Pi-Journal #24

Pi

Pi is a name given to the ratio of the circumference of a circle to the diameter. That means, for any circle, you can divide the circumference (the distance around the circle) by the diameter and always get exactly the same number. It doesn't matter how big or small the circle is, Pi remains the same. Pi is often written using the symbol and is pronounced "pie", just like the dessert.

A Brief History of Pi

Ancient civilizations knew that there was a fixed ratio of circumference to diameter that was approximately equal to three. The Greeks refined the process and Archimedes is credited with the first theoretical calculation of Pi.

In 1761 Lambert proved that Pi was irrational, that is, that it can't be written as a ratio of integer numbers.
In 1882 Lindeman proved that Pi was transcendental, that is, that Pi is not the root of any algebraic equation with rational coefficients. This discovery proved that you can't "square a circle", which was a problem that occupied many mathematicians up to that time.


How many digits are there? Does it ever end?


Because Pi is known to be an irrational number it means that the digits never end or repeat in any known way. But calculating the digits of Pi has proven to be an fascination for mathematicians throughout history. Some spent their lives calculating the digits of Pi, but until computers, less than 1,000 digits had been calculated. In 1949, a computer calculated 2,000 digits and the race was on. Millions of digits have been calculated, with the record held (as of September 1999) by a supercomputer at the University of Tokyo that calculated 206,158,430,000 digits. (first 1,000 digits)


Amount of Pi

Archimedes calculated that Pi was between 3 10/71 and 3 1/7 (also written 223/71 < < 22/7 ). 22/7 is still a good approximation. 355/113 is a better one.

Magnitude Estimate-Journal #23

Magnitude Estimate

A rough estimate ,it tells which place value the answer will be in tens,hundreds,thousends,etc.

Example:
Find magnitude estimate for:
           56*32
Round 56 to 60 and 32 to 30
             60*30=1800
The magnitude estimate
                                                                                 will be in the thousends

Range,Mean,Median,Mode Definition-Journal#22

Range,Mean,Median,Mode Definition

Range:The difference between the highest and the lowest numbers in a set of numbers

Mean:The mean, or average, of a set of numbers is found by dividing the sum of the numbers by the amount of numbers added.

Median:The middle number when numbers are arranged in order. If there are two middle numbers, the median is the average of the two.

Mode:The number or numbers that occur most often in a set of numbers

Outlier-Journal#21

Outlier Numbers

Outlier. For a set of numerical data, any value that is markedly smaller or larger than other values. Mathematically, outliers are considered any number that is more than 1.5 times the interquartile range away from the median. For example, in the data set {3, 5, 4, 4, 6, 2, 25, 5, 6, 2} the value of 25 is an outlier.
Basically the one that doesn't belong.
Basically, the outlier is the number that:

stands out.

So let's say we have these numbers: 2,3,1,17. 17 is our outlier. Why? Because, it's obvious.17 is separate away from the all the other numbers. An outlier is an element of a data set that distinctly stands out from the rest of the data.

Lattice Method Multiplication-Journal#20

Lattice Method Multiplication

A method ,or algorithm,for  multiplying that uses the multiplication of basic facts one place at a time and then records the answers to get the products.


                                                    For Example:14*56=784

Or:    247*38=9386

Prisms-Journal#19

Prisms

A Prism has the same cross -section all along its length.
A cross- section is the shape we get when cutting straight across an object.

The cross section of this object is a triangle .It has the same cross section all along its length and so it's a Triangular Prism.










These are all Prisms:

             Cube                                                                              Cross- Section

                                                                                                                                                                                                                                                                                                                                 

                                                Cross-Section

Triangular Prism                                               Cross -Section                       

                                                                                         

Painted Cube Problem-Journal #18

  Painted Cube Problem:      
                                       

                               


When a 3*3*3 cube made up o 1*1*1 is dipped into red paint ,investigate the number of smaller cubes:
 with 3 faces ,2 faces,1 face,0 face painted red as a fuction of edge length of larger cube.


