, p.8

needed and felt that you might like, too.

Chapter One

1,000 Logarithms 3 [email protected],000 Logarithms 3 [email protected]

The Common Logarithm

A common logarithm looks like this:Log10 251 = 2.399667 � .399667 is the mantissa

� 2 is the characteristic.

251 is the Antilog of 2.399667

102.399667 ≈ 251

Properties of Common Logarithms

Log (AB) = Log A + Log B

Log (A/B) = Log A - Log B

Log Ab = b Log A

Log (1/A) = - Log A

Implications of the properties of Logarithms

1. Multiplying two numbers can be done using logarithms by adding the Logs of the two numbers together and finding the Antilog.

2. Dividing two numbers is as simple as subtracting the Log of one number from the Log of the other, then finding the Antilog.

3. Powers or roots of numbers can be found by multiplying the logarithm by the power or root. For example, the fifth root of 20 is the Antilog of 1⁄5Log 20.

4. Negative logarithms represent fractional numbers.

Proceed to the next page.


The Common Logarithm

5. The mantissa indicates what digits are in the antilog. The characteristic indicates the number of primary digits (minus one).

Same Mantissa: Log 251 = 2.399667Log 2.51 = 0.399667 � The mantissa is the same. Note the decimal place in the second number.

Same Characteristic:Log 251 = 2.399667Log 322 = 2.507853 � The chartacteristic is the same.Three primary digits.


Scientific Notation

Scientific notation is a special way to represent numbers. This notation makes some calculations easier.

The number "987654" written in scientific notation:9.87654 x 10598.7654 x 104987.654 x 103

9, 98 and 987 - the numbers to the left of the decimal place - are the primary digits. The total digits - 987654 - are the significant figures.

Scientific notation makes it easy to represent both large and small numbers. For example, 4.3 trillion is 4.3 x 1012.

Choose carefully when selecting primary digits. The square root of 98.765 and the square root of 9.8765 are completely different.

Using an even exponent makes finding square roots easier.

For example:√(4 x 1012) = 2 x 106√(16 x 10-12) = 4 x 10-6


Squaring on a soroban

This is a special method for performing multiplication on a soroban. Examine this photograph straight from my notebooks:

Note the text at left, scanned from my notebook. Here the number 9.6236 is being squared. “10)” refers to a stage using a method in this book. Ignore the “10)”.

This method is a special case of long multiplication. Using standard long multiplication, multiplying would begin with the least significant digit to the most significant, right to left. Each level would be shifted left, further and further. Last, the numbers would be added and a result obtained.

I decided that the standard method of multiplication on the soroban was wasteful

of digits. Also, the shifting of long multiplication on paper was wasteful of space. This method corrects both deficiencies.

The levels of significant digits are noted on the left - highest significance to least. The results of multiplying each digit are in the middle. The column of the first digit of each answer is on the right. Each type of digit - “6” for example - needs be multiplied only once. You may note the column of the answers as you do them. Reset the initial number on the soroban for each digit. Once this little table is complete, use the columns as a guide and simply add the answers on the soroban. With practice, it is quite fast. The final answer will appear naturally.

Check the final answer - as mistakes can happen. Correcting mistakes is easy, because the answers can be checked line by line. There is no need to perform the entirety of the multiplication over again. Note that I truncate answers to five digits, but this procedure is good for any number of digits. It is possible to use every digit on the soroban this way - and an entire sheet of paper if you so choose.


Napier’s Bones

Napier’s Bones is a special tool for multiplying one number with several digits by a single digit. The bones are fast and accurate, with little room for error. Napier’s Bones would make an excellent aid to the soroban. Using my special method of squaring with the bones, and adding the results on a soroban would be fast, if not as fast as a calculator.

Unfortunately, Napier’s Bones is not the type of aid you find at the local office supply store. The bones are simple to make, and could even be made out of paper. A chart duplicating the benefits of the bones can be drawn on paper as needed. Although, if the bones chart is drawn, performing the long multiplication instead is probably faster.

Here is what using Napier’s Bones might look like, if you had that tool available:

Let’s find 9.62362 using my instant bones chart:

X 9 6 2 3 6

9

6

2

3

6

0

12

3

45

6

78

9

18 4

5

45 1

21

83

6

18

12 4

06

12

27

18

06

0 18

364

5 12

18

36

= 866124

= 577416

= 192472

= 288708

= 577416

Using a real set of Napier’s Bones, answers would be quick and easy.

One of my upcoming personal projects is making a set of Bones out of ¾-inch hollow aluminum rodstock with an engraver.


1012

Finding square roots is important for using one of Euler’s methods for approximating logarithms. Square roots are also generally useful in

Chapter Two


Finding Square Roots - Babylonian Method

This example shows how to find the square root of 5821 using the babylonian method.

Step one - Rewrite 5821 in scientific notation.5821 is 58.21 x 102

Step two - Find the closest perfect square to the primary digits.The primary digits are 58. 49 is the closest square.

Step three - Note the square root and adjust for scale.√49 = 7√102 = 101 = 107 x 10 = 70

Step four - Find the first estimate.702 = 4900The first estimate is 70.

This specific estimate is 91.74% correct.

Step five - Divide the original number by the estimate.5821/70 = 83.16 (more or less)

Step six - Add the two numbers together70 + 83.16 = 153.16

Step seven - Divide by 2.153.16/2 = 76.5876.58 is approximately 7676 is the new estimate

Step eight - Divide the original number by the new estimate.5821/76 = 76.59The primary digits converge as your estimates improve.76.59 is 99.23% correct.

Proceed to the next page.1,000 Logarithms 10 [email protected]

Finding Square Roots - Babylonian Method

Step nine - Add the two numbers together.76 + 76.59 = 152.59

Step ten - Divide by 2.152.59/2 = 76.29576.295 is the new estimate.


Step eleven - Divide 5821 by the new estimate.5821/76.295 = 76.2959 76.29592 = 5821.06476.29552 = 5821.003

And so it goes.


Finding Square Roots - Bakhshali Approximation

This section demonstrates use of the Bakhshali approximation for square roots.

This example will show how to find the square root of 5821.

Part one - Making the first estimate

Step one - Rewrite the number in scientific notation. Sum = 58.21 x 102

Step two - Find the closest perfect square to the sum. The closest perfect square is 49 x 102.

Step three - Find the first estimate.Find the square root of 4900. This is the first estimate for the square root of 5821.The first estimate is 7 x 101 (or 70).


Part two - Finding the second estimate

Step four - Find the difference between the sum and the first estimate.The difference between the two numbers isSum - estimate = 58.21 - 49.00 = 9.21 (x 102)

Step five - Find the first adjustment.The first improvement to the estimate is the difference divided by twice the closest square root.

First adjustment = difference/(2)(root)= (9.21 x 102)/(2)(7 x 101)= 92.1/14= 6.58 (more or less)



Finding Square Roots - Bakhshali Approximation

Step six - Find the second estimate.The second estimate is the first estimate plus the adjustment amount.

First estimate = 70Plus the adjustment amount: 6.5870 + 6.58 = 76.58The second estimate is 76.58

This estimate is 99.6% accurate in six easy steps.

Part three - Finding the third estimate

Step seven - Find the second adjustment.The second adjustment is:(First adjustment)2 divided by 2 times the second estimate.6.582 /(2)(76.58) = .283 (more or less)

Step eight - Find the third estimate.The third estimate is the second estimate minus the second adjustment.

Second estimate = 76.58Second adjustment = .28376.58 - .283 = 76.297The second estimate is 76.297

This specific estimate is 99.998% accurate.

Part four - Improving the estimateEven better estimates could be made at this point by continuing the calculations using the Babylonian method.


