D. Haugli, Lecturer Aer E 161 Aerospace Engineering
2/3/2005 Binary Conversion: Integers, Page 1 Iowa State University
Binary Conversion: Integers
This document demonstrates how to convert a base-10 integer to binary and back. In this
document, 2
n represents a binary number, 10
n a base-10 integer, and
γ
the highest, integer power
of 2 that fits into 10
n.
A search of the web indicates that two procedures are typically used to convert a base-10 number
to binary. The first procedure involves subtracting powers of 2 from the base 10 number, and the
second involves dividing the base-10 number repeatedly by 2.
The conversion from a base-10 integer to binary is exact and does not involve round-off errors.
Subtraction Procedure. The subtraction procedure depends on determining
γ
such that
γ
2 will
fit into the base-10 number (or its remainder) without exceeding it. Mathematically, this
statement is equivalent to:
1
210 =
γ
n, (1)
where any digits to the right of the decimal have been truncated.
Rearrange this equation and apply the property of logs to find
γ
.
,log2log
,log2log
,2
10
10
10
x
x
x
=
=
=
γ
γ
γ
2log
log 10
x
=
γ
. (2)
The subtraction procedure is:
1. Determine
γ
from Equation (2).
2. Evaluate the remainder:
γ
2
1010 −= nn .
3. Return to Step 1, computing the next highest power of 2 that will fit into the remainder.
Repeat Steps 1 and 2 until the remainder is 0. Keep track of each power of 2 that fits into
each remainder.
4. Assemble the binary number, 2
n, beginning with the highest power of two on the left.
The procedure is most easily completed with a table, as shown in Example 1.
