SD & D Representing 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 104 thousands 103 hundreds 102 tens 101 units 100 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 bitoff = 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: 104 103 102 101 100 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: 24 23 22 21 20 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 binary digit 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 3264128
- 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. 128 64 32 16 0 1 0 1 Next 16. Does 16 go into 22? Yes. Converting Decimal Numbers to Binary 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
- 12. Credits Higher Computing – Data Representation – Representation of Positive Numbers Produced by P. Greene and adapted by R. G. Simpson for the City of Edinburgh Council 2004 Adapted by M. Cunningham 2010 All images licenced under Creative Commons 3.0 •Hard Disk B by Christian Jansky •CD-ROM by Vincent1969 •Voltometer by Brandi Sims •Numbers by Procsilas Moscas