To represent numbers in base n the numbers through are available for. each digit (in base 10, we use, in base 2, ). Each digit has a different place

Transcription

1 To represent numbers in base n the numbers through are available for each digit (in base 10, we use, in base 2, ). Each digit has a different place holder value: the first digit to the left of the decimal point represents n 0 place, the next digit to the left represents the n 1 place, the next to the left the n 2 place, etc. The pattern also extends in the other direction. The first digit to the right of the decimal point represents the n 1 place, the next the n 2 place, etc. 1 Decimal System For the decimal system, we re working in base 10 (base is also known as radix). As we said earlier, the numbers 0-9 are used, and each digit is in a powered place: the 1000s place, the 100s place, the 10s place, the 1s place, the 0.1 place, etc = 4 } + 8 {{ }} + 6 {{ } Integer P art F raction P art 2 Octal System The Octal System is base 8, so we use numbers 0-7. Now digits are in the 64s place, the 8s place, 1s place, 1/8 place, 1/64 place, etc = = Note: this example is just to show decimal (or octal in this case) points are used. In standard form, octal usually represents only integers 1

2 3 Binary System Each binary (base 2) digit is known as a bit. Only two digits, 0 and 1, are used. This system is especially important in digital logic, which will be discussed in detail tomorrow and in your electrical engineering class. Typically one byte is 8 bits, and 4 bytes together makes a word (on a 32-bit computer system). The digits are in the places corresponding to powers of = = = = 4 Human Computer Activity 5 Converting To Decimal We ve already begun to do this when we talked about octal and binary numbers. To complete the conversion, simply evaluate the sum: 2

3 563 8 = = = = 6 Converting From Decimal For the new base n, remember to only use digits 0 through n-1 as we convert a number x to decimal. Determine the highest place holder value (largest power of n that is smaller than x), and subtract it out. Continue subtracting out the lower place holder values until you get to n 0. For example, convert to base is less than 6 4 (6 4 = 1296), so try as the largest place holder. 7 Converting From Decimal Shortcut There is an additional method to easily convert numbers from decimal to a different base. We repeatedly divide the decimal number by the base in which it is to be converted, until 3

4 the quotient becomes zero. As the number is divided, the remainders - in reverse order - form the digits of the number in the other base. For example, if we want to convert to base 7: = R 7 = R 7 = R The result is. To check our work: 8 Hexadecimal System Hexadecimal is base. The numbers 0 though 9 are used, and also the letters A-F. (Remember, each digit, even if its value is greater than 10, can only use one place, and 10 through 15 each use two places, so we need new symbols for these values.) Other bases greater than 10 must use letters in addition to digits as well. Each hexadecimal digit represents 0 to 15. Each binary digits also represent 0 to 15. We can easily translate binary to hexadecimal by dividing the binary number into blocks of digits, and translating each of these blocks to hexadecimal. As an example, let s convert to hexadecimal without having to go through decimal }{{} 0110 }{{} 0111 }{{} 1001 }{{} Block1 Block2 Block3 Block4 4

5 = = = = So in hexadecimal, this number is. We can do the same thing translating between binary and any other base that is a power of 2, using a corresponding block size, for example, blocks of size 3 for translating to base (2 3 = ) and blocks of size for base 4. One common use of the hexadecimal system is the definition of RGB colors (e.g. in html pages, proceeded by the pound sign). Each color is expressed as a combination of Red (R), Green (G) and Blue (B). Each red, green or blue component is a numerical value that represents the amount of each color present and is a number between and. A shorter way to express a color is by dedicating two hexadecimal digit per color: for example DD0F9A means: R = DD 16 = 10, G = 0F 16 = 10 and B = 9A 16 = 10 which is: 10 of red, 10 of green and 10 of blue. Some standard colors: red is encoded as 16, green as 16 and blue as 16; white is 16, black is 16, yellow is 16, magenta is 16. 5

6 Figure 1: The color DD0F9A 6