1. How Computers Represent Numbers Friday, Week 5

2. Binary Code A series of 1’s and 0’s Place value is in powers of 2

3. The Decimal System Analyze the number 2,473 2,473 = 2 * 1000 + 4 * 100 + 7 * 10 + 3 *1 2,473 = 2 * 10^3 + 4 * 10^2 + 7 * 10^1 + 3 * 10^0 Each position in our number represents a different power of 10 Decimal is a base 10 system

4. Binary as Base 2 1011001 = 1*2^6 + 0*2^5 + 1*2^4 + 1*2^3 + 0*2^2 + 0*2^1 +1*2^0 1011001 = 1*64 + 0*32 + 1*16 + 1*8 + 0*4 + 0*2 + 1*1 1011001 = 64 + 16 + 8 + 1 1011001 = 89 (decimal)

5. Base 3 1011001 = 1*3^6 + 0*3^5 + 1*3^4 + 1*3^3 + 0*3^2 + 0*3^1 +1*3^0 1011001 = 1*729 + 0*243 + 1*81 + 1*27 + 0*9 + 0*3 + 1*1 1011001 = 729 + 81 + 27 + 1 1011001 = 838 (decimal)

6. Exercise Find the decimal equivalent of 100111 100111 = 1*2^5 + 1*2^2 + 1*2^1 + 1*2^0 100111 = 1*32 + 1*4 + 1*2 + 1*1 100111 = 32 + 4 + 2 + 1 100111 = 39 (decimal)

7. Hexadecimal System Base-16 system Needs digits 0 through 15 - we don’t have numbers for 10 - 15. We use the letters A - F to represent the numbers 10 - 15.

8. Exercise What would 3B in hexadecimal be in decimal? 3B = 3 * 16 ^ 1 + 11 * 16 ^ 0 3B = 3 * 16 + 11 * 1 3B = 48 + 11 3B = 59

9. More on hexadecimal 4 binary digits equal one hexadecimal number 0101 (binary) = 5 (hex) 1101 (binary) = D (hex) 1011101 (binary) = 93 (decimal) = 5D (hex)

10. Decimal to Binary Divide by 2 and keep track of the remainders. 39 (decimal) = 100111 (binary) 39/2 =19Rem 1 19/2 =9Rem 1 9/2 =4Rem 1 4/2 =2Rem 0 2/2 =1Rem 0 1/2 =0Rem 1

11. Exercise Convert 89 (decimal) to binary 89 (decimal) = 1011001 (binary) 89/2 =44R 1 44/2 =22R 0 22/2 =11R 0 11/2 =5R 1 5/2 =2R 1 2/2 =1R 0 1/2 =0R 1

12. Why use binary? Binary uses more digits than decimal, so why do we use it? Electronic hardware can either be ‘on’ or ‘off’ - nothing in between. Binary fits this pattern - ‘on’ state is 1 in binary and ‘off’ state is 0 in binary.

13. Numeric Representation of Letters and Digits In a computer, letters and digits are represented by numeric codes. Example Code: What does this say? 8 9 3 12 1 19 19 ABCDEFGHIJKLM 12345678910111213 NOPQRSTUVWXYZ 14151617181920212223242526

14. ASCII and Unicode ASCII (American Standard Code for Information Interchange) - 7 bit code that represents commony used English characters. Unicode - Newer 16 bit code that is able to encode large asian alphabets. (ASCII is included as a subset.)

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