Method #1, p.15Method #2, p.16Method #3, p.19Method #4, p.22Method #5, p.23Method #6, p.25

logarithm is now available to you, without a calculator. Although, certainly, most people will still use the magic plastic box.

Chapter Three


Finding Logarithms - Method #1

This method uses the log table in this book.

Finding Log 21242This number was chosen because $21,242 represents a yearly income.

Step one - Rewrite the number in scientific notation.21242 = 212.42 x 102

Step two - Round off to the nearest three primary digits.212.42 is close to 212

Step three - Use the log chart in this book to find the logarithm of 212.Log 212 = 2.326333

Step four - Use the properties of logarithms to find the logarithm of 21200.Log 21200 = Log 212 + 2 = 2.326333 + 2= 4.326333



Finding Logs - Method #2

This method approximates a logarithm ex nihilo (from nothing).


Step one - Find the characteristic of the answer.Rewrite the number in scientific notation so that there is only one primary digit.

21482 = 2.1482 x 104 �Exponent

The exponent of the number is the characteristic of the answer. Here it is 4.

Step two - Make a seven stage chart with a remainder.1)2)3)4)5)6)7)Remainder:

This chart generates the mantissa.

Step three - Fill the chart.Each stage of the chart is the square of the previous stage. Truncate answers to five digits. If the answer is greater than 10, divide it by 10 at the next stage.

1) 2.14822 = 4.61472) 4.61472 = 17.344 (too big! Divide by 10)3) 1.73442 = 3.00814) 3.00812 = 9.04865) 9.04862 = 81.877 (too big! Divide by 10)

Proceed to the next page. 1,000 Logarithms 16 [email protected]


6) 8.18172 = 67.038 (too big! Divide by 10)7) 6.70382 = 44.940 (too big! Divide by 10)Remainder: 4.4940

Step four - Examine the chart. For each stage that require division, add ½stage to the mantissa. The mantissa starts at zero.

2) 4.61472 = 17.344mantissa = mantissa + ½2 (stage 2)




Step five - Add the characteristic to the mantissa. This is the first estimate of the logarithm.

Characteristic = 4Mantissa = ½2 + ½5 + ½6 + ½7 = .3046875First estimate = 4.3046875


Optional steps

Step six - Find the first estimate of the logarithm of the remainder. Because the remainder is always one digit, the characteristic is always zero.

Proceed to the next page. 1,000 Logarithms 17 [email protected]


1) 4.4942 = 20.196 (too big! Divide by 10)2) 2.01962 = 4.07873) 4.07872 = 16.635 (too big! Divide by 10)4) 1.66352 = 2.76725) 2.76722 = 7.65736) 7.65732 = 58.634 (too big! Divide by 10)7) 5.86342 = 34.379 (too big! Divide by 10)There is no remainder for this step.

Mantissa = ½1 + ½3 + ½6 + ½7= .6484375

Log 4.4940 = .6484375 (more or less)

Step seven - Find the adjustment to the first estimate.Multiply the logarithm of the remainder by ½7.

½7 x Log 4.4940= 5.066 x 10-3This number is the adjustment.

Step eight - Add the adjustment to the first estimate.4.3046875 + 5.066 x 10-3= 4.3097535This number is the final estimate




This method is the same as method two, except that step six is not optional and there are 14 stages. Every prime number in my chart was done with this method. The initial three primes were done with this method on a soroban.


Step one - Find the characteristic of the answer.Rewrite the number in scientific notation so that there is only one primary digit.

21482 = 2.1482 x 104 � Exponent The exponent of the number 10 is the characteristic of the answer.

Step two - Make a 14 stage chart with a remainderThis chart is identical to the chart in method #2, step 2, except that the chart is 14 stages instead of 7.

Step three - Fill the chart.Each stage of the chart is the square of the previous stage. Truncate answers to five digits. If the answer is greater than 10, divide it by 10 at the next stage.

1) 2.14822 = 4.61472) 4.61472 = 17.344 (too big! Divide by 10)3) 1.73442 = 3.00814) 3.00812 = 9.04865) 9.04862 = 81.877 (too big! Divide by 10)6) 8.18172 = 67.038 (too big! Divide by 10)7) 6.70382 = 44.940 (too big! Divide by 10)8) 4.49402 = 20.196 (too big! Divide by 10)9) 2.01962 = 4.078710) 4.07872 = 16.635 (too big! Divide by 10)11) 1.66352 = 2.767212) 2.76722 = 7.657313) 7.65732 = 58.634 (too big! Divide by 10)



14) 5.86342 = 34.379 (too big! Divide by 10)Remainder: 3.4379

Step four - Examine the chart. For each stage that require division, add ½stage to the mantissa. The mantissa starts at zero.





8) 4.49402 = 20.196 mantissa = mantissa + ½8 (stage 8)

10) 4.07872 = 16.635 mantissa = mantissa + ½10 (stage 10)



The mantissa is .30975341

Step five - Find the first estimate.Add the characteristic and the mantissa.




Characteristic = 4Mantissa = .30975341Characteristic + mantissa = first estimate.First estimate = 4.30975341


Step six - Find the first estimate of the logarithm of the remainder.Although you can repeat the 14-stage tabulation, I suggest finding another way. Use your best judgment, and make an informed, but quick, estimation. Accuracy is not essential at this stage. One method for estimating the remainder is to use the logarithm chart in this book.

Log 4 = .602060Log 3 = .477120Log 3.4379 ≈ (Log 4 + Log 3) /2 (≈ .5396)

Step seven - Find the adjustment.Multiply the remainder's logarithm by ½14.(½14)(.5396) ≈ 3.293 x 10-5

Rounding to four digits makes for simpler math at this stage.

Step eight - Find the second estimate.Add the adjustment to the first estimate.4.30975341 + 3.293 x 10-5= 4.30978634This is the second estimate.


For numbers under four digits this method is close to 99.9995% accurate. The lesson here is that accuracy is relative. Select your tools carefully.



Estimating a logarithm using known logarithms without factoring. This method uses the logarithm chart in this book.


Step one - Identify the upper and lower bounds of the estimate.

Log 21400 and Log 21500 are good choices for bounds, because each one is similar to a known logarithm + 2. For example, Log 21400 = Log 214 + 2.

The lower bound for the logarithm of 21482 is 4.330406 (Log 21400).

The upper bound for the logarithm of 21482 is 4.332435 (Log 21500).

Step two - Find the difference between the upper and lower bounds.Log 21500 - Log 21400 is 2.029 x 10-3. 100% of the difference is 2.029 x 10-3.

The difference between 21482 and 21400 is 82. The difference between 21400 and 21500 is 100.

Step three - Find the first adjustment.82⁄100 = 82% (2.029 x 10-3) x 82% = 1.663 x 10-3This is the first adjustment.

Step four - Find the second estimate.The second estimate is the first estimate plus the adjustment.4.330406 + 1.663 x 10-3 = 4.332069This is the second estimate.




Estimating a logarithm using known logarithms using factoring. This method uses the logarithm chart in this book. The key is breaking down large numbers quickly into factors that you know or can find easily.


Step one - Find the first estimateFactor 21482Sometimes numbers have obnoxious factors. This step could take lots of time!Remember the square root methods earlier in this book?√21482 = 146 (more or less) and 146*147 = 21462

Therefore, due to the properties of logarithms ...Log 146 + Log 147 = Log 21462 = 4.331663This is the first estimate

This specific estimate is 99.99% accurate. But $20 is a lot of error.

Step two - Improving the estimate

Better factoring will give better answers. Common factors of numbers are the prime numbers.Good choices are 2, 3, 5, 7, 9, 11, 13, 17, 19 and 23.

