Why a Zero Exponent Equals One, and Changing Number BasesDate: 9/26/95 at 19:24:48 From: Anonymous Subject: Business Dear Dr. Math: Why is any number to the zero power equal to one? Also, could I have some information on hexadecimal and binary for my classes. Thank you! Carol Lyons Date: 9/30/95 at 15:52:21 From: Doctor Ken Subject: Re: Business Hello! About your first question: I think the right way to think about exponents is to say that a^k is equal to 1 times k x's. So 2^3 = 1*2*2*2 = 8, 2^(-4) = 1*1/2*1/2*1/2*1/2 = 1/16 since multiplying something once should be the opposite of dividing something once. With this concept in mind, any number to the zero power is "1 times no repetitions of that number." So any number to the zero power is 1. Also, look at the following chart: 2^5 2^4 2^3 2^2 2^1 2^0 2^-1 2^-2 32 16 8 4 2 1 1/2 1/4 Every time you go one higher in the exponent, you multiply by 2 (and every time you go one lower, you divide by 2). Here's how hexadecimals and binaries work. What we usually deal with is base ten. It's just a method of notation, a way that's convenient for us to write down numbers. It means that if we have the number 9745, that's 9*10^3 + 7*10^2 + 4*10^1 + 5*10^0. If there are numbers after the decimal point, you just continue the pattern: 234.95 = 2*10^2 + 3*10^1 + 4*10^0 + 9*10^-1 + 5*10^-2 So it's actually very related to your first question. The only difference between base 10 and base anything else is that we replace 10 (as in 9*10^3) with the new number, and instead of using 10 digits, we now use however many digits our base is. So in base 2 (binary) we use 2 digits, 0 and 1, and in base 16 (hexadecimal) we use 16 digits, 0,1,2,3,4,5,6,7,8,9,a,b,c,d,e, and f. To convert the number 984 to hexadecimal, we'd try to write it as w*16^3 + x*16^2 + y*16^1 + z*16^0 where w,x,y, and z are between 0 and 15(f). Since 16^3 = 4096, we have w=0, since 1 would be too big. 16^2 = 256. How many times does 256 go into 984? Three. That's x. So now we have 984 = 3*16^2 + y*16^1 + z*16^0. So 216 = y*16^1 + z*16^0. Since 16 goes into 216 13 times, y is 13, i.e. "d". So we have 984 = 3*16^2 + 13*16^1 + z*16^0. So 8 = z*16^0, and z=8. So 984 in hexadecimal is 3d8. How would you write 984 in binary? -Doctor Ken, The Geometry Forum |
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