# 4.3 Notes

```Math in Our World
Section 4.3
Base Number Systems
Learning Objectives
 Convert between base 10 and other bases.
Base Number System
If a system uses some of our “regular”
numerals, but is based on powers other than
10, we will call it a base number system.
In the base 10 positional system, a number like 453 can be
expanded out as
453 = 4 x 100 + 5 x 10 + 3 x 1
= 4 x 102 + 5 x 101 + 3 x 100
We understand that a 5 in the second digit from the right
means five 10s. We can expand numbers in positional
systems with bases other than 10 in the same way.
Base Five System
In a base five system it is not necessary
to have 10 numerals as in the HinduArabic system; only five numerals
(symbols) are needed. A base five
number system can be formed using only
the numerals 0, 1, 2, 3, and 4. Just as
each digit in the Hindu-Arabic system
represents a power of 10, each digit in a
base five system represents a power of 5.
The place values for the digits in base
five are:
Base Five System
When writing numbers in base five, we use the subscript
“five” to distinguish them from base 10 numbers, because
a numeral like 453 in base 5 corresponds to a different
number than the numeral 453 in base 10. The table below
shows some base 10 numbers also written in base five.
Converting Base Five to Base 10
Base five numbers can be converted to base 10
numbers using the place values of the base five
numbers and expanded notation. For example,
the number 242five can be expanded as
242five = 2 x 52 + 4 x 51 + 2 x 50
= 2 x 25 + 4 x 5 + 2 x 1
= 50 + 20 + 2
= 72
EXAMPLE 1
Converting Numbers from
Base Five to Base 10
Write each number in base 10.
(a) 42five
(b) 134five
(c) 4213five
EXAMPLE 1
Converting Numbers from
Base Five to Base 10
SOLUTION
The place value chart for base five is used in each case.
(a)42five = 4 x 51 + 2 x 1 = 20 + 2 = 22
(b) 134five = 1 x 52 + 3 x 5 + 4 x 1
= 1 x 25 + 3 x 5 + 4 x 1
= 25 + 15 + 4 = 44
(c) 4213five = 4 x 53 + 2 x 52 + 1 x 5 + 3 x 1
= 4 x 125 + 2 x 25 + 1 x 5 + 3 x 1
= 500 + 50 + 5 + 3 = 558
Converting Base 10 to Base 5
Base 10 numbers can be written in the base five
system using the place values of the base five
system and successive division. This method is
illustrated in Examples 2 and 3.
EXAMPLE 2
Converting Numbers from
Base 10 to Base Five
Write 84 in the base five system.
SOLUTION
Step 1 Identify the largest place value number (1, 5, 25, 125, etc.) that
will divide into the base 10 number. In this case, it is 25.
Step 2 Divide 25 into 84, as shown.
This tells us that there are three 25s in 84.
Step 3 Divide the remainder by the next lower place
value. In this case, it is 5.
Step 4 Continue dividing until the remainder is less than 5. In this case, it
is 4, so the division process is stopped. In other words, four 1s are left.
The answer is 314five. In 84, there are three 25s, one 5, and four 1s.
EXAMPLE 3
Converting Numbers from
Base 10 to Base Five
Write 653 in the base five system.
SOLUTION
Step 1 Identify the largest place value number (1, 5, 25, 125, etc.) that
will divide into the base 10 number. In this case, it is 625.
Divide 625 into 653, as shown.
Step 2 Divide the remainder by the
next lower place value, which is 125.
Even though 125 does not divide into the 28, the zero must be written to
hold its place value in the base five number system.
EXAMPLE 3
Converting Numbers from
Base 10 to Base Five
Write 653 in the base five system.
SOLUTION
Step 3 Divide the remainder by the next lower place value, which is 25.
Step 4 Divide by 5.
Since we’ve reached the
ones place value, 3 is
our last digit in the
After reviewing the results, the solution is 10103five.
Check: 1 x 625 + 0 x 125 + 1 x 25 + 0 x 5 + 3 x 1 = 653.
Other Number Bases
Once we understand the idea of alternative
bases, we can define new number
systems with as few as two symbols, or
digits. (Remember, we only needed digits
zero through four for base five numbers.)
Binary System
For example, a base two, or binary system
(used extensively in computer programming)
uses only two digits, 0 and 1. The place values
of the digits in the base two numeration system
are powers of two:
Octal System
The base eight or octal
system consists of eight digits,
0, 1, 2, 3, 4, 5, 6, and 7. The
place values of the digits in the
base eight system are powers
of eight:
EXAMPLE 4
Converting Numbers to
Base 10
Write each number in base 10.
(a) 132six
(b) 10110two
(c) 1532eight
(d) 2102three
EXAMPLE 4
Converting Numbers to
Base 10
SOLUTION
(a) 132six = 1 x 62 + 3 x 61 + 2 x 1
= 1 x 36 + 3 x 6 + 2 x 1
= 36 + 18 + 2 = 56
(b)10110two = 1 x 24 + 0 x 23 + 1 x 22 + 1 x 21 + 0 x 1
= 1 x 16 + 0 x 8 + 1 x 4 + 1 x 2 + 0 x 1
= 16 + 0 + 4 + 2 + 0 = 22
(c)1532eight = 1 x 83 + 5 x 82 + 3 x 81 + 2 x 1
= 1 x 512 + 5 x 64 + 3 x 8 + 2 x 1
= 512 + 320 + 24 + 2 = 858
(d) 2102three = 2 x 33 + 1 x 32 + 0 x 31 + 2 x 1
= 2 x 27 + 1 x 9 + 0 x 3 + 2 x 1
= 54 + 9 + 0 + 2 = 65
EXAMPLE 5
Converting Numbers to Bases
Other Than 10
(a) Write 48 in base three.
(b) Write 51 in base two.
```