If you are in a hurry, find the next nearest factors instead. Step three - Estimating the next nearest factors

21482 / 2 = 1074110741 / 2 = 5370 (more or less)5370/10 = 537 (Log 537 is on the chart in this book)537 x 10 x 2 x 2 = 21480 21480 is within 2 points of 21482.



Since 21482 represents $21,482, a difference of $2 is small.… but two dollars is two dollars.

The actual factors of 21482 are 23 and 934.Skip to step 5 if you want to see the accuracy that $2 can buy.

Step four - Find the second estimate based on the next nearest factors.

Log 537 = 2.729969Log 10 = 1Log 2 = 0.30103Log 537 + Log 10 + Log 2 + log 2= 4.332029

This is the second estimate.This estimate is 99.9989% accurate.

Step five - Find the third estimate based on the actual factors.The actual factors of 21482 are 23 and 934.

Log 934 = 2.970334Log 23 = 1.361727Log 21482 = Log 934 + Log 23= 4.332061This is the third estimate. This estimate is 99.9997% accurate.

All three estimates have their purpose. Even the worst estimate was 99.99% correct. Application helps determine the acceptable error, but that is the topic of another book.


Finding Logarithms - Method #6 - Euler's method

This is a method for finding logarithms based mostly on a method demonstrated by Euler.

Finding Log 21482This number was chosen because it represents a yearly income.

Step one - Find an upper and lower bounds based on known logarithms.Upper bound: 21500This log is known because 215 is on the chart in this book.Log 21500 = Log 215 + 2

Lower bound: 21400Log 21400 = Log 214 + 2

Step two - Start a list using the chosen bounds.A - 21500B - 21400

Step three - Find the mean of the upper and lower bounds.The mean of A and B is √(AB)√(AB) = 21449.94172

Add C to the list:√(AB) C = 21449.94172

Note that Log C is unknown at this stage.

Step four - Set the new upper and lower bounds based on the new list.

C is less than 21482 but greater than B.C is the new lower bound.

Step five - Find the mean of the upper and lower bounds.√(AC) D = 21474.95627Add D to the list.



Step six - Set the new upper and lower bounds based on the new list.

D is less than 21482 but greater than C.D is the new lower bound.

Step seven - Find the mean of the upper and lower bounds.

√(AD) E = 21487.47449Add E to the list

Step eight - Set the new upper and lower bounds based on the new list.

E is greater than 21482 but less than A.E is the new upper bound.

Step nine - Find the mean of the upper and lower bounds.

√(ED) F = 21481.21447Add F to the list.

Note that the upper and lower bounds converge towards 21482.21481 is close enough to 21482 for demonstration purposes.

The list now looks like this:A = 21500B = 21400

√AB C = 21449.94172√AC D = 21474.95627√AD E = 21487.47449√ED F = 21481.21447

Step 10 - Add logarithms of A and B to the list.A = 21500 Log A = 4.332435B = 21400 Log B = 4.330406



Step 11 - Find the logarithm of C.

Log C = (Log A + Log B)/2= 4.3314205Add Log C to the list

√AB C = 21449.94172 Log C = 4.3314205

Step 12 - Find the logarithm of D.

Log D = (Log A + Log C)/2= 4.33192775Add Log D to the list

√AC D = 21474.95627 Log D = 4.33192775

Step 13 - Find the logarithm of E.

Log E = (Log A + Log D)/2= 4.332181375Add Log E to the list

√AD E = 21487.47449 Log E = 4.332181375

Step 14 - Find the logarithm of F.

Log F = (Log E + Log D)/2= 4.332054563Add Log F to the list

√ED F = 21481.21447 Log F = 4.332054563

Log F is approximately 4.332055

Even though F, 21481, is not equal to 21482, this estimate is 99.9995% correct.


Table #1, 1,000 logarithms, p.29Table #2, small logarithms, p.40Perfect squares, p.41Table #3, multiplication, p.41Table #4, powers of ½, p.42

Table #1 contains the logarithms of the numbers from 1 to 1000 in an open, of space and digits. Table #2 contains

the value of Pi and two small values near 1. Table #4 was created on a soroban to silly levels of precision.

Chapter Four


Table 1: 1,000 Logarithms

1 02 0 . 3 0 1 0 3 03 0 . 4 7 7 1 2 04 0 . 6 0 2 0 6 05 0 . 6 9 8 9 7 06 0 . 7 7 8 1 5 07 0 . 8 4 5 0 9 68 0 . 9 0 3 0 9 09 0 . 9 5 4 2 4 010 1

31 1 . 4 9 1 3 6 132 1 . 5 0 5 1 5 033 1 . 5 1 8 5 0 934 1 . 5 3 1 4 7 835 1 . 5 4 4 0 6 636 1 . 5 5 6 3 0 037 1 . 5 6 8 1 9 638 1 . 5 7 9 7 8 139 1 . 5 9 1 0 6 340 1.602060

41 1 . 6 1 2 7 8 042 1 . 6 2 3 2 4 643 1 . 6 3 3 4 6 544 1 . 6 4 3 4 4 945 1 . 6 5 3 2 1 046 1 . 6 6 2 7 5 747 1 . 6 7 2 0 9 448 1 . 6 8 1 2 4 049 1 . 6 9 0 1 9 250 1.698970

51 1 . 7 0 7 5 6 852 1 . 7 1 6 0 0 353 1 . 7 2 4 2 7 354 1 . 7 3 2 3 9 055 1 . 7 4 0 3 5 956 1 . 7 4 8 1 8 657 1 . 7 5 5 8 7 158 1 . 7 6 3 4 2 759 1 . 7 7 0 8 4 860 1.778150

61 1 . 7 8 5 3 2 262 1 . 7 9 2 3 9 163 1 . 7 9 9 3 3 664 1 . 8 0 6 1 8 065 1 . 8 1 2 9 1 366 1 . 8 1 9 5 3 967 1 . 8 2 6 0 7 368 1 . 8 3 2 5 0 869 1 . 8 3 8 8 4 770 1.845096

71 1 . 8 5 1 2 5 472 1 . 8 5 7 3 3 073 1 . 8 6 3 3 2 174 1 . 8 6 9 2 2 675 1 . 8 7 5 0 6 076 1 . 8 8 0 8 1 177 1 . 8 8 6 4 8 578 1 . 8 9 2 0 9 379 1 . 8 9 7 6 2 680 1.903090

81 1 . 9 0 8 4 8 082 1 . 9 1 3 8 1 083 1 . 9 1 9 0 7 784 1 . 9 2 4 2 7 685 1 . 9 2 9 4 1 886 1 . 9 3 4 4 9 587 1 . 9 3 9 5 1 788 1 . 9 4 4 4 7 989 1 . 9 4 9 3 8 890 1.954240

21 1 . 3 2 2 2 1 622 1 . 3 4 2 4 1 923 1 . 3 6 1 7 2 724 1 . 3 8 0 2 0 025 1 . 3 9 7 9 4 026 1 . 4 1 4 9 7 327 1 . 4 3 1 3 6 028 1 . 4 4 7 1 5 629 1 . 4 6 2 3 9 730 1.477120

11 1 . 0 4 1 3 8 912 1 . 0 7 9 1 8 013 1 . 1 1 3 9 4 314 1 . 1 4 6 1 2 615 1 . 1 7 6 0 9 016 1 . 2 0 4 1 2 017 1 . 2 3 0 4 4 818 1 . 2 5 5 2 7 019 1 . 2 7 8 7 5 120 1.301030



91 1 . 9 5 9 0 3 992 1 . 9 6 3 7 8 793 1 . 9 6 8 4 8 194 1 . 9 7 3 1 2 495 1 . 9 7 7 7 2 196 1 . 9 8 2 2 7 097 1 . 9 8 6 7 6 598 1 . 9 9 1 2 2 299 1 . 9 9 5 6 2 9100 2

