Common Number Systems Number Systems


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1 5/29/204 Common Number Systems Number Systems System Base Symbols Used by humans? Used in computers? Decimal 0 0,, 9 Yes No Binary 2 0, No Yes Octal 8 0,, 7 No No Hexadecimal 6 0,, 9, A, B, F No No Number System Concepts Digital computers internally use the binary (base 2) number system to represent data and perform arithmetic calculations. Thebinary number system is very efficient for computers, but not for humans. Representing even relatively small numbers with the binary system requires working with long strings of ones and zeroes. Why You Need to Know About... Numbering Systems. Computers store programs and data as binary digits 2. Hexadecimal number system Provides convenient representation Written into error messages (helps debug error messages) 3. Octal and hex are a convenient way to represent binary numbers, as used by computers. 4. Computer mechanics often need to write out binary quantities, but in practice writing out a binary number such as is tedious, and prone to errors. 5. Therefore, binary quantities are written in a base8 ("octal") or, much more commonly, a base6 ("hexadecimal" or "hex") number format. 6. Helpful as students interact with computers in later computing courses and throughout their careers 4 DECIMAL NUMBER SYSTEMS This is referred to as Base 0. Symbols are : 0,,2,3,4,5,6,7,8,9 Written as: e.g Used by humansnot computers! BINARY NUMBER SYSTEM Referred to as Base 2. A Binary Number is made up of only 0s and s Example of a Binary Number
2 5/29/204 OCTAL NUMBER SYSTEM Referred to as Base 8. Used by humans for ease of communication, because binary is too long. Symbols are: 0,,2,3,4,5,6,7 Example: 3 8 HEXADECIMAL SYSTEM Referred to as base 6. Used by humans for ease of communication, because binary is too long. Symbols are: 0,,2,3,4,5,6,7,8,9 AND A,B,C,D,E,F Decimal HEX A 0 B C 2 D 3 E 4 F 5 Example: A9 6 Conversion Among Bases The possibilities: Decimal Octal Binary Hexadecimal Converting To and From Decimal Quick Example Decimal = 00 2 = 3 8 = 9 6 Octal 8 Hexadecimal Binary A B C D E F Base 0 2
3 5/29/204 Review: Decimal Binary a) Divide the decimal number by 2; the remainder is the LSB (Least Significant Bit) of the binary number. b) If the quotation is zero, the conversion is complete. Otherwise repeat step (a) using the quotation as the decimal number. The new remainder is the next most significant bit of the binary number. Example: From Decimal to Binary 43 0 = 43/2= 2 Remainder 2/2= 0 remainder 0/2=5 remainder 0 5/2= 2 remainder 2/2= remainder 0 /2=0 remainder a) Multiply each bit of the binarynumber by its corresponding bitweighting factor (i.e., Bit =; Bit 2 =2; Bit =4; etc). b) Sum up all of the products in step (a) to get the decimal number. 3 Answer: 43 0 = 00 2 Example: Binary to Decimal Bit => x 2 0 = x 2 = 2 0 x 2 2 = 0 x 2 3 = 8 0 x 2 4 = 0 x 2 5 = 32 BINARY CALCULATIONS 43 0 BINARY ADDITION Rules of Binary Addition In fourth case, a binary addition is creating a sum of (+=0) i.e. 0 is write in the given column and a carry of over to the next column. Binary Addition Example Example : Add binary 0 to Col ) Add + 0 = Write Col 2) Add + 0 = Write Col 3) Add + = 2 (0 in binary) Write 0, carry Col 4) Add = 2 Write 0, carry Col 5) Add + + = 3 ( in binary) Write, carry Col 6) Add = 2 Write 0, carry Col 7) Bring down the carried Write 3
4 5/29/204 What is actually happened when we carried in binary? Binary Addition Explanation In the first two columns, there were no carries. In column 3, we add + = 2 Since 2 is equal to the base, subtract the base from the sum and carry. In column 4, we also subtract the base from the sum and carry. In column 5, we also subtract the base from the sum and carry. In column 6, we also subtract the base from the sum and carry. In column 7, we just bring down the carried Binary Addition You can always check your answer by converting the figures to decimal, doing the addition, and comparing the answers = = 83 0 Binary Addition Example 2 Example 2: Add to = There four rules of the binary Subtraction = 73 0 Explanation In binary, the base unit is 2 So when you cannot subtract, you borrow from the column to the left. The amount borrowed is 2. The 2 is added to the original column value, so you will be able to subtract. Example Example : Subtract binary 00 from Col ) Subtract 0 = Col 2) Subtract 0 = Col 3) Try to subtract 0 can t. Must borrow 2 from next column. But next column is 0, so must go to column after next to borrow. Add the borrowed 2 to the 0 on the right. Now you can borrow from this column (leaving remaining). Add the borrowed 2 to the original 0. Then subtract 2 = Col 4) Subtract = 0 Col 5) Try to subtract 0 can t. Must borrow from next column. Add the borrowed 2 to the remaining 0. Then subtract 2 = Col 6) Remaining leading 0 can be ignored. 4
5 5/29/204 Subtract binary 00 from 00: = = 23 0 Example 2 Example 2: Subtract binary 000 from = = 2 0 Binary Binary multiplication is similar to decimal multiplication. It is simpler than decimal multiplication because only 0s and s are involved.there four rules of the binary multiplication. 5
Oct: 50 8 = 6 (r = 2) 6 8 = 0 (r = 6) Writing the remainders in reverse order we get: (50) 10 = (62) 8
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