Base Number Systems

 Convert between base 10 and other bases.

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 10 2 + 5 x 10 1 + 3 x 10 0
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.

In a base five system it is not necessary to have 10 numerals as in the Hindu-
Arabic 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:

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.

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 242 five can be expanded as
242 five
= 2 x 5 2 + 4 x 5 1
= 2 x 25 + 4 x 5 + 2 x 1
+ 2 x 5 0
= 50 + 20 + 2
= 72

EXAMPLE 1 Converting Numbers from
Base Five to Base 10
Write each number in base 10.
(a) 42 five
(b) 134 five
(c) 4213 five

EXAMPLE 1 Converting Numbers from
Base Five to Base 10
SOLUTION
The place value chart for base five is used in each case.
(a)42 five
= 4 x 5 1 + 2 x 1 = 20 + 2 = 22
(b) 134 five
= 1 x 5 2 + 3 x 5 + 4 x 1
= 1 x 2 5 + 3 x 5 + 4 x 1
= 25 + 15 + 4 = 44
(c) 4213 five
= 4 x 5 3 + 2 x 5 2 + 1 x 5 + 3 x 1
= 4 x 125 + 2 x 25 + 1 x 5 + 3 x 1
= 500 + 50 + 5 + 3 = 558

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 314 five
. 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 10103 five
.
Check: 1 x 625 + 0 x 125 + 1 x 25 + 0 x 5 + 3 x 1 = 653.

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.)

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:

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) 132 six
(b) 10110 two
(c) 1532 eight
(d) 2102 three

EXAMPLE 4 Converting Numbers to
Base 10
SOLUTION
(a) 132 six
= 1 x 6 2 + 3 x 6
= 1 x 36 + 3 x 6 + 2 x 1
1 + 2 x 1
= 36 + 18 + 2 = 56
(b)10110two = 1 x 2 4 + 0 x 2 3 + 1 x 2 2 + 1 x 2 1 + 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 8 3 + 5 x 8 2 + 3 x 8 1 + 2 x 1
= 1 x 512 + 5 x 64 + 3 x 8 + 2 x 1
= 512 + 320 + 24 + 2 = 858
(d) 2102 three
= 2 x 3 3 + 1 x 3 2 + 0 x 3
= 2 x 27 + 1 x 9 + 0 x 3 + 2 x 1
1
= 54 + 9 + 0 + 2 = 65
+ 2 x 1

EXAMPLE 5 Converting Numbers to Bases
Other Than 10
(a) Write 48 in base three.
(b) Write 51 in base two.