Number Systems and Base Conversions

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1 Number Systems and Base Conversions As you know, the number system that we commonly use is the decimal or base- 10 number system. That system has 10 digits, 0 through 9. While it's very convenient for humans, base 10 is less convenient for computers. In a computer, the smallest unit of memory is called a bit, for binary digit. A bit contains either a 0 or a 1. That may seem strange until you realize that a computer uses electricity and magnetism to store results. Electricity (alternating current) typically flows in one of two directions, and magnets have two poles, so it's reasonable that the smallest storage unit should have one of two values. That makes the binary or base-2 system the natural one for computers. Other fairly common systems that humans use when working with computers are octal (base-8) and hexadecimal (base-16). All number systems have one thing in common: if the number system is base-r, it contains r digits, starting at 0. If r is 10 or less, then the digits in the system are 0 through r-1. As an example, any base-2 number contains only the digits 0 and 1, so numbers in base-2 (or binary) look like 101 or Similarly, base-8 or octal numbers contain the digits 0 through 7, so 643 and 4205 are examples of valid base-8 numbers, but is not (because 8 and 9 are not valid digits in base-8.) But what happens in base 16 (or any base greater than 10)? We know it will contain 16 different digits, but what are they? By convention, we use the digits 0 through 9, followed by the first six letters of the alphabet (A through F). Positional Notation Because humans are so used to base-10, if we need to work with numbers in some other base, we typically want to know what those numbers are in base 10. It's easy to convert from any other base to base 10, if we remember that the position of a digit determines its significance. For example, consider the decimal (base-10) number 375. We speak of this as having a hundreds' digit (which is 3), a tens' digit (7) and a ones' (or units') digit (5), and we know this represents 3 100's, 7 10's and 5 1's. Another way to think of this is as a sum of products, where each product is a coefficient and a power of 10 (because 10 is the base we're using here). If we remember that any positive number raised to the 0 th power is 1, we can think of 375 as 375 = 3 * * * 10 0 We can use positional notation to represent numbers in any base; we just need to use that base as the base of the exponents. This is particularly useful if we want to convert a number from some other base to base 10.

2 Example 1 Convert the binary (base-2) number to base 10. We know that = 1 * * * * * * * 2 0 so now we can just "do the math" to figure the conversion = 1 * * * * * * * 2 0 = = 115 (base 10) Example 2 Convert octal (base-8) to base = 6 * * * 8 0 = 6 * * * 1 = = 419 in base 10 Example 3 Convert hexadecimal (base-16) 1EA to base 10. Here we need to know what the letters A through F in base 16 represent. A represents 10, B represents 11, C is 12, D is 13, E is 14 and F is 15. With that in mind, we have 1EA 16 = 1 * * * 16 0 = 1 * * * 1 = = 490 in base 10 Converting Base-10 Numbers to Other Bases Let's say you want to go the other way, to convert a decimal number to some other base r. Here s how you do it. 1. Divide the base-10 number by r. Remember the quotient and the remainder of this division. 2. Divide the quotient from the first division by r, again remembering the quotient and the remainder. 3. Keep dividing your new quotient by r until you get a quotient of 0. Each time, keep track of the remainder. 4. When you reach a quotient of 0, the remainders of all the divisions, written in reverse order, will be the equivalent base-r number. By reverse order, I mean that the first remainder that you got in step 1 will be the least significant digit of the base-r number.

3 Example 4 (the reverse of Example 1) Convert decimal (base-10) 115 to base First divide 115 by 2. Quotient is 57, remainder is Then divide 57 by 2. New quotient is 28, new remainder is Then divide 28 by 2. New quotient is 14, new remainder is Then divide 14 by 2. New quotient is 7, new remainder is Then divide 7 by 2. New quotient is 3, new remainder is Then divide 3 by 2. New quotient is 1, new remainder is Then divide 1 by 2. New quotient is 0, new remainder is 1. We stop dividing, because the quotient is now Writing the remainder in reverse order, we get , which is what we started with in Example 4 (so it must be correct!). Example 5 (the reverse of Example 2) Convert decimal (base-10) 419 to base First divide 419 by 8. Quotient is 52, remainder is Then divide 52 by 8. New quotient is 6, new remainder is Then divide 6 by 8. New quotient is 0, new remainder is 6. We stop dividing, because the quotient is now Writing the remainders in reverse order, we get 643 8, which is what we started with in Example 5. Example 6 (the reverse of Example 3) Convert decimal (base-10) 490 to hexadecimal (base 16). 1. First divide 490 by 16. Quotient is 30, remainder is 10 BUT 10 is represented in base-16 as A. 2. Then divide 30 by 16. Quotient is 1, remainder is 14 BUT 14 is represented in base-16 as E. 3. Then divide 1 by 16. Quotient is 0, remainder is 1. Stop dividing. 4. Writing the remainders in reverse order, we get 1EA 16.

4 Number Conversions between Bases 2, 8 and 16 Base 10 Number Base 2 Number Base 8 Number Base 16 Number A B C D E F This table can be used to simplify number conversions between bases 2, 8 and 16 (but not between base 10 and any of these bases). A base 16 number can be converted to base 2 by simply replacing each base 16 digit with its four-digit base 2 equivalent. Hence 3BE 16 is (or you could drop the leading zeroes for ). Conversely, you can convert a base 2 number to base 16 by starting from the right and separating the base 2 number into groups of four digits, padding on the left with zeroes if necessary. Then simply convert each group of four binary digits to its hexadecimal equivalent. Example: to convert to base 16, we separate it into the following groups (padding with two zeroes on the left): so the base 16 equivalent is 2D8 16. You can do similar conversions between base 8 and base 2 if you use three base 2 digits for each base 8 digit. So is and is

5 Integer Representation Integers are usually represented in either 16 bits or 32 bits. Positive integers are simply represented in base 2, with extra 0's added on the left to pad the number out to the appropriate number of bits. So 35 (which is in base 2) would be represented as inside the computer (assuming 16 bits for an integer). Negative integers are represented in twos complement. To determine the representation of a negative integer, do the following: 1. first find the representation of the absolute value of the number 2. interchange all the 0's and 1's 3. add 1 (using base-2 addition). So, to find the representation of -35, we do this: 1. The representation of 35 is , so we 2. Interchange the 0's and 1's to get Then we add 1 to get To find -36, we'd first determine that +36 is represented as We'd then interchanged the 0's and 1's to get To add 1, we do this: And just ignore the leading 1 (since it doesn't fit in 16 bits). Character Representation Here you need your ASCII character chart and you also need to know that characters are represented in 8 bits. To find how a particular character is represented, see what number is used to represent it. Then change that number to base 2 and place it in 8 bits, padding with 0's on the left, if necessary. For example, the character # is represented by the number 35. As we've seen above, 35 is in base 2, so the representation of the character # is

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