Lesson 1, Topic 2
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# Converting from other Bases to Base 10

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Here, we will learn the methods to convert from other bases to Base 10

### Example 1

Convert

(i) 7603