09-15-2009Representing NumbersThe representation system for numbers that you are used to is called a positional base 10 or decimalsystem. The idea is that when you write down the number 256 the digit 6 in the “ones” position tellsyou that there are 6 things, the digit 5 in the “tens” position tells you there are 50 things, and the digit2 in the “hundreds” position tells you there are 200 things. In total 200 + 50 + 6 = 256.Example 1:• 1341 = 1000 + 300 + 40 + 1 = 1 · 103+ 3 · 102+ 4 · 101+ 1 · 100• One thousand, three hundreds, four tens, and one one.It turns out that in certain applications (particularly computer science) it can be useful to representnumbers in a different base.Example 2: Let’s explore the idea of witting numbers in base 5.• In base 5 we have the digits 0,1,2,3,4. Our “positions” are ones, fives, fives squared, fives cubed,...• From what we know about base 10 systems, let’s try to write 37 in base 5. Now we know 52= 25,51= 5 and 50= 1. 25 divides into 37 one time with a remainder of 12, so in the 52place we shouldhave a 1. 5 divides into 12 two times with a remainder of 2, so in the 5 place we should have a 2.Since 1 divides 2 two times we should have a 2 in the ones place.37 = 1 · 52+ 2 · 51+ 2 · 50= (122)5• (3124)five= 3 · 53+ 1 · 52+ 2 · 51+ 4 · 50• 97 = (342)fiveExample 3: Binary representation of numbers.• At the most basic level computer memory is a vast series of “switches” which can either be “off”(0) or “on” (1).• Every number can be encoded by a sequence of on and off switches that is as a number in base 2(binary).• Let’s write 11 in binary. 11 ÷ 23= 1R3, 3 ÷ 22= 0R3, 3 ÷ 21= 1R1, and 1 ÷ 20= 1. So11 = 1 · 23+ 0 · 22+ 1 · 21+ 1 · 20= (1011)two• (10010)two= 0 · 20+ 1 · 2 + 0 · 22+ 0 · 23+ 1 · 24= 2 + 24= 18Example 4: Base 12.• In base 12 we have the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B where A is the digit for 10 and B is thedigit for 11. The positions in this system are ones, twelves, twelves squared, twelves cubed...• (23)twelve= 2 · 12 + 3 · 1 = 27.• (11A)twelve= 122+ 12 + 10 = 144 + 12 + 10 =
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