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 EBCDIC The 2 most common codes used in computers are ASCII (American Standard Code for Information Interchange) and EBCDIC (Extended Binary Coded Decimal Interchange Code). They provide codes for letters, digits, punctuation marks, and other special characters.

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