Number Base System
Number Base System
In todays blog we will learn about the number base system useful concept in CAT quantitative exam. number base system like binary and hexadecimal seem a bit strange at first. The key is understanding how different number base systems “tick over” like an odometer when they are full. Base 10, our decimal system, “ticks over” when it gets 10 items, creating a new digit. We wait 60 seconds before “ticking over” to a new minute. Hex and binary are similar, but tick over every 16 and 2 items, respectively
Decimal system:
In our customary baseten system, we have digits for the numbers zero through nine. We do not have a singledigit numeral for “ten”. (The Romans did, in their character “X“.) Yes, we write “10“, but this stands for “1 ten and 0 ones”. This is two digits; we have no single solitary digit that stands for “ten”.
Instead, when we need to count to one more than nine, we zero out the ones column and add one to the tens column. When we get too big in the tens column — when we need one more than nine tens and nine ones (“99“), we zero out the tens and ones columns, and add one to the tentimesten, or hundreds, column. The next column is the tentimestentimesten, or thousands, column. And so forth, with each bigger column being ten times larger than the one before. We place digits in each column, telling us how many copies of that power of ten we need.
The only reason baseten math seems “natural” and the other bases don’t is that you’ve been doing baseten since you were a child. And (nearly) every civilization has used baseten math probably for the simple reason that we have ten fingers. If instead we lived in a cartoon world, where we would have only four fingers on each hand (count them next time you’re watching TV or reading the comics), then the “natural” base system would likely have been baseeight, or “octal”.
Binary System:
Let’s look at basetwo, or binary, numbers. How would you write, for instance, 12_{10} (“twelve, base ten”) as a binary number? You would have to convert to basetwo columns, the analogue of baseten columns. In base ten, you have columns or “places” for 10^{0} = 1, 10^{1} = 10, 10^{2} = 100, 10^{3} = 1000, and so forth. Similarly in base two, you have columns or “places” for 2^{0} = 1, 2^{1} = 2, 2^{2}= 4, 2^{3} = 8, 2^{4} = 16, and so forth.
The first column in basetwo math is the units column. But only “0” or “1” can go in the units column. When you get to “two”, you find that there is no single solitary digit that stands for “two” in basetwo math. Instead, you put a “1” in the twos column and a “0” in the units column, indicating “1 two and 0 ones”. The baseten “two” (2_{10}) is written in binary as 10_{2}.
A “three” in base two is actually “1 two and 1 one”, so it is written as 11_{2}. “Four” is actually twotimestwo, so we zero out the twos column and the units column, and put a “1” in the fours column; 4_{10} is written in binary form as 100_{2}. Here is a listing of the first few numbers:
Converting binary to decimal system
Converting between binary and decimal numbers is fairly simple, as long as you remember that each digit in the binary number represents a power of two.

Convert 101100101_{2} to the corresponding baseten number.
I will list the digits in order, as they appear in the number they’ve given me. Then, in another row, I’ll count these digits off from the RIGHT, starting with zero:
digits: 
1 
0 
1 
1 
0 
0 
1 
0 
1 
numbering: 
8 
7 
6 
5 
4 
3 
2 
1 
0 
The first row above (labelled “digits”) contains the digits from the binary number; the second row (labelled “numbering”) contains the power of 2 (the base) corresponding to each digit. I will use this listing to convert each digit to the power of two that it represents:
1×2^{8} + 0×2^{7} + 1×2^{6} + 1×2^{5} + 0×2^{4} + 0×2^{3} + 1×2^{2} + 0×2^{1} + 1×2^{0}
= 1×256 + 0×128 + 1×64 + 1×32 + 0×16 + 0×8 + 1×4 + 0×2 + 1×1
= 256 + 64 + 32 + 4 + 1
= 357
Then 101100101_{2} converts to 357_{10}.
Octal system:
An older computerbased number system is “octal”, or base eight. The digits in octal math are 0, 1, 2, 3, 4, 5, 6, and 7. The value “eight” is written as “1 eight and 0ones”, or 10_{8}.
In technical terms, there are very many different computerlanguage protocols for octal, but we’ll just be using the simple mathematical system.
A few NewWorld tribes used base8 numbering systems; they counted by using the eight spaces between their fingers, rather than the ten fingers themselves. The blue natives in the movie “Avatar” used octal because their hands had only four fingers.

Convert 357_{10} to the corresponding baseeight number.
I will do the usual sequential division, this time dividing by 8 at each step:
Once I got to the “5” on top, I had to stop, because 8 doesn’t divide into 5.
Then the corresponding octal number is 545_{8}.
Hexa decimal number system:
As mentioned before, decimal math does not have one single solitary digit that represents the value of “ten”. Instead, we use two digits, a 1 and a 0: “10“. But in hexadecimal math, the columns stand for multiples of sixteen! That is, the first column stands for how many units you have, the second column stands for how many sixteens, the third column stands for how many two hundred fiftysixes (sixteentimessixteens), and so forth.
In base ten, we had digits 0 through 9. In base eight, we had digits 0 through 7. In base 4, we had digits 0 through 3. In any base system, you will have digits 0 through onelessthanyourbase. This means that, in hexadecimal, we need to have “digits” 0 through 15. To do this, we would need single solitary digits that stand for the values of “ten”, “eleven”, “twelve”, “thirteen”, “fourteen”, and “fifteen”. But we don’t. So, instead, we use letters. That is, counting in hexadecimal, the sixteen “numerals” are:
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F
In other words, A is “ten” in “regular” numbers, B is “eleven”, C is “twelve”, D is “thirteen”, E is “fourteen”, and “F” is fifteen. It is this use of letters for digits that makes hexadecimal numbers look so odd at first. But the conversions work in the usual manner.This system contains sixteen symbols to represent its numbers, Which are: {0,1, 2, 3 ,4,5,6,7,8,9,A,B,C,D,E,F} Where. A represent the value (10)10, B represent (11)10, C represent (12)10, D represent (13)10, E represent (14)10, F represent (15)10. ASCII Code.
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