Higher Computing – Data Representation – Representation of Positive Numbers

2. Decimal Numbers In every day life we use 10 decimal numbers to represent numerical data 0 1 2 3 4 5 6 7 8 9 The number 42678 can be read as ten thousands 10 4 thousands 10 3 hundreds 10 2 tens 10 1 units 10 0 4 2 6 7 8 The number that forms the basis of any system is called the base . So the decimal number system is called base 10 .

3. The Binary Number System The lowest number base that can be used to represent numbers is base 2 . Using only two numbers to represent what happens inside a computer is called the binary system . on = 1 bit off = 0 bit

4. Why design computers as 2 state devices? 1. Simplicity in only having to generate and detect two voltage levels. 2. Good tolerance, because a degraded positive voltage (1 bit) is still recognisable as a positive voltage

5. Why design computers as 2 state devices? 3. Calculations are kept simple. There are four rules for addition / subtraction / multiplication / division in the binary number system. There are one hundred rules for addition / subtraction / multiplication / division in the decimal number system.

6. Why design computers as 2 state devices? 4. Magnetic and optical media are suited to two state systems.

7. Binary Numbers We have seen that the number 42678 can be represented as: 10 4 10 3 10 2 10 1 10 0 4 2 6 7 8 Binary (base 2) works in a similar way. The binary number 1 1 0 0 1 can be represented as: 2 4 2 3 2 2 2 1 2 0 1 1 0 1 1 Or 16 8 4 2 1 1 1 0 1 1

8. Converting Binary Numbers to Decimal The binary number 1 1 0 0 1 16 8 4 2 1 1 1 0 0 1 = ( 1 * 16) + ( 1 * 8) + ( 0 * 4) + ( 0 * 2) + ( 1 * 1) = 16 + 8 + 0 + 0 + 1 = 25

9. Converting a Byte to Decimal The term bit is short for b inary dig it A byte is made up of 8 bits The byte 1 0 1 0 1 1 0 1 can be converted to decimal as: 16 8 4 2 1 1 0 1 0 1 1 0 1 = ( 1 * 128) + ( 1 * 32) + ( 1 * 8) + ( 1 * 4) + ( 1 * 1) = 128 + 32 + 8 + 4 + 1 = 173 32 64 128

10. Converting Decimal Numbers to Binary Example: Convert 86 to binary 128 0 Next 64 . Does 64 go into 86? Yes. 128 64 0 1 Subtract the 64 from 86, leaving 22. Next 32 . Does 32 go into the remaining 22? No 128 64 32 0 1 0 Start with 128 . Does 128 go into 86? No.

11. Converting Decimal Numbers to Binary 128 64 32 16 0 1 0 1 Next 16 . Does 16 go into 22? Yes. Subtract the 16 from 22, leaving 6. Next 8 . Does 8 go into the remaining 6? No 128 64 32 16 8 0 1 0 1 0 Next 4 . Does 4 go into the remaining 6? Yes Subtract 4 from 6, leaving 2. Next 2 . Does 2 go into the remaining 2? Yes Subtract 2 from 2, leaving 0. Finally 1 . Does 1 go into the remaining 0? No 128 64 32 16 8 4 2 1 0 1 0 1 0 1 1 0