Number Systems (2.1.1)

Transcription

1 Computer Studies 0 Syllabus Number Systems (..) Representation of numbers in binary. Conversion between decimal and binary, and between binary and hexadecimal. Use of subscripts, 0 and 6 for bases. The decimal number system is a base 0 system; this means that it is made up of 0 basic numbers which are 0,,, 3,, 5, 6, 7, and 9. By using and combining these numbers an infinite list of numbers can be created. In order to represent numbers bigger than 9, tens, hundreds, thousands are used. For instance the decimal number 3 is the total of 300, 0 and. This can be represented as follows: Placement Number 3 Calculation (00 x 3) (0 x ) ( x ) Value Total = In the above table one can see the 0 x in the placement part. Starting from the right side 0 0 is used because it represents the units, then the exponent x is increased by for each number to the left. In computing we ve seen that the binary system is used, so the decimal system cannot be used to store values, so the binary system which is base is used. This means that only 0 and can be used to represent numbers. The following table shows a binary representation of the decimal numbers from 0 to 9. Decimal Binary Decimal Binary It s important to write down the base (in subscript) at the end of each number to show which numeric representation is being used. For example 9 0 is 9 in decimal format, 0 is 0 in binary format. Then there is also the hexadecimal number system which has a base of 6. This is used in computing usually Ms.P.Sullivan Page of 7

2 Computer Studies 0 Syllabus to refer to the computer memory locations. Since hexadecimal has a base of 6, it uses 0-9 for the first 0 digits then the letters A, B, C, D, E and F are used for the remaining 6 digits. Hexadecimal Symbol Decimal Representation A 0 B C D 3 E F 5 Converting from Binary to Decimal A binary number can be converted to decimal by using the placement values, but in this case instead of 0 x, x is used since binary is base. So let s change the number 00 into decimal. Placement 3 0 Number 0 0 Calculation (6 x ) ( x 0) ( x ) ( x 0) ( x ) Value Total =6++ = 0 So the equivalent value of 00 is 0. This involves:. Writing down the placement value on top of each number starting with 0 from the right-most number, then increase the exponent by each number to the left.. Multiply the top value with the binary number. 3. Add all totals together. Ms.P.Sullivan Page of 7

3 Converting from Decimal to Binary Computer Studies 0 Syllabus A decimal number can be converted to binary by using division and writing down only the remainder. The decimal number must be divided by (since binary is base ) and the result along with the remainder is listed down, then the whole number of the result is divided by again and the same process is repeated until at the end is divided by. The remainder is written down (from bottom moving upwards) and the binary number is obtained. So let s convert 6 0 to binary. 6 / = 3 r 0 3 / = r / = 5 r 5 / = r / = r 0 / = 0 r So the equivalent value of 6 0 is 00. This involves:. Divide the original number by to obtain a new whole number and its remainder.. Keep on dividing the whole number by until at the end is divided by. 3. Write down the remainders next to each other starting from the bottom moving upwards. Ms.P.Sullivan Page 3 of 7

4 Converting from Binary to Hexadecimal Computer Studies 0 Syllabus In this case the binary number has to be split in groups of. Each group is then converted into its hexadecimal value by first converting it to decimal. So let s consider (x) (x0) (x0) (x) (x) (x) (x) (x) =9 =5 9F 6 So the equivalent value of 00 is 9F 6. This involves:. Split the binary number in individual groups of.. For each group write the x on top of each number starting from the right (remember to increase the exponent each number to the left). 3. Multiply each binary number with its placement value.. For each group, calculate the total. 5. Write down totals next to each other, if there s a total that is greater than 9, convert it to the appropriate hexadecimal letter. Ms.P.Sullivan Page of 7

5 Computer Studies 0 Syllabus In the case of not having enough numbers for groups of, 0s are added to the front of the smallest number. For instance converting 000 to hexadecimal will be: (x0) (x0) (x) (x) (x) (x0) (x0) (x0) + =3 = 3 6 Converting from Hexadecimal to Binary For this conversion each hexadecimal number is taken on its own and individually converted to binary by using placement values. So let s convert C3 6 to binary. C So the equivalent value of C3 6 is This involves:. Write down each individual number of the hexadecimal number.. Use the binary placement values for each hexadecimal number and list down s under the placement values to obtain the value represented by the hexadecimal number. 3. Write down the binary number as a single number. Ms.P.Sullivan Page 5 of 7

6 Convert from Decimal to Hexadecimal Computer Studies 0 Syllabus To convert from decimal to hexadecimal, the decimal number must be divided by 6 until there s nothing more to divide with. Then the remainder is written down bottom up, converting any number greater than 9 to a hexadecimal letter where required. So 65 0 in hexadecimal would be worked out as follows: 65 / 6 = 9 r 9 / 6 = r 3 / 6 = 0 r So the equivalent value of 65 0 is D 6.This involves:. Divide the original number by 6 to obtain a new whole number and its remainder.. Keep on dividing the whole number by 6 until the whole number obtained is Write down the remainders next to each other starting from the bottom, changing numbers greater than 9 to hexadecimal letters where required. Ms.P.Sullivan Page 6 of 7

7 Convert from Hexadecimal to Decimal Computer Studies 0 Syllabus In order to convert a hexadecimal number to decimal, the placement values of base 6 are placed on top of each number starting from the right with 6 0 then incrementing the exponent by for each left move. The results of the multiplications are then added up to obtain the decimal number. So let s convert 3A 6 to decimal: A (56x) (6x3) (x0) 0 0 =0++0 =0 0 So the equivalent value of 3A 6 is 0 0. This involves:. Writing down the placement value on top of each number starting with 6 0 from the right-most number, then increase the exponent by each time to the left.. Multiply the top value with the hexadecimal number. 3. Add all the results of the multiplications together. *** Ms.P.Sullivan Page 7 of 7