The document discusses binary numbers and how computers use binary to represent data. It explains that computers use a two-state system of 1s and 0s because it is simple, tolerant to degradation, has simple calculation rules, and magnetic media can easily represent two states. It provides examples of converting binary numbers to decimal and decimal numbers to binary, illustrating how binary works in a similar place value system as decimal.

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