121 2 . 0 8 2 7 7 8122 2 . 0 8 6 3 5 2123 2 . 0 8 9 9 0 0124 2 . 0 9 3 4 2 1125 2 . 0 9 6 9 1 0126 2 . 1 0 0 3 6 6127 2 . 1 0 3 7 9 9128 2 . 1 0 7 2 1 0129 2 . 1 1 0 5 8 5130 2.113943

131 2 . 1 1 7 2 6 7132 2 . 1 2 0 5 6 9133 2 . 1 2 3 8 4 7134 2 . 1 2 7 1 0 3135 2 . 1 3 0 3 3 0136 2 . 1 3 3 5 3 8137 2 . 1 3 6 7 1 6138 2 . 1 3 9 8 7 7139 2 . 1 4 3 0 1 5140 2.146126

141 2 . 1 4 9 2 1 4142 2 . 1 5 2 2 8 4143 2 . 1 5 5 3 3 2144 2 . 1 5 8 3 6 0145 2 . 1 6 1 3 6 7146 2 . 1 6 4 3 5 1147 2 . 1 6 7 3 1 2148 2 . 1 7 0 2 5 6149 2 . 1 7 3 1 8 5150 2.176090

151 2 . 1 7 8 9 7 3152 2 . 1 8 1 8 4 1153 2 . 1 8 4 6 8 8154 2 . 1 8 7 5 1 5155 2 . 1 9 0 3 3 1156 2 . 1 9 3 1 2 3157 2 . 1 9 5 8 9 9158 2 . 1 9 8 6 5 6159 2 . 2 0 1 3 9 3160 2.204120

161 2 . 2 0 6 8 2 3162 2 . 2 0 9 5 1 0163 2 . 2 1 2 1 8 7164 2 . 2 1 4 8 4 0165 2 . 2 1 7 4 7 9166 2 . 2 2 0 1 0 7167 2 . 2 2 2 7 1 3168 2 . 2 2 5 3 0 6169 2 . 2 2 7 8 8 6170 2.230448

171 2 . 2 3 2 9 9 1172 2 . 2 3 5 5 2 5173 2 . 2 3 8 0 4 6174 2 . 2 4 0 5 4 7175 2 . 2 4 3 0 3 6176 2 . 2 4 5 5 0 9177 2 . 2 4 7 9 6 8178 2 . 2 5 0 4 1 8179 2 . 2 5 2 8 4 9180 2.255270

111 2 . 0 4 5 3 1 6112 2 . 0 4 9 2 1 6113 2 . 0 5 3 0 7 2114 2 . 0 5 6 9 0 1115 2 . 0 6 0 6 9 7116 2 . 0 6 4 4 5 7117 2 . 0 6 8 1 8 3118 2 . 0 7 1 8 7 8119 2 . 0 7 5 5 4 4120 2.079180

101 2 . 0 0 4 3 1 7102 2 . 0 0 8 5 9 8103 2 . 0 1 2 8 3 6104 2 . 0 1 7 0 3 3105 2 . 0 2 1 1 8 6106 2 . 0 2 5 3 0 3107 2 . 0 2 9 3 7 6108 2 . 0 3 3 4 2 0109 2 . 0 3 7 4 2 4110 2.041389



181 2 . 2 5 7 6 6 6182 2 . 2 6 0 0 6 9183 2 . 2 6 2 4 4 2184 2 . 2 6 4 8 1 7185 2 . 2 6 7 1 6 6186 2 . 2 6 9 5 1 1187 2 . 2 7 1 8 3 7188 2 . 2 7 4 1 5 4189 2 . 2 7 6 4 5 6190 2.278751

211 2 . 3 2 4 2 7 8212 2 . 3 2 6 3 3 3213 2 . 3 2 8 3 7 4214 2 . 3 3 0 4 0 6215 2 . 3 3 2 4 3 5216 2 . 3 3 4 4 5 0217 2 . 3 3 6 4 5 7218 2 . 3 3 8 4 5 4219 2 . 3 4 0 4 4 1220 2.342419

221 2 . 3 4 4 3 9 1222 2 . 3 4 6 3 4 6223 2 . 3 4 8 3 0 0224 2 . 3 5 0 2 4 6225 2 . 3 5 2 1 8 0226 2 . 3 5 4 1 0 2227 2 . 3 5 6 0 2 3228 2 . 3 5 7 9 3 1229 2 . 3 5 9 8 3 2230 2.361727

231 2 . 3 6 3 6 0 5232 2 . 3 6 5 4 8 7233 2 . 3 6 7 3 5 0234 2 . 3 6 9 2 1 3235 2 . 3 7 1 0 6 4236 2 . 3 7 2 9 0 8237 2 . 3 7 4 7 4 6238 2 . 3 7 6 5 7 4239 2 . 3 7 8 3 9 0240 2.380200

241 2 . 3 8 2 0 1 2242 2 . 3 8 3 8 0 8243 2 . 3 8 5 6 0 0244 2 . 3 8 7 3 8 2245 2 . 3 8 9 1 6 2246 2 . 3 9 0 9 3 0247 2 . 3 9 2 6 9 4248 2 . 3 9 4 4 5 1249 2 . 3 9 6 1 9 7250 2.397940

251 2 . 3 9 9 6 6 7252 2 . 4 0 1 3 9 6253 2 . 4 0 3 1 1 6254 2 . 4 0 4 8 2 9255 2 . 4 0 6 5 3 8256 2 . 4 0 8 2 4 0257 2 . 4 0 9 9 3 0258 2 . 4 1 1 6 1 5259 2 . 4 1 3 2 9 2260 2.414973

261 2 . 4 1 6 6 3 7262 2 . 4 1 8 2 9 7263 2 . 4 1 9 9 5 1264 2 . 4 2 1 5 9 9265 2 . 4 2 3 2 4 3266 2 . 4 2 4 8 7 7267 2 . 4 2 6 5 0 8268 2 . 4 2 8 1 3 3269 2 . 4 2 9 7 4 9270 2.431360

201 2 . 3 0 3 1 9 3202 2 . 3 0 5 3 4 7203 2 . 3 0 7 4 9 3204 2 . 3 0 9 6 2 8205 2 . 3 1 1 7 5 0206 2 . 3 1 3 8 6 6207 2 . 3 1 5 9 6 7208 2 . 3 1 8 0 6 3209 2 . 3 2 0 1 4 0210 2.322216

191 2 . 2 8 1 0 2 7192 2 . 2 8 3 3 0 0193 2 . 2 8 5 5 4 6194 2 . 2 8 7 7 9 5195 2 . 2 9 0 0 3 3196 2 . 2 9 2 2 5 2197 2 . 2 9 4 4 6 3198 2 . 2 9 6 6 5 9199 2 . 2 9 8 8 4 9200 2.301030



271 2 . 4 3 2 9 6 5272 2 . 4 3 4 5 6 8273 2 . 4 3 6 1 5 9274 2 . 4 3 7 7 4 6275 2 . 4 3 9 3 2 9276 2 . 4 4 0 9 0 7277 2 . 4 4 2 4 7 7278 2 . 4 4 4 0 4 5279 2 . 4 4 5 6 0 1280 2.447156

301 2 . 4 7 8 5 6 1302 2 . 4 8 0 0 0 3303 2 . 4 8 1 4 3 7304 2 . 4 8 2 8 7 1305 2 . 4 8 4 2 9 2306 2 . 4 8 5 7 1 8307 2 . 4 8 7 1 3 4308 2 . 4 8 8 5 4 5309 2 . 4 8 9 9 5 6310 2.491361

