Does the concept of 2\'s complement work for any number system? Is there such a

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YES To understand binary numbers, begin by recalling elementary school math. When we first learned about numbers, we were taught that, in the decimal system, things are organized into columns: H | T | O 1 | 9 | 3 such that "H" is the hundreds column, "T" is the tens column, and "O" is the ones column. So the number "193" is 1-hundreds plus 9-tens plus 3-ones. Years later, we learned that the ones column meant 10^0, the tens column meant 10^1, the hundreds column 10^2 and so on, such that 10^2|10^1|10^0 1 | 9 | 3 the number 193 is really {(1*10^2)+(9*10^1)+(3*10^0)}. As you know, the decimal system uses the digits 0-9 to represent numbers. If we wanted to put a larger number in column 10^n (e.g., 10), we would have to multiply 10*10^n, which would give 10^(n+1), and be carried a column to the left. For example, putting ten in the 10^0 column is impossible, so we put a 1 in the 10^1 column, and a 0 in the 10^0 column, thus using two columns. Twelve would be 12*10^0, or 10^0(10+2), or 10^1+2*10^0, which also uses an additional column to the left (12). The binary system works under the exact same principles as the decimal system, only it operates in base 2 rather than base 10. In other words, instead of columns being 10^2|10^1|10^0 they are 2^2|2^1|2^0 Instead of using the digits 0-9, we only use 0-1 (again, if we used anything larger it would be like multiplying 2*2^n and getting 2^n+1, which would not fit in the 2^n column. Therefore, it would shift you one column to the left. For example, "3" in binary cannot be put into one column. The first column we fill is the right-most column, which is 2^0, or 1. Since 3>1, we need to use an extra column to the left, and indicate it as "11" in binary (1*2^1) + (1*2^0). Examples: What would the binary number 1011 be in decimal notation? Click here to see the answer Try converting these numbers from binary to decimal: 10 111 10101 11110 Remember: 2^4| 2^3| 2^2| 2^1| 2^0 | | | 1 | 0 | | 1 | 1 | 1 1 | 0 | 1 | 0 | 1 1 | 1 | 1 | 1 | 0 Click here to see the answer Return to Table of Contents Binary Addition Consider the addition of decimal numbers: 23 +48 ___ We begin by adding 3+8=11. Since 11 is greater than 10, a one is put into the 10's column (carried), and a 1 is recorded in the one's column of the sum. Next, add {(2+4) +1} (the one is from the carry)=7, which is put in the 10's column of the sum. Thus, the answer is 71. Binary addition works on the same principle, but the numerals are different. Begin with one-bit binary addition: 0 0 1 +0 +1 +0 ___ ___ ___ 0 1 1 1+1 carries us into the next column. In decimal form, 1+1=2. In binary, any digit higher than 1 puts us a column to the left (as would 10 in decimal notation). The decimal number "2" is written in binary notation as "10" (1*2^1)+(0*2^0). Record the 0 in the ones column, and carry the 1 to the twos column to get an answer of "10." In our vertical notation, 1 +1 ___ 10 The process is the same for multiple-bit binary numbers: 1010 +1111 ______ Step one: Column 2^0: 0+1=1. Record the 1. Temporary Result: 1; Carry: 0 Step two: Column 2^1: 1+1=10. Record the 0, carry the 1. Temporary Result: 01; Carry: 1 Step three: Column 2^2: 1+0=1 Add 1 from carry: 1+1=10. Record the 0, carry the 1. Temporary Result: 001; Carry: 1 Step four: Column 2^3: 1+1=10. Add 1 from carry: 10+1=11. Record the 11. Final result: 11001

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