The Handy Math Answer Book

How do you convertfrom the binaryto the decimal systemand vice versa?

Besides the decimal system, one of the most familiar number systems is the binary

numeration system; this is mainly because of its use in computers (for more about

computers, see “Math in Computing”). In a binary numeration system, only 1 and 0

are used—or a base 2 system. Converting between binary and decimal systems is fairly

simple; just remember that each digit in the binary number represents a power of two.

The first column in the base 2 math is the units column,then the twos, fours,

eights, etc. columns, all of which can only be filled with 0s or 1s. Since there is no sin-

gle digit that stands for “2” in base 2, when you get to what stands for 2, you put a 1 in

the 2’s column and a 0 in the units column, creating one 2 and no 1s. Thus, the base

ten “two” (2 10 , or just 2 in decimal form) is written in the binary as 10 2 ; a 3 (3 10 , or just

3 in decimal form) in base 2 is actually “one 2 and one 1,” or 11 2. The number 4 is

actually 2 2, so you eliminate the 2 and unit columns and put a 1 in the 4s column.

Thus, 4 10 (or just 4 in decimal form) is written in binary form as 100 2. To see how

computers “translate” decimal to binary numbers, here are the first ten conversions:

Decimal Binary Explanation

0 0 no 1s

1 1 one 1

2 10 one 2 and no 1s

3 11 one 2 and one 1

4 100 one 4, no 2s, and no 1s

5 101 one 4, no 2s, and one 1

6 110 one 4, one 2, and no 1s

7 111 one 4, one 2, and one 1

8 1000 one 8, no 4s, no 2s, and no 1s

9 1001 one 8, no 4s, no 2s, and one 1

10 1010 one 8, no 4s, one 2, and no 1s

How do you expresscertain numbers in powers of tennotation?
A number in power of ten notation is represented as a base number (or mantissa) times
ten raised to some power (or exponent). This means the mantissa is multiplied by ten
times the number of times the power indicates. For example, in the equation 32  104 ,
the 32 is the mantissa multiplied by 10  10  10 10. This would equal 320,000.
Power of ten notation can also be used to express the same number in various
ways. For example, the distance from the Earth to the Sun averages 93,000,000 miles.
This can be represented by the following:
93,000,000
93  106
9.3  107
88 or 0.93 ^108 , and so on.