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Appendix G
Hexadecimal Numbers
Humans normally use the base 10 numbering system. This is called the decimal system for the Greek word deka for ten. Our ancient ancestors learned it by counting with their 10 fingers. Computers have only 2 fingers (i.e., there are only 2 possible values for each bit), so the binary system works well for computers. But the trouble with binary numbers is that their representations require long strings of bits. For example, 1996 is represented as 11111001100 in binary. Most humans find long strings like that difficult to process. Binary numbers are easy to convert to other bases if the base is a power of 2. For example, conversion between binary and octal (base 8 = 23) merely requires grouping the binary bits into groups of 3 and interpreting each triplet as an octal digit. For example, to convert the binary numeral 11111001100 write 11,111,001,100 = 3714. Here, 11 converts to 3, 111 converts to 7, 001 converts to 1, and 100 converts to 4. Conversion from octal back to binary is just as simple. For example, 2650 converts to 10110101000, which is 1448 in decimal. Note that octal numerals use only the first 8 decimal digits: 0, 1, 2, 3, 4, 5, 6, 7. After 8, the next power of 2 is 16. Using that base makes the numerals even shorter. This is called the hexadecimal system (from the Greek hex + deka for six + ten ). Conversion between binary and hexadecimal is just as simple as it is between binary and octal. For example, to convert the binary numeral 10111010100 to hexadecimal, group the bits into groups of 4 (from right to left) and then translate each group into the corresponding hexadecimal digit: 101,1101,0100 = 5d4. Here, 101 converts to 5, 1101 converts to 11, and 0100 converts to 4. The hexadecimal digits 10, 11, 12, 13, 14, and 15 are denoted by the first six letters of the alphabet: a, b, c, d, e, f. Most operating systems provide a calculator utility that converts number representations between hexadecimal, decimal, octal, and binary. For example, the Calculator utility in Microsoft Windows is located in Start > Programs > Accessories. In that application, to convert from hexadecimal to decimal, select Scientific from its View menu, select the Hex radio button, enter the hexadecimal representation of the number, and then select the Dec radio buttons. The example here shows that 0x0064fdbc is hexadecimal notation for 6,618,556.
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HEXADECIMAL NUMBERS
[APP. G
The output manipulators dec, hex, and oct are used for converting different bases, as the next example illustrates. EXAMPLE G.1 Using Output Manipulators
This shows how both the value and the address of a variable can be printed: int main() { int n = 1492; // base 10 cout << "Base 8: n = " << oct << n << endl; cout << "Base 10: n = " << n << endl; cout << "Base 16: n = " << hex << n << endl; } Base 8: n = 2724 Base 10: n = 1492 Base 16: n = 5d4 Here the manipulator oct is used to convert the next output to octal form. Note that the output reverts back to decimal until the hex manipulator is used. The next example shows how to input integers in octal and hexadecimal. Octal numerals are denoted with a 0 prefix, and hexadecimal numerals are denoted with a 0x prefix.
EXAMPLE G.2 Using Input Manipulators
This shows how both the value and the address of a variable can be printed: int main() { int n; cout << "Enter an octal numeral (use 0 prefix): "; cin >> oct >> n; cout << "Base 8: n = " << oct << n << endl; cout << "Base 10: n = " << dec << n << endl; cout << "Base 16: n = " << hex << n << endl; cout << "Enter a decimal numeral: "; cin >> dec >> n; cout << "Base 8: n = " << oct << n << endl; cout << "Base 10: n = " << dec << n << endl; cout << "Base 16: n = " << hex << n << endl; cout << "Enter a hexadecimal numeral (use 0x prefix): "; cin >> hex >> n; cout << "Base 8: n = " << oct << n << endl; cout << "Base 10: n = " << dec << n << endl; cout << "Base 16: n = " << hex << n << endl; } Enter an octal numeral (use 0 prefix): 0777 Base 8: n = 777 Base 10: n = 511 Base 16: n = 1ff Enter a decimal numeral: 511 Base 8: n = 777 Base 10: n = 511
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