311 2 . 4 9 2 7 6 1312 2 . 4 9 4 1 5 3313 2 . 4 9 5 5 4 3314 2 . 4 9 6 9 2 9315 2 . 4 9 8 3 0 6316 2 . 4 9 9 6 8 6317 2 . 5 0 1 0 2 9318 2 . 5 0 2 4 2 3319 2 . 5 0 3 7 8 6320 2.505150

321 2 . 5 0 6 4 9 6322 2 . 5 0 7 8 5 3323 2 . 5 0 9 1 9 9324 2 . 5 1 0 5 4 0325 2 . 5 1 1 8 8 3326 2 . 5 1 3 2 1 7327 2 . 5 1 4 5 4 4328 2 . 5 1 5 8 7 0329 2 . 5 1 7 1 9 0330 2.518509

331 2 . 5 1 9 8 2 0332 2 . 5 2 1 1 3 7333 2 . 5 2 2 4 3 6334 2 . 5 2 3 7 4 3335 2 . 5 2 5 0 4 3336 2 . 5 2 6 3 3 6337 2 . 5 2 7 6 0 4338 2 . 5 2 8 9 1 6339 2 . 5 3 0 1 9 2340 2.531478

341 2 . 5 3 2 7 5 0342 2 . 5 3 4 0 2 1343 2 . 5 3 5 2 8 8344 2 . 5 3 6 5 5 5345 2 . 5 3 7 8 1 7346 2 . 5 3 9 0 7 6347 2 . 5 4 0 3 0 9348 2 . 5 4 1 5 7 7349 2 . 5 4 2 8 1 9350 2.544066

351 2 . 5 4 5 3 0 3352 2 . 5 4 6 5 3 9353 2 . 5 4 7 7 5 5354 2 . 5 4 8 9 9 8355 2 . 5 5 0 2 2 4356 2 . 5 5 1 4 4 8357 2 . 5 5 2 6 6 4358 2 . 5 5 3 8 7 9359 2 . 5 5 5 0 9 1360 2.556300

291 2 . 4 6 3 8 8 5292 2 . 4 6 5 3 8 1293 2 . 4 6 6 8 6 7294 2 . 4 6 8 3 4 2295 2 . 4 6 9 8 1 8296 2 . 4 7 1 2 8 6297 2 . 4 7 2 7 4 9298 2 . 4 7 4 2 1 5299 2 . 4 7 5 6 7 0300 2.477120

281 2 . 4 4 8 7 0 2282 2 . 4 5 0 2 4 4283 2 . 4 5 1 7 8 2284 2 . 4 5 3 3 1 4285 2 . 4 5 4 8 4 1286 2 . 4 5 6 3 6 2287 2 . 4 5 7 8 7 6288 2 . 4 5 9 3 9 0289 2 . 4 6 0 8 9 6290 2.462397



361 2 . 5 5 7 5 0 2362 2 . 5 5 8 6 9 6363 2 . 5 5 9 8 9 8364 2 . 5 6 1 0 9 9365 2 . 5 6 2 2 9 1366 2 . 5 6 3 4 7 2367 2 . 5 6 4 6 4 8368 2 . 5 6 5 8 4 7369 2 . 5 6 7 0 2 0370 2.568196

391 2 . 5 9 2 1 7 5392 2 . 5 9 3 2 8 2393 2 . 5 9 4 3 8 7394 2 . 5 9 5 4 9 3395 2 . 5 9 6 5 9 6396 2 . 5 9 7 6 8 9397 2 . 5 9 8 7 7 3398 2 . 5 9 9 8 7 9399 2 . 6 0 0 9 6 7400 2.602060

401 2 . 6 0 3 1 3 7402 2 . 6 0 4 2 2 3403 2 . 6 0 5 3 0 4404 2 . 6 0 6 3 7 7405 2 . 6 0 7 0 5 0406 2 . 6 0 8 5 2 3407 2 . 6 0 9 5 8 5408 2 . 6 1 0 6 5 8409 2 . 6 1 1 7 1 8410 2.612780

411 2 . 6 1 3 8 3 6412 2 . 6 1 4 8 9 6413 2 . 6 1 5 9 4 4414 2 . 6 1 6 9 9 7415 2 . 6 1 8 0 4 7416 2 . 6 1 9 0 9 3417 2 . 6 2 0 1 3 5418 2 . 6 2 1 1 7 0419 2 . 6 2 2 2 1 1420 2.623246

421 2 . 6 2 4 2 7 7422 2 . 6 2 5 3 0 8423 2 . 6 2 6 3 3 4424 2 . 6 2 7 3 6 3425 2 . 6 2 8 3 8 8426 2 . 6 2 9 4 0 4427 2 . 6 3 0 4 1 8428 2 . 6 3 1 4 3 6429 2 . 6 3 2 4 5 2430 2.633465

431 2 . 6 3 4 4 7 0432 2 . 6 3 5 4 8 0433 2 . 6 3 6 4 7 0434 2 . 6 3 7 4 8 7435 2 . 6 3 8 4 8 7436 2 . 6 3 9 4 8 4437 2 . 6 4 0 4 7 8438 2 . 6 4 1 4 7 1439 2 . 6 4 2 4 5 5440 2.643449

441 2 . 6 4 4 4 3 2442 2 . 6 4 5 4 2 1443 2 . 6 4 6 3 9 0444 2 . 6 4 7 3 7 6445 2 . 6 4 8 3 5 8446 2 . 6 4 9 3 3 0447 2 . 6 5 0 3 0 5448 2 . 6 5 1 2 7 6449 2 . 6 5 2 2 4 0450 2.653210

381 2 . 5 8 0 9 1 9382 2 . 5 8 2 0 5 7383 2 . 5 8 3 1 8 3384 2 . 5 8 4 3 3 0385 2 . 5 8 5 4 5 5386 2 . 5 8 6 5 7 6387 2 . 5 8 7 7 0 5388 2 . 5 8 8 8 2 5389 2 . 5 8 9 9 4 3390 2.591063

371 2 . 5 6 9 3 6 9372 2 . 5 7 0 5 4 1373 2 . 5 7 1 6 8 8374 2 . 5 7 2 8 6 7375 2 . 5 7 4 0 3 0376 2 . 5 7 5 1 8 4377 2 . 5 7 6 3 4 0378 2 . 5 7 7 4 8 6379 2 . 5 7 8 6 3 2380 2.579781



451 2 . 6 5 4 1 6 9452 2 . 6 5 5 1 3 2453 2 . 6 5 6 0 9 3454 2 . 6 5 7 0 5 3455 2 . 6 5 8 0 0 9456 2 . 6 5 8 9 6 1457 2 . 6 5 9 9 0 3458 2 . 6 6 0 8 6 2459 2 . 6 6 1 8 0 8460 2.662757

481 2 . 6 8 2 1 3 9482 2 . 6 8 3 0 4 2483 2 . 6 8 3 9 4 3484 2 . 6 8 4 8 3 8485 2 . 6 8 5 7 3 5486 2 . 6 8 6 6 3 0487 2 . 6 8 7 5 1 8488 2 . 6 8 8 4 1 2489 2 . 6 8 9 3 0 7490 2.690192

491 2 . 6 9 1 0 7 7492 2 . 6 9 1 9 6 0493 2 . 6 9 2 8 4 5494 2 . 6 9 3 7 2 4495 2 . 6 9 4 5 9 9496 2 . 6 9 5 4 8 1497 2 . 6 9 6 3 5 0498 2 . 6 9 7 2 2 7499 2 . 6 9 8 0 9 9500 2.698970

501 2 . 6 9 9 8 3 3502 2 . 7 0 0 6 9 7503 2 . 7 0 1 5 5 6504 2 . 7 0 2 4 2 6505 2 . 7 0 3 2 8 7506 2 . 7 0 4 1 4 6507 2 . 7 0 5 0 0 6508 2 . 7 0 5 8 5 9509 2 . 7 0 6 7 1 4510 2.707568