Edge length                   Number of smaller cubes with 3 faces painted

        X                                                                           Y
        1                                                                            0
        2                                                                            8
        3                                                                            8
        4                                                                            8
        5                                                                            8   
                                                                                                       

Edge of length                                   Number of smaller cubes with 2 faces painted

       X                                                                              Y                                                     
        1                                                                              0
        2                                                                              0                                                       
        3                                                                             12
        4                                                                             24
        5                                                                             36
        6                                                                             48

Y=12(X-2),X>1

Edge of length                                       Number of  smaller cubes with 1 face painted

           X                                                                              Y
            1                                                                              0
            2                                                                              0
            3                                                                              6
            4                                                                             24
            5                                                                             54

Y=6(X-2)^2  ,X>1

Edge of length                                     Number of smaller cubes with 0 face painted

           X                                                                              Y
           1                                                                               0
           2                                                                               0
           3                                                                               1
           4                                                                               8
           5                                                                              27

                                                                        
Y=(X-2)^3, X>1                                                                          
                                                                                                                                                                                                                                                                                                                                             
                                                                                             
                                                                                             

                                                                            

Fibonacci Numbers -Journal #17

Fibonacci numbers

The Fibonacci numbers are the numbers in the following Integer Sequence:



0, 1, 1, 2, 3, 5, 8, 13, 21, 21, 34, 55, 89, 144

By definition, the first  two numbers in the Fibonacci sequence are 0 and 1, and each subsequent number is the sum of the previous two.

In mathematical terms, the sequence F_n of Fibonacci numbers is defined as follow:

F(n) = F(n-1) + F(n-2)


with seed values


F(0) = 0  ,  F(1) = 1

A Fibonacci spiral created by drawing circular arcs connecting the opposite corners of squares in the Fibonacci tiling; this one uses squares of sizes 1, 1, 2, 3, 5, 8, 13, 21, and 34.

Fibonacci considers the growth of an idealized (biologically unrealistic) rabbit population, assuming that: a newly born pair of rabbits, one male, one female, are put in a field; rabbits are able to mate at the age of one month so that at the end of its second month a female can produce another pair of rabbits; rabbits never die and a mating pair always produces one new pair (one male, one female) every month from the second month on. The puzzle that Fibonacci posed was: how many pairs will there be in one year?

          

           At the end of the first month, they mate, but there is still only 1 pair.

           At the end of the second month the female produces a new pair, so now there are 2 pairs of rabbits in the field.

·        At the end of the third month, the original female produces a second pair, making 3 pairs in all in the field.

·        At the end of the fourth month, the original female has produced yet another new pair, the female born two months ago produces her first pair also, making 5 pairs
    
      At the end of the nth month, the number of pairs of rabbits is equal to the number of new pairs (which is the number of pairs in month n − 2) plus the number of pairs alive last month (n − 1). This is the nth Fibonacci number.

Roman Numbers-Journal # 16

Roman Numbers : Roman numerals were originated in Ancient Rome. It is based on certain letters which  are given values as numerals. Roman Numerals are widely used nowadays, in clocks, books, numberlists, etc.,. 1 I 2 II 3 III 4 IV 5 V 6 VI 7 VII 8 VIII 9 IX 10 X 11 XI 12 XII 13 XIII 14 XIV 15 XV 16 XVI 17 XVII 18 XVIII 19 XIX 20 XX 21 XXI 22 XXII 23 XXIII 24 XXIV 25 XXV 26 XXVI 27 XXVII 28 XXVIII 29 XXIX 30 XXX 31 XXXI 32 XXXII 33 XXXIII 34 XXXIV 35 XXXV 36 XXXVI 37 XXXVII 38 XXXVIII 39 XXXIX 40 XL 41 XLI 42 XLII 43 XLIII 44 XLIV 45 XLV 46 XLVI 47 XLVII 48 XLVIII 49 XLIX 50 L 51 LI 52 LII 53 LIII 54 LIV 55 LV 56 LVI 57 LVII 58 LVIII 59 LIX 60 LX 61 LXI 62 LXII 63 LXIII 64 LXIV 65 LXV 66 LXVI 67 LXVII 68 LXVIII 69 LXIX 70 LXX 71 LXXI 72 LXXII 73 LXXIII 74 LXXIV 75 LXXV 76 LXXVI 77 LXXVII 78 LXXVIII 79 LXXIX 80 LXXX 81 LXXXI 82 LXXXII 83 LXXXIII 84 LXXXIV 85 LXXXV 86 LXXXVI 87 LXXXVII 88 LXXXVIII 89 LXXXIX 90 XC 91 XCI 92 XCII 93 XCIII 94 XCIV 95 XCV 96 XCVI 97 XCVII 98 XCVIII 99 XCIX 100 C 200 CC ,300 CCC ,400 CD ,500 D,600 DC,700 DCC ,800 DCCC, 900 CM ,1000 M An accurate way to write the roman numbers is to first take the thousanda,hundreda, tens and units. Example: 1999, one thousand is M, nine hundred is CM, ninety is XC, nine is IX. Combine all these: MCMXCIX