SECTION 2.1 Elementary Number Theory 91
The coefficients are called the (decimal)digits.
Arguably the second-most popular number base is 2, givingbinary
numbers(or binary representations of numbers). In this case the
binary digits include only “0” and “1”. As an example, we can convert
a binary number such as 1001101 into its equivalent decimal number
by computing the relevant powers of 2:
1001101 = 1· 20 + 0· 21 + 1· 22 + 1· 23 + 0· 24 + 0· 25 + 1· 26 = 77.
Another way of expressing this fact is by writing 77 2 = 1001101,mean-
ing that the binary representation of the decimal number 77 is 1001101.
Example 1. Find the binary representation of the decimal number
93.
Solution. First notice that the highest power of 2 less than or equal
to 93 is 2^6. Next, the highest power of two less than or equal to 93− 26 is
24. Continuing, the highest power of 2 less than or equal to 93− 26 − 24
is 2^3. Eventually we arrive at 93 = 2^6 + 2^4 + 2^3 + 2^2 + 1, meaning that
932 = 1011101.
Example 2. Find the binary representation of 11111. Note first that
if n is the number of binary digits required, then after a moment’s
thought one concludes thatn− 1 ≤log 211111 < n.Since log 2 11111 =
ln 11111
ln 2
≈ 13 .44, we conclude that 11111 will require 14 binary digits.
That is to say, 11111 = 2^13 + lower powers of 2.Specifically, one shows
that
11111 = 2^13 + 2^11 + 2^9 + 2^8 + 2^6 + 2^5 + 2^2 + 2 + 1.That is to say, 11111 2 = 10101101100111.
As one would expect, there areb-aryrepresentations for any base.
For example atrinaryrepresentation would be a representation base 3,
and the numbernof trinary digits needed to represent m would satisfy
n− 1 ≤log 3 m≤n.