Get your BASE right

A main reason for our friction with quant is that we learn numbers... not number systems. Many years ago, when our life was not complex as today, basic representation of numbers was enough to handle our daily transactions. Don’t know when! But some point during our evolution someone felt the need to count and arrange things. May be some CroMagnon dude want to show off saying I caught 'more' fish or I am the 'tallest' in the group... and various number systems were developed to meet the purpose (No! not to abet the CroMagnon show off... for counting and arranging :) ) In this article we will see some interesting stuffs about numbers and number system.
Decimal system (base 10) is a positional number system which uses ten numeric digits (0, 1, 2 ... 9) to create any number we want. In a positional system, position of a digit is used to signify the power of its base that the digit is to be multiplied with. Decimal is a base 10 system means each position in a decimal number is BASEd on a power of 10. So we will start from zero and count till nine once we have one more than 9 rather than having a new numeric digit we say as have a 10 now (1 * 10^{1} + 0 * 10^{0}). The concept of using zero as a number and not merely a numeric symbol is attributed to India.
In a general way, a positional baseb numeral system (with b a natural number greater than 1 known as the radix), b basic symbols (or digits) corresponding to the first b natural numbers including zero are used. To generate the rest of the numerals, the position of the symbol in the figure is used. The symbols in the last position has its own value, and as it moves to the left its value is multiplied by b.To avoid confusion while using different number systems, the base of each individual number may be specified by writing it as a subscript of the number. If no base is mentioned treat it as a decimal system (base 10)
For example, when we write 1875 in decimal system we are saying there is one 10^{3}, eight 10^{2}, seven 10^{1} and five 10^{0}
During your exam if you see some questions like "In some island numbers are represented using base8" don’t lose heart. it just means that in that island there are only 8 symbols ( 0 to 7) and after counting till 7 they will say 10 ( means 1 * 8^1 + 0 * 8^0.. which is equal to 8 in our decimal system). There is a general way to convert any base representation to a decimal number.
Number d_{1}d_{2}d_{3}....d_{n} in base b shall be converted to base 10 as d_{1} * b^{n1 }+ d_{2} + b^{n2 }+ ... + d_{n} * b^{0}
Convert 1432001 in base 5 to a decimal number
(1432001)_{5} = (1 * 5^{6} + 4 * 5^{5} + 3 * 5^{4} + 2 * 5^{3} + 0 * 5^{2} + 0 * 5^{1} + 1 *5^{0})
Now how will we convert a decimal number to another base representation? Before going to the details we know a base b system has b number of symbols to represent it numbers. Like base 2 has 2 (0 and 1), base 3 has three digits (0, 1, 2) and so on. Now a decimal system has 10 symbols (0 to 9) and another base say (base5) has only 5 symbols (0 to 4) so base 5 people don’t understand what is 5,6,7,8 or 9. So when we convert we cannot put any of these symbols to the base 5 number. Here is the trick. We will carry forward for every 5... That is 4 + 1 in base 5 is 10 (one 5 and zero 1). While discussing remainder concept we saw that when we divide a number with another number n, the possible remainder values are 0 to (n1). Means division by 2 can yield only two remainders values, zero or one; division by 3 can give only zero, one or two as remainder and so on.
To convert a decimal number to another base, divide the given number with the base value... And then repeatedly divide the quotient with the base value till the quotient is zero. Write the remainder in each step and then use these remainders as place values and use the last remainder as most significant digit and the first one as the least significant digit.
To convert 461 to base 8
We get 461 = 715(base 8 )
Want to cross check..??? 715(base 8 ) = 7 * 8^2 + 1 * 8 + 5 * 8^0 = 461 :)
Convert 366 to binary (base 2)
Clear...??? Now convert 45.68 to base 2.
For the portion before decimal point we can do the same way we did before and we can get 45 = 101101. Now 0.68, instead of dividing, multiply with the base till you get the fractional part of the result as zero.
0.625 * 2 = 1.25 integer part is 1
0.25 * 2 = 0.50 integer part is 0
0.50 * 2 = 1.00 integer part is 1, fractional part is zero. We can stop :)
0.625 = 0.101
45.625 = 101101.101
How can we convert (101101.101)2 back to decimal...?
101101 the same way we did before...
