Lesson 3, Topic 3
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# Converting Numbers in Other Bases to Base Ten

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A given number can be converted from any base to base 10 using the Expansion Method.

Once you are able to work out the place values, you can easily convert other base numbers to binary.

Let’s take a look at the following examples;

### Example 1

Write the following numbers in expanded form.

1. 356eight
2. 10321four

Solution

1st Step: To use the expansion method for conversion we first find the place numbers.

1. 356eight

Students often find it difficult to locate the place values. The easiest way is to start from the last digit to the first digit. The last Digit always starts with a 0, the digit that follows a 1, and so on. (0, 1 2, 3, etc)

2nd Step: Multiply each digit by its base number raised to the power of its place value

356eight  = 3 x 82 + 5 x 81 + 6 x 80

Base eight is also called the octal system

2. 10321four

Multiply each digit by its base number raised to the power of its place value

10321four = 1 x 44 + 0 x 43 + 3 x 42 + 2 x 41 + 1 x 40

To convert any number base to base ten, simply express it as the power of the given base. As discussed earlier the place value is important here.

### Example 2

Express the following binary numbers as numbers in base ten.

a. 110100two
b. 534six
c. 4307eight

a. 110100two

Solution

Using the expanding method

110100ten = 1 x 25 + 1 x 24 + 0 x 23 + 1 x 22 + 0 x 21 + 0 x 20

= (1 x 25) + (1 x 24) + (0 x 23) + (1 x 22) + (0 x 21) + (0 x 20)

= (1 x 32) + (1 x 16) + 0 + (1 x 4) + 0 + 0

= 32 + 16 + 0 + 4 + 0 + 0

= 32 + 16 + 4

= 52ten

110100two = 52ten

b. 534six

Solution

Using the expansion method:

534six = 5 x 62 + 3 x 61 + 4 x 60

= (5 x 36) + (3 x 6) + (4 x 1)

= 180 + 18 + 4

= 202ten

534six  = 202ten

Note
62 = 6 x 6 = 36
61 = 6
60 = 1

c. 4307eight

Solution

Using the expanding method: (same approach as Examples above)

4307eight = 4 x 83 + 3 x 82 + 0 x 81 + 7 x 80

= (4 x 512) + (3 x 64) + (0 x 8) + (7 x 1)

= 2048 + 192 + 0 + 7

= 2247ten

⇒ 4307eight = 2247ten

### Example 3

Find the value of (111two)2 in base two

Solution

1st Step: The first thing to do is to convert the number to base ten.
2nd Step: After converting to base ten we will then square the value.
3rd Step: Finally we will convert the squared value back to base two.

1st Step:

Using the expansion method;

111two = (1 x 22) + (1 x 21) + (1 x 20

= 1 x 4 + 1 x 2 + 1 x 1

= 4 + 2 + 1

= 7ten

2nd Step:

Recall that (111two)2 = (7ten)2 = 7 x 7 = 49

= 49ten

3rd Step:

Then convert 49ten to base two (binary) using the method of repeated division

⇒ 49ten = 110001two

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