1. CPS120: Introduction to Computer Science Computer Math: Converting to Decimal

2. 9 Decimal is base 10 and has 10 digits: 0,1,2,3,4,5,6,7,8,9 Binary is base 2 and has 2 digits: 0,1 For a number to exist in a given number system, the number system must include those digits. For example: The number 284 only exists in base 9 and higher. Binary

3. Codes Given any positive integer base (RADIX) N, there are N different individual symbols that can be used to write numbers in the system. The value of these symbols range from 0 to N-1 All systems we use in computing are positional systems 495 = 400 + 90 +5

4. Number Systems We use the DECIMAL (10 system Computers use BINARY (2 or some shorthand for it like OCTAL (8 or HEXADECIMAL (16

5. BinaryOctalDecimal 00000 00111 01022 01133 10044 10155 11066 11177 100108 1001119 10101210 16 Power of 2 Number System

6. 10 How are digits in bases higher than 10 represented? Base 16: 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E, and F Bases Higher than 10

7. Conversions

8. Decimal Equivalents Assuming the bits are unsigned, the decimal value represented by the bits of a byte can be calculated as follows: 1. Number the bits beginning on the right using superscripts beginning with 0 and increasing as you move left Note: 2 0, by definition is 1 2. Use each superscript as an exponent of a power of 2 3. Multiply the value of each bit by its corresponding power of 2 4. Add the products obtained

9. What is the decimal equivalent of the octal number 642? 6 x 8² = 6 x 64 = 384 + 4 x 8¹ = 4 x 8 = 32 + 2 x 8º = 2 x 1 = 2 = 418 in base 10 11 Converting Octal to Decimal

10. What is the decimal equivalent of the hexadecimal number DEF? D x 16² = 13 x 256 = 3328 + E x 16¹ = 14 x 16 = 224 + F x 16º = 15 x 1 = 15 = 3567 in base 10 Remember, base 16 is 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F Converting Hexadecimal to Decimal

11. What is the decimal equivalent of the binary number 010110? 1 x 2 6 = 1 x 64 = 64 + 1 x 2 5 = 1 x 32 = 32 + 0 x 2 4 = 0 x 16 = 0 + 1 x 2 3 = 1 x 8 = 8 + 1 x 2 2 = 1 x 4 = 4 + 1 x 2 1 = 1 x 2 = 2 + 0 x 2º = 0 x 1 = 0 = 112 in base 10 13 Converting Binary to Decimal

12. Horner’s Method Another procedure to calculate the decimal equivalent of a binary number Note: This method works with any base Horner’s Method: Step 1: Start with the first digit on the left Step 2: Multiply it by the base Step 3: Add the next digit Step 4: Multiply the sum by the base Step 5: Continue the process until you add the last digit