511 2 . 7 0 8 4 1 7512 2 . 7 0 9 2 7 0513 2 . 7 1 0 1 1 1514 2 . 7 1 0 9 6 0515 2 . 7 1 1 8 0 6516 2 . 7 1 2 6 4 5517 2 . 7 1 3 4 8 3518 2 . 7 1 4 3 2 2519 2 . 7 1 5 1 6 6520 2.716003

521 2 . 7 1 6 8 3 4522 2 . 7 1 7 6 6 7523 2 . 7 1 8 4 9 3524 2 . 7 1 9 3 2 7525 2 . 7 2 0 1 5 6526 2 . 7 2 0 9 8 1527 2 . 7 2 1 8 0 9528 2 . 7 2 2 6 2 9529 2 . 7 2 3 4 5 4530 2.724273

531 2 . 7 2 5 0 8 8532 2 . 7 2 5 9 0 7533 2 . 7 2 6 7 2 3534 2 . 7 2 7 5 3 8535 2 . 7 2 8 3 4 6536 2 . 7 2 9 1 6 3537 2 . 7 2 9 9 6 9538 2 . 7 3 0 7 7 9539 2 . 7 3 1 5 8 1540 2.732390

471 2 . 6 7 3 0 1 9472 2 . 6 7 3 9 3 8473 2 . 6 7 4 8 5 4474 2 . 6 7 5 7 7 6475 2 . 6 7 6 6 9 1476 2 . 6 7 7 6 0 4477 2 . 6 7 8 5 1 3478 2 . 6 7 9 4 2 0479 2 . 6 8 0 3 3 1480 2.681240

461 2 . 6 6 3 6 9 5462 2 . 6 6 4 6 3 5463 2 . 6 6 5 5 7 0464 2 . 6 6 6 5 1 7465 2 . 6 6 7 4 5 1466 2 . 6 6 8 3 8 0467 2 . 6 6 9 3 0 4468 2 . 6 7 0 2 4 3469 2 . 6 7 1 1 6 9470 2.672094



541 2 . 7 3 3 1 9 5542 2 . 7 3 3 9 9 5543 2 . 7 3 4 7 8 6544 2 . 7 3 5 5 9 8545 2 . 7 3 6 3 9 4546 2 . 7 3 7 1 8 9547 2 . 7 3 7 9 8 1548 2 . 7 3 8 7 7 6549 2 . 7 3 9 5 6 2550 2.740359

571 2 . 7 5 6 6 1 8572 2 . 7 5 7 3 9 2573 2 . 7 5 8 1 4 7574 2 . 7 5 8 9 0 6575 2 . 7 5 9 6 6 7576 2 . 7 6 0 4 2 0577 2 . 7 6 1 1 6 2578 2 . 7 6 1 2 9 6579 2 . 7 6 2 6 6 6580 2.763427

581 2 . 7 6 4 1 7 3582 2 . 7 6 4 9 1 5583 2 . 7 6 5 6 6 2584 2 . 7 6 6 4 1 1585 2 . 7 6 7 1 5 3586 2 . 7 6 7 8 9 7587 2 . 7 6 8 6 2 8588 2 . 7 6 9 3 7 2589 2 . 7 7 0 1 1 2590 2.770848

591 2 . 7 7 1 5 8 3592 2 . 7 7 2 3 1 6593 2 . 7 7 3 0 4 7594 2 . 7 7 3 7 7 9595 2 . 7 7 4 5 1 4596 2 . 7 7 5 2 4 5597 2 . 7 7 5 9 6 9598 2 . 7 7 6 7 0 0599 2 . 7 7 7 4 1 9600 2.778150

601 2 . 7 7 8 8 6 6602 2 . 7 7 9 5 9 1603 2 . 7 8 0 3 1 3604 2 . 7 8 1 0 3 3605 2 . 7 8 1 7 4 8606 2 . 7 8 2 4 6 7607 2 . 7 8 3 1 7 5608 2 . 7 8 3 9 0 1609 2 . 7 8 4 6 1 3610 2.785322

611 2 . 7 8 6 0 3 7612 2 . 7 8 6 7 4 8613 2 . 7 8 7 4 4 9614 2 . 7 8 8 1 6 4615 2 . 7 8 8 8 7 0616 2 . 7 8 9 5 7 5617 2 . 7 9 0 2 7 2618 2 . 7 9 0 9 8 6619 2 . 7 9 1 6 8 7620 2.792391

621 2 . 7 9 3 0 8 7622 2 . 7 9 3 7 9 1623 2 . 7 9 4 4 8 4624 2 . 7 9 5 1 8 3625 2 . 7 9 5 8 8 0626 2 . 7 9 6 5 7 3627 2 . 7 9 7 2 6 0628 2 . 7 9 7 9 5 9629 2 . 7 9 8 6 4 4630 2.799336

561 2 . 7 4 8 9 5 7562 2 . 7 4 9 7 3 2563 2 . 7 5 0 4 9 3564 2 . 7 5 1 2 7 4565 2 . 7 5 2 0 4 2566 2 . 7 5 2 8 1 2567 2 . 7 5 3 5 7 6568 2 . 7 5 4 3 4 4569 2 . 7 5 5 1 0 7570 2.755871

551 2 . 7 4 1 1 4 8552 2 . 7 4 1 9 3 7553 2 . 7 4 2 7 2 2554 2 . 7 4 3 5 0 7555 2 . 7 4 4 2 8 6556 2 . 7 4 5 0 7 5557 2 . 7 4 5 8 4 7558 2 . 7 4 6 6 3 1559 2 . 7 4 7 4 0 8560 2.748186



631 2 . 8 0 0 0 2 5632 2 . 8 0 0 7 1 6633 2 . 8 0 1 3 9 8634 2 . 8 0 2 0 5 9635 2 . 8 0 2 7 6 9636 2 . 8 0 3 4 5 3637 2 . 8 0 4 1 3 5638 2 . 8 0 4 8 1 6639 2 . 8 0 5 4 9 4640 2.806180

661 2 . 8 2 0 1 9 0662 2 . 8 2 0 8 5 0663 2 . 8 2 1 5 1 1664 2 . 8 2 2 1 6 7665 2 . 8 2 2 8 1 7666 2 . 8 2 3 4 6 6667 2 . 8 2 4 1 2 4668 2 . 8 2 4 7 7 3669 2 . 8 2 5 4 2 0670 2.826073

671 2 . 8 2 6 7 1 1672 2 . 8 2 7 3 6 6673 2 . 8 2 8 0 0 3674 2 . 8 2 8 6 3 4675 2 . 8 2 9 3 0 0676 2 . 8 2 9 9 4 6677 2 . 8 3 0 5 7 9678 2 . 8 3 1 2 2 2679 2 . 8 3 1 8 6 1680 2.832508

681 2 . 8 3 3 1 4 3682 2 . 8 3 3 7 8 0683 2 . 8 3 4 4 1 2684 2 . 8 3 5 0 5 1685 2 . 8 3 5 6 8 6686 2 . 8 3 6 3 1 8687 2 . 8 3 6 9 5 2688 2 . 8 3 7 5 8 5689 2 . 8 3 8 2 1 6690 2.838847

691 2 . 8 3 9 4 7 2692 2 . 8 4 0 1 0 6693 2 . 8 4 0 7 2 5694 2 . 8 4 1 3 3 9695 2 . 8 4 1 9 8 5696 2 . 8 4 2 6 0 7697 2 . 8 4 3 2 2 8698 2 . 8 4 3 8 4 9699 2 . 8 4 4 4 7 0700 2.845096

