Math in Our World
Section 4.3
Base Number Systems
Learning Objectives
 Convert between base 10 and other bases.
 Convert between binary, octal, and
hexadecimal.
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
answer.
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:
Hexadecimal System
When the base number is greater
than 10, new digits must be created
to make the numbers. For example,
base 16 (called the hexadecimal
system) is used in computer
technology. We need 16 digits for
this system; the digits in base 16
are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B,
C, D, E, and F, where A represents
10, B represents 11, etc. The place
values of the digits in base 16 are
powers of 16:
Base Number Systems
For any base system,
1. The number of symbols used is equal to the
base.
2. The place values of any base are
…b6, b5, b4, b3, b2, b1, b0
3. In order to convert from numbers written in
bases other than 10 to base 10 numbers,
expanded notation is used.
EXAMPLE 4
Converting Numbers to
Base 10
Write each number in base 10.
(a) 132six
(b) 10110two
(c) 1532eight
(d) 2102three
(e) 5BD8sixteen
EXAMPLE 4
Converting Numbers to
Base 10
SOLUTION
(a) The place values of the digits in base six are powers of 6:
132six = 1 x 62 + 3 x 61 + 2 x 1
= 1 x 36 + 3 x 6 + 2 x 1
= 36 + 18 + 2 = 56
(b) The place values of the digits in base two are powers of 2:
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) The place values of the digits in base eight are powers of 8:
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
EXAMPLE 4
Converting Numbers to
Base 10
SOLUTION
(d) The place values of the digits in base three are powers of 3:
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
(e) The place values of the digits in base 16 are powers of 16:
5BD8sixteen = 5 x 163 + 11 x 162 + 13 x 161 + 8 x 1
= 5 x 4,096 + 11 x 256 + 13 x 16 + 8 x 1
= 20,480 + 2,816 + 208 + 8 = 23,512
EXAMPLE 5
Converting Numbers to Bases
Other Than 10
(a) Write 48 in base three.
(b) Write 51 in base two.
(c) Write 19,443 in base 16.
EXAMPLE 5
Converting Numbers to Bases
Other Than 10
SOLUTION
(a) Write 48 in base three.
Step 1 The place values for base three are powers of three. The
largest power of three less than 48 is 33, or 27, so we divide 48
by 27.
Step 2 Divide the remainder by 32 or 9.
Step 3 Divide the remainder by 31 or 3.
So, 48 is 1 x 33 + 2 x 32 + 1 x 31 + 0 x 30, which makes it 1210three.
EXAMPLE 5
Converting Numbers to Bases
Other Than 10
SOLUTION
(b) Write 51 in base two.
The place values for base two are 1, 2, 4, 8, 16, 32, etc.
Use successive division, as shown.
So, 51 = 110011two.
EXAMPLE 5
Converting Numbers to Bases
Other Than 10
SOLUTION
(c) Write 19,443 in base 16.
The place values in base 16 are 1, 16, 256 (162), 4096 (163),
etc. Use successive division as shown. (Remember, in base
16, B plays the role of 11 and F plays the role of 15.)
So 19,443 = 4BF3sixteen.
EXAMPLE 6
Converting Between
Octal and Binary
(a) Convert the binary number 1001110110two
to octal.
(b) Convert the octal number 7643eight to binary.
EXAMPLE 6
Converting Between
Octal and Binary
SOLUTION
(a) Convert the binary number 1001110110two to octal.
Starting at the rightmost digit, group the
digits of the binary number into groups of
three (if there are not three digits that
remain at the left of the number, fill them
in with zeros). Then use the table to
change each group to an octal digit as
follows.
001 001 110 110
1 1 6 6
So, 1001110110two = 1166eight.
EXAMPLE 6
Converting Between
Octal and Binary
SOLUTION
(b) Convert the octal number 7643eight to binary.
First, convert each octal digit into a
three-digit binary digit using the table,
and then string them together to form
a binary number.
7 6 4
3
111 110 100 011
So, 7643eight = 111110100011two.
EXAMPLE 7
Converting Between
Binary and Hexadecimal
(a) Convert the binary number 1110011001111two
to hexadecimal.
(b) Convert the hexadecimal number 9D7A3sixteen
to binary.
EXAMPLE 7
Converting Between
Binary and Hexadecimal
SOLUTION
(a) Convert the binary number 1110011001111two to
hexadecimal.
Starting at the rightmost digit, group the
binary number into groups of four (adding
zeros in front as needed) and then
convert each group of four to a
hexadecimal digit using the table.
0001 1100 1100 1111
1
C
C
F
So 1110011001111two = 1CCFsixteen.
EXAMPLE 7
Converting Between
Binary and Hexadecimal
SOLUTION
(b) Convert the hexadecimal number 9D7A3sixteen to
binary.
First, convert each hexadecimal
number to a four-digit binary number,
and then string them together to form
a binary number.
9
D
7
A
3
1001 1101 0111 1010 0011
So 9D7A3sixteen =
10011101011110100011two.