Why numbering system

Find the next lower power of two equal to or less than the octal value. Repeat this until all octal value equals 0. Thus we have an easy to memorize chart:. Now line these bits up: ,,, or and you have your binary number. Numbering Systems Numbering systems are just symbolic ways to represent the numbers.

In number systems, its often helpful to think of recurring sets , where a set of values is repeated over and over again. Considering the decimal number system, it has a set of values which range from 0 to 9. This basic set is repeated over and over, creating large numbers. Note how the set of values 0 to 9 is repeated, and for each repeat, the column to the left is incremented from 0 to 1, then 2.

Each increase in value occurs, till the value of the largest number in the set is reached 9 , at which stage the next value is the smallest in the set 0 and a new value is generated in the left column ie, the next value after 9 is Base Values The base value of a number system is the number of different values the set has before repeating itself. For example, decimal has a base of ten values, 0 to 9. Weighting Factor The weighting factor is the multiplier value applied to each column position of the number.

For instance, decimal has a weighting factor of TEN, in that each column to the left indicates a multiplication value increase of 10 over the previous column on the right, ie; each column move to the left increases in a multiply factor of The set values used in decimal are.

The digit or column on the left has the greatest value, whilst the digit on the right has the least value. When doing a calculation, if the highest digit 9 is exceeded, a carry occurs which is transferred to the next column to the left. Positional Values [Units, Tens, Hundreds, Thousands etc Columns] We probably got taught at school about positional values, in that columns represent powers of This is expressed to us as columns of ones 0 - 9 , tens groups of 10 , hundreds groups of and so on.

The values are,. Columns are used in the same way as in the decimal system, in that the left most column is used to represent the greatest value. As we have seen in the decimal system, the values in the set 0 and 1 repeat, in both the vertical and horizontal directions.

In a computer, a binary variable capable of storing a binary value 0 or 1 is called a BIT. In the decimal system, columns represented multiplication values of That was because there were 10 values 0 - 9 in the set. Example 8 0.

Last position in an octal number represents a x power of the base 8. Example 8 x where x represents the last position - 1. Each position in a hexadecimal number represents a 0 power of the base Example, 16 0.

Last position in a hexadecimal number represents a x power of the base Example 16 x where x represents the last position - 1. Step 2: Multiply each digit of the given number, starting from the rightmost digit, with the exponents of the base. The exponents should start with 0 and increase by 1 every time as we move from right to left.

Since the base is 2 here, we multiply the digits of the given number by 2 0 , 2 1 , 2 2 , and so on from right to left.

Here, the sum is the equivalent number in the decimal number system of the given number. Or, we can use the following steps to make this process simplified. The steps are shown on how to convert a number from the decimal system to the octal system.

Step 1: Identify the base of the required number. Since we have to convert the given number into the octal system, the base of the required number is 8.

Step 2: Divide the given number by the base of the required number and note down the quotient and the remainder in the quotient-remainder form. Repeat this process dividing the quotient again by the base until we get the quotient less than the base. Step 3: The given number in the octal number system is obtained just by reading all the remainders and the last quotient from bottom to top.

Step 1: Convert this number to the decimal number system as explained in the above process. Step 2: Convert the above number which is in the decimal system , into the required number system. Here, we have to convert 10 into the hexadecimal system using the above-mentioned process. It should be noted that in the hexadecimal system, the numbers 11 and 12 are written as B and C respectively. Solution: 10 is in the decimal system.

We divide by 2 and note down the quotient and the remainder. We will repeat this process for every quotient until we get a quotient that is less than 2. The equivalent number in the binary system is obtained by reading all the remainders and just the last quotient from bottom to top as shown above. Solution: 5BC 16 is in the hexadecimal system. So we get the equivalent number in the decimal system using the following process:.