701 2 . 8 4 5 7 1 2702 2 . 8 4 6 3 3 3703 2 . 8 4 6 9 4 7704 2 . 8 4 7 5 6 9705 2 . 8 4 8 1 8 4706 2 . 8 4 8 7 8 5707 2 . 8 4 9 4 1 3708 2 . 8 5 0 0 2 8709 2 . 8 5 0 6 4 2710 2.851254

711 2 . 8 5 1 8 6 6712 2 . 8 5 2 4 7 8713 2 . 8 5 3 0 8 8714 2 . 8 5 3 6 9 4715 2 . 8 5 4 3 0 2716 2 . 8 5 4 9 0 9717 2 . 8 5 5 5 1 0718 2 . 8 5 6 1 2 1719 2 . 8 5 6 7 2 4720 2.857330

651 2 . 8 1 3 5 7 7652 2 . 8 1 4 2 4 7653 2 . 8 1 4 9 0 1654 2 . 8 1 5 5 7 4655 2 . 8 1 6 2 3 7656 2 . 8 1 6 9 0 0657 2 . 8 1 7 5 6 1658 2 . 8 1 8 2 2 0659 2 . 8 1 8 8 7 7660 2.819539

641 2 . 8 0 6 8 5 5642 2 . 8 0 7 5 2 6643 2 . 8 0 8 2 0 3644 2 . 8 0 8 8 8 3645 2 . 8 0 9 5 5 5646 2 . 8 1 0 2 2 9647 2 . 8 1 0 8 9 5648 2 . 8 1 1 5 7 0649 2 . 8 1 2 2 3 7650 2.812913



721 2 . 8 5 7 9 3 2722 2 . 8 5 8 5 3 2723 2 . 8 5 9 1 3 2724 2 . 8 5 9 7 2 6725 2 . 8 6 0 3 3 7726 2 . 8 6 0 9 2 8727 2 . 8 6 1 5 3 0728 2 . 8 6 2 1 2 9729 2 . 8 6 2 7 2 0730 2.863321

751 2 . 8 7 5 6 3 4752 2 . 8 7 6 2 1 4753 2 . 8 7 6 7 8 7754 2 . 8 7 7 3 7 0755 2 . 8 7 7 9 4 3756 2 . 8 7 8 5 1 6757 2 . 8 7 9 0 8 7758 2 . 8 7 9 6 6 2759 2 . 8 8 0 2 3 6760 2.880811

761 2 . 8 8 1 3 7 7762 2 . 8 8 1 9 4 9763 2 . 8 8 2 5 2 0764 2 . 8 8 3 0 8 7765 2 . 8 8 3 6 5 8766 2 . 8 8 4 2 1 3767 2 . 8 8 4 7 9 1768 2 . 8 8 5 3 6 0769 2 . 8 8 5 9 2 1770 2.886485

771 2 . 8 8 7 0 5 0772 2 . 8 8 7 6 0 6773 2 . 8 8 8 1 7 4774 2 . 8 8 8 7 3 5775 2 . 8 8 9 3 0 1776 2 . 8 8 9 8 5 5777 2 . 8 9 0 4 1 2778 2 . 8 9 0 9 7 3779 2 . 8 9 1 5 3 1780 2.892093

781 2 . 8 9 2 6 4 3782 2 . 8 9 3 2 0 5783 2 . 8 9 3 7 5 7784 2 . 8 9 4 3 1 2785 2 . 8 9 4 8 6 9786 2 . 8 9 5 4 1 7787 2 . 8 9 5 9 6 6788 2 . 8 9 6 5 2 3789 2 . 8 9 7 0 7 1790 2.897626

791 2 . 8 9 8 1 6 8792 2 . 8 9 8 7 1 9793 2 . 8 9 9 2 6 5794 2 . 8 9 9 8 0 3795 2 . 9 0 0 3 6 3796 2 . 9 0 0 9 0 9797 2 . 9 0 1 4 5 0798 2 . 9 0 1 9 9 7799 2 . 9 0 2 5 4 2800 2.903090

801 2 . 9 0 3 6 2 8802 2 . 9 0 4 1 6 7803 2 . 9 0 4 7 1 0804 2 . 9 0 5 2 5 3805 2 . 9 0 5 7 9 3806 2 . 9 0 6 3 3 4807 2 . 9 0 6 8 6 9808 2 . 9 0 7 4 0 7809 2 . 9 0 7 9 4 4810 2.908480

741 2 . 8 6 9 8 1 4742 2 . 8 7 0 3 9 9743 2 . 8 7 0 9 8 1744 2 . 8 7 1 5 7 1745 2 . 8 7 2 1 5 5746 2 . 8 7 2 7 1 8747 2 . 8 7 3 3 1 7748 2 . 8 7 3 8 9 7749 2 . 8 7 4 4 7 2750 2.875060

731 2 . 8 6 3 9 1 3732 2 . 8 6 4 5 0 2733 2 . 8 6 5 0 9 6734 2 . 8 6 5 6 7 8735 2 . 8 6 6 2 8 2736 2 . 8 6 6 8 7 7737 2 . 8 6 7 4 6 2738 2 . 8 6 8 0 5 0739 2 . 8 6 8 6 4 1740 2.869226



811 2 . 9 0 9 0 1 7812 2 . 9 0 9 5 5 3813 2 . 9 1 0 0 8 5814 2 . 9 1 0 6 1 5815 2 . 9 1 1 1 5 7816 2 . 9 1 1 6 8 8817 2 . 9 1 2 2 1 6818 2 . 9 1 2 7 4 8819 2 . 9 1 3 2 7 9820 2.913810

841 2 . 9 2 4 7 9 4842 2 . 9 2 5 3 0 7843 2 . 9 2 5 8 2 2844 2 . 9 2 6 3 3 8845 2 . 9 2 6 8 5 6846 2 . 9 2 7 3 6 4847 2 . 9 2 7 8 7 4848 2 . 9 2 8 3 9 3849 2 . 9 2 8 9 0 2850 2.929418

851 2 . 9 2 9 9 2 3852 2 . 9 3 0 4 3 4853 2 . 9 3 0 9 4 5854 2 . 9 3 1 4 4 8855 2 . 9 3 1 9 6 1856 2 . 9 3 2 4 6 6857 2 . 9 3 2 9 7 7858 2 . 9 3 3 4 8 2859 2 . 9 3 3 9 8 9860 2.934495

861 2 . 9 3 4 9 9 6862 2 . 9 3 5 5 0 0863 2 . 9 3 6 0 0 5864 2 . 9 3 6 5 1 0865 2 . 9 3 7 0 1 6866 2 . 9 3 7 5 0 0867 2 . 9 3 8 0 1 6868 2 . 9 3 8 5 1 7869 2 . 9 3 9 0 1 5870 2.939517

871 2 . 9 4 0 0 1 6872 2 . 9 4 0 5 1 4873 2 . 9 4 1 0 0 5874 2 . 9 4 1 5 0 8875 2 . 9 4 2 0 0 6876 2 . 9 4 2 5 0 1877 2 . 9 4 2 9 9 3878 2 . 9 4 3 4 8 5879 2 . 9 4 3 9 8 7880 2.944479

881 2 . 9 4 4 9 7 2882 2 . 9 4 5 4 6 2883 2 . 9 4 5 9 5 5884 2 . 9 4 6 4 5 1885 2 . 9 4 6 9 3 8886 2 . 9 4 7 4 2 0887 2 . 9 4 7 9 1 8888 2 . 9 4 8 4 0 6889 2 . 9 4 8 8 9 5890 2.949388

891 2 . 9 4 9 8 6 9892 2 . 9 5 0 3 6 0893 2 . 9 5 0 8 4 5894 2 . 9 5 1 3 3 5895 2 . 9 5 1 8 1 9896 2 . 9 5 2 3 0 6897 2 . 9 5 2 7 9 0898 2 . 9 5 3 2 7 0899 2 . 9 5 3 7 5 8900 2.954240

