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A number system defines a set of values used to represent quantity. We talk about the number of people attending class, the number of modules taken per student, and also use numbers to represent grades achieved by students in tests. Quantifying values and items in relation to each other is helpful for us to make sense of our environment. We do this at an early age; figuring out if we have more toys to play with, more presents, more lollies and so on. The study of number systems is not just limited to computers.
We apply numbers every day, and knowing how numbers work will give us an insight into how a computer manipulates and stores numbers. Mankind through the ages has used signs or symbols to represent numbers. The early forms were straight lines or groups of lines, much like as depicted in the film Robinson Crusoe , where a group of six vertical lines with a diagonal line across represented one week. Its difficult representing large or very small numbers using such a graphical approach.
This was a major advance, because it reduced the number of symbols required. For instance, 12 could be represented as a 10 and two units three symbols instead of 12 that was required previously. The Romans devised a number system which could represent all the numbers from 1 to 1,, using only seven symbols. The number system in most common use today is the Arabic system.
It was first developed by the Hindus and was used as early as the 3rd century BC. The introduction of the symbol 0, used to indicate the positional value of digits was very important. We thus became familiar with the concept of groups of units, tens of units, hundreds of units, thousands of units and so on.
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. 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.
In this binary system, there are only two values 0 - 1 in the set, so columns represent multiplication values of 2. Converting Decimal to Binary There are a number of ways to convert between decimal and binary. Lets start with converting the decimal value to binary. Divide the number by 2, then divide what's left by 2, and so on until there is nothing left 0.
Write down the remainder which is either 0 or 1 at each division stage. Once there are no more divisions, list the remainder values in reverse order. This is the binary equivalent. Each column represents a power of 2, so use this as a basis of calculating the number.
It is sometimes referred to as the 8: Write down the binary number. Where a 1 appears in the column, add the column value as a power of 2 to a total. Hexadecimal is often used to represent values [numbers and memory addresses] in computer systems. Converting hexadecimal to decimal Problem: Convert in hexadecimal to decimal. Converting binary to hexadecimal Problem: Convert to hexadecimal. Converting decimal to hexadecimal Problem: Convert decimal to hexadecimal.
To avoid confusion, we often add a suffix to indicate the number base. Representing Positive and negative Numbers in Binary When a number of bits is used to store values, the most significant bit [the bit which has the greatest value, in the left most column] is used to store the sign [positive or negative] of the number.
The remaining bits hold the actual value. If the number is negative, the sign is 1 , and for positive numbers, the sign is 0. What is the range of numbers available when 8 bits are used. Because of problems doing addition and subtraction, negative numbers are usually stored in a different format to positive numbers. Ones Complement 1's complement is a method of storing negative values.
It simply inverts all 0's to 1's and all 1's to 0's. Twos Complement 2's complement is another method of storing negative values.
It is obtained by adding 1 to the 1's complement value. Another way of generating a 2's complement number is to start at the least significant bit, and copy down all the 0's till the first 1 is reached. Copy down the first 1, then invert all the remaining bits. The following table depicts both 1's and 2's complement using a range of 4 bits. See how in the 1's complement case there are two representations for 0.
Gray Code This is a variable weighted code and is cyclic. This means that it is arranged so that every transition from one value to the next value involves only one bit change. The gray code is sometimes referred to as reflected binary , because the first eight values compare with those of the last 8 values, but in reverse order.
A unit distance code derives its name from the fact that there is only one bit change between two consecutive numbers. The excess 3 gray code is such a code, the values for zero and nine differ in only 1 bit, and so do all values for successive numbers.
Outputs from linear devices or angular encoders may be coded in excess 3 gray code to obtain multi-digit BCD numbers. Weighting 8 4 2 1 Answer Binary Value 1 0 1 1 Weighting 8 4 2 1 Answer Binary Value 0 1 1 1 7. Weighting 32 16 8 4 2 1 Answer Binary Value 1 1 1 0 1 1 Weighting 32 16 8 4 2 1 Answer Binary Value 1 0 1 0 1 0 Decimal - Binary - Hexadecimal Decimal Binary Hexadecimal 0 0 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 A 11 B 12 C 13 D 14 E 15 F.
Original Number Binary value 1's Complement value 7 32 Original Number Binary value 1's Complement value 2's Complement value 7 32 Table of Complements Binary 1's Complement 2's Complement Unsigned 7 7 7 6 6 6 5 5 5 4 4 4 3 3 3 2 2 2 1 1 1 0 0 0 -0 -1 15 -1 -2 14 -2 -3 13 -3 -4 12 -4 -5 11 -5 -6 10 -6 -7 9 -7 -8 8. Decimal Binary Gray 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Decimal Excess 3 Gray 0 1 2 3 4 5 6 7 8 9