What are the examples of number system conversion?

Answer 1

The first idea to understand in number system conversion is that each number system is simply a different way to represent numerical values. The most common number systems, decimal, binary, hexadecimal, and octal, all use different "base" numbers in their representations. These are 10, 2, 16, and 8, respectively. Now, these "base" numbers simply represent the number of symbols that the number system has available to represent numbers. Decimal has ten symbols (0 - 9), binary has two (0 and 1), hexadecimal has sixteen (0 - 9 and A - F), and octal has eight (0 - 7). The base number, then, also represents how many times a place value can "count up" in a number. For example, the number 13, in decimal (or "base ten"), is represented by the symbols 1 and 3 in different place values. The 1, in the second place value, represents the quantity of ten, because the decimal number system cannot represent a quantity greater than nine using only ten symbols (0 - 9). So, the second place value is used to represent a multiple of the base number. 1 in the second place value makes ten, and 3 in the first place value makes three, so ten plus three is thirteen, the quantity we originally set out to represent in decimal. From this, we can develop a general formula for the quantity represented by a symbol in a number system:

q = v(b^n)

Where q is the quantity, v is the symbol value, b is the base number, and n is the place value, starting at zero. If this equation is used for all symbols in a number, and the results are summed together, you get the total value of the number. If the equation is used in this manner in a computer program, you can easily convert from any number system to a computer-friendly decimal number, given that you know the base number.

Some examples:

The number 100110, base two to base ten

q = 1(2^5) + 0(2^4) + 0(2^3) + 1(2^2) + 1(2^1) + 0(2^0)
q = 1(32) + 0(16) + 0(8) + 1(4) + 1(2) + 0(1)
q = 32 + 0 + 0 + 4 + 2 + 0
q = 38

The number A5, base sixteen to base ten

q = 10(16^1) + 5(16^0)
q = 10(16) + 5(1)
q = 160 + 5
q = 165

It even works with fractions!
The number 101.101

q = 1(2^2) + 0(2^1) + 1(2^0) + 1(2^-1) + 0(2^-2) + 1(2^-3)
q = 1(4) + 0(2) + 1(1) + 1(1/2) + 0(1/4) + 1(1/8)
q = 4 + 1 + .5 + .125
q = 5.625