831 2 . 9 1 9 5 9 7832 2 . 9 2 0 1 2 3833 2 . 9 2 0 6 4 0834 2 . 9 2 1 1 6 5835 2 . 9 2 1 6 8 6836 2 . 9 2 2 2 0 0837 2 . 9 2 2 7 2 1838 2 . 9 2 3 2 4 1839 2 . 9 2 3 7 5 9840 2.924276

821 2 . 9 1 4 3 3 8822 2 . 9 1 4 8 6 6823 2 . 9 1 5 3 9 5824 2 . 9 1 5 9 2 6825 2 . 9 1 6 4 4 9826 2 . 9 1 6 9 7 4827 2 . 9 1 7 4 9 9828 2 . 9 1 8 0 2 7829 2 . 9 1 8 5 5 0830 2.919077



901 2 . 9 5 4 7 2 1902 2 . 9 5 5 1 9 9903 2 . 9 5 5 6 8 1904 2 . 9 5 6 1 6 2905 2 . 9 5 6 6 3 6906 2 . 9 5 7 1 2 3907 2 . 9 5 7 6 0 2908 2 . 9 5 8 0 8 3909 2 . 9 5 8 5 5 7910 2.959039

931 2 . 9 6 8 9 4 3932 2 . 9 6 9 4 1 0933 2 . 9 6 9 8 8 1934 2 . 9 7 0 3 3 4935 2 . 9 7 0 8 0 7936 2 . 9 7 1 2 7 3937 2 . 9 7 1 7 3 3938 2 . 9 7 2 1 9 9939 2 . 9 7 2 6 6 3940 2.973124

941 2 . 9 7 3 5 8 6942 2 . 9 7 4 0 4 9943 2 . 9 7 4 5 0 7944 2 . 9 7 4 9 6 8945 2 . 9 7 5 4 2 6946 2 . 9 7 5 8 8 4947 2 . 9 7 6 3 4 5948 2 . 9 7 6 8 0 6949 2 . 9 7 7 2 6 4950 2.977721

951 2 . 9 7 8 1 4 9952 2 . 9 7 8 6 3 4953 2 . 9 7 9 0 8 9954 2 . 9 7 9 5 4 3955 2 . 9 7 9 9 9 7956 2 . 9 8 0 4 5 0957 2 . 9 8 0 9 0 6958 2 . 9 8 1 3 6 1959 2 . 9 8 1 8 1 2960 2.982270

961 2 . 9 8 2 7 2 2962 2 . 9 8 3 1 6 9963 2 . 9 8 3 6 1 6964 2 . 9 8 4 0 7 2965 2 . 9 8 4 5 1 6966 2 . 9 8 4 9 7 3967 2 . 9 8 5 4 1 3968 2 . 9 8 5 8 6 8969 2 . 9 8 6 3 1 9970 2.986765

971 2 . 9 8 7 2 1 7972 2 . 9 8 7 6 6 0973 2 . 9 8 8 1 1 1974 2 . 9 8 8 5 4 8975 2 . 9 8 9 0 0 3976 2 . 9 8 9 4 4 2977 2 . 9 8 9 8 7 7978 2 . 9 9 0 3 3 7979 2 . 9 9 0 7 7 7980 2.991222

981 2 . 9 9 1 6 6 4982 2 . 9 9 2 1 0 7983 2 . 9 9 2 5 4 9984 2 . 9 9 2 9 9 0985 2 . 9 9 3 4 3 3986 2 . 9 9 3 8 7 5987 2 . 9 9 4 3 1 0988 2 . 9 9 4 7 5 4989 2 . 9 9 5 1 9 2990 2.995629

921 2 . 9 6 4 2 5 4922 2 . 9 6 4 7 2 5923 2 . 9 6 5 1 9 7924 2 . 9 6 5 6 6 5925 2 . 9 6 6 1 3 6926 2 . 9 6 6 6 0 0927 2 . 9 6 7 0 7 6928 2 . 9 6 7 5 4 7929 2 . 9 6 8 0 1 3930 2.968481

911 2 . 9 5 9 5 1 6912 2 . 9 5 9 9 9 1913 2 . 9 6 0 4 6 6914 2 . 9 6 0 9 3 3915 2 . 9 6 1 4 1 2916 2 . 9 6 1 8 9 2917 2 . 9 6 2 3 6 3918 2 . 9 6 2 8 3 8919 2 . 9 6 3 3 1 3920 2.963787



991 2 . 9 9 6 0 7 2992 2 . 9 9 6 5 1 1993 2 . 9 9 6 9 4 0994 2 . 9 9 7 3 8 0995 2 . 9 9 7 8 1 9996 2 . 9 9 8 2 5 7997 2 . 9 9 8 6 9 2998 2 . 9 9 9 1 2 9999 2 . 9 9 9 5 5 61000 3

1.01 0.0043171.1 0.041389π 0.497323 (using π = 22⁄7, about 3.14285)

Table 2: Small Logarithms


Perfect squares

Perfect squares are squares of any positive integer:

0 4 9 16 25 36 49 64 81 100 121 144These numbers are the squares of the numbers 0 - 12.

The closest perfect square is the number closest and lower than a given number. For example, the closest perfect square to 58 is 49.

Table 3: Multiplication Chart

123456789

101112

1123456789

101112

22468

1012141618202224

3369

121518212427303336

448

12162024283236404448

55

1015202530354045505560

66

1218243036424854606672

77

1421283542495663707784

88

1624324048566472808896

99

18273645546372819099

118

10102030405060708090

100110120

11112233445566778899

110121132

121224364860728496

118120132144


Table 4: Powers of 1/2

1) .52) .253) .1254) 6.25 x 10-25) 3.125 x 10-26) 1.5625 x 10-27) 7.8125 x 10-38) 3.90625 x 10-39) 1.953125 x 10-310) 9.765625 x 10-411) 4.8828125 x 10-412) 2.44140625 x 10-413) 1.220703125 x 10-414) 6.103515625 x 10-515) 3.0517578125 x 10-516) 1.52587890625 x 10-517) 7.62939453125 x 10-618) 3.814697265625 x 10-619) 1.90734863228125 x 10-620) 9.5367431640625 x 10-7

- from my notebooks


About Me ... a slice of my life

There are not enough hours in the day for me - or, at least,

I do not feel the need to manage the time better.

Jim [email protected]

November 29, 2009


Source References

Scientific Notation:Scientific notation. (2009, November 19). In Wikipedia, The Free Encyclopedia. Retrieved 00:40, November 25, 2009, from http://en.wikipedia.org/w/index.php?title=Scientific_notation&oldid=326668059

Square Roots:Methods of computing square roots. (2009, November 19). In Wikipedia, The Free Encyclopedia. Retrieved 22:57, November 24, 2009, from http://en.wikipedia.org/w/index.php?title=Methods_of_computing_square_roots&oldid=326774986

Logarithm:Logarithm. (2009, November 22). In Wikipedia, The Free Encyclopedia. Retrieved 12:44, November 27, 2009, from http://en.wikipedia.org/w/index.php?title=Logarithm&oldid=327308217

Euler’s Method:Column: “How Euler Did It”Subtitle “Finding logarithms by hand”Source document “How Euler Did It 21 logs.PDF”Author: Ed Sandifer, July 2005Hosted on the MAA website.Date retrieved from the MAA website: 11/26/2009MAA = The Mathematical Association of AmericaSource document URL:http://www.maa.org/editorial/euler/How%20Euler%20Did%20It%2021%20logs%20.pdf

My thanks to the many contributors to Wikipedia, and to Ed Sandifer, for their generous contributions to the online community.

- Jim Hall, 2009


Documents

1000 logs book