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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