1 * 2^{5 }+ 0 * 2 ^{4} + 1 * 2^{3} + 1 * 2^{2} + 0 * 2^{1} + 1 * 2^{0} =
32 + 0 + 8 + 4 + 1 = 45.
For decimal part, decrement the power by 1 as we go right from decimal point.
Here, 0.1010 = 1 * 2^{1} + 0 * 2^{2} + 1 * 2^{3} + 0 * 2^{4} = 0.5 + 0.125 = 0.625
101101.101 = 45.625
As an alternate method, in case of finite fractions, move the decimal point to the right end of the number. Count how many places we moved the decimal point. Convert the resulting integer to binary using normal method, and divide by 2^n, where n is the number of places we moved the decimal point.
For 101101.101,
101101101 (we moved decimal point to three places) = 365.
Now divide by 2^{3} = 8.
101101.101 = 365/8 = 45.625
We will see some other systems which were used to represent numbers. May be after reading about them you will fall in love with decimal number system :)
Finger numerals were used by the ancient Greeks, Romans, Europeans of the Middle Ages, and later the Asiatics.
The Mayan number system dates back to the fourth century and they used a base 20 system. Now the reason behind this is said as we have 20 fingers! The Mayan system used a combination of two symbols. A dot (.) was used to represent the units (one through four) and a dash () was used to represent five. For further study: The 360 day calendar also came from the Mayan's who actually used base 18 when dealing with the calendar. Each month contained 20 days with 18 months to a year. This left five days at the end of the year which was a month in itself that was filled with danger and bad luck. In this way, the Mayans had invented the 365 day calendar which revolved around the solar system.
The Egyptians used a written numeration that was changed into hieroglyphic writing, which enabled them to note whole numbers to 1,000,000. It had a decimal base and allowed for the additive principle. In this notation there was a special sign for every power of ten. This hieroglyphic numeration was a written version of a concrete counting system using material objects. To represent a number, the sign for each decimal order was repeated as many times as necessary. To make it easier to read the repeated signs they were placed in groups of two, three, or four and arranged vertically.
The Greek numbering system was uniquely based upon their alphabet. The Greek alphabet came from the Phoenicians around 900 B.C. When the Phoenicians invented the alphabet, it contained about 600 symbols. Those symbols took up too much room, so they eventually narrowed it down to 22 symbols. The Greeks borrowed some of the symbols and made up some of their own. But the Greeks were the first people to have separate symbols, or letters, to represent vowel sounds. Our own word "alphabet" comes from the first two letters, or numbers of the Greek alphabet  "alpha" and "beta." Using the letters of their alphabet enabled them to use these symbols in a more condensed version of their old system, called Attic. The Attic system was similar to other forms of numbering systems of that era. It was based on symbols lined up in rows and took up a lot of space to write. This might not be too bad, except that they were still carving into stone tablets, and the symbols of the alphabet allowed them to stamp values on coins in a smaller, more condensed version.
The Roman numerical system is still used today although the symbols have changed from time to time. The Romans wrote four as IV, I from V and they wrote six as VI, I to V. Today the Roman numerals are used to represent numerical chapters of books or for the main divisions of outlines. The earliest forms of Roman numeral values are:
Another very common number system we use today is base 2 (binary) which is based on two digits 0 and 1 which can be represented easily using electronic states ( ON and OFF). This is the base of all software and electronic gadgets we enjoy today :)
Last one we discuss here is Hexadecimal number system which is base 16. Now we need 16 symbols. We can take 10 from decimal... What about the other 6... ? First 6 Alphabets are included as symbols (A, B, C, D, E and F) to represent 10 to 15. Hexadecimal is commonly used to represent computer memory addresses.
2AF3 = (2 × 16^{3}) + (10 × 16^{2}) + (15 × 16^{1}) + (3 × 16^{0}) = 10995
Bored of theory and history ? try to solve this famous puzzle to refresh your brain cells.. :)
On the island of Knights and Knaves, every inhabitant is either a knight or a knave. Knights always tell the truth. Knaves never tell the truth; any sentence uttered by a knave is false. A stranger came to the island and encountered three inhabitants, A, B, and C. He asked A, "Are you a knight, or a knave?" A mumbled an answer that the stranger could not understand. The stranger then asked B, "What did he say?" B replied, "A said that there is exactly one knight among us." Then C burst out, "Don't believe B, he is lying!" What are B and C?
Happy learning :)
[reference: www.math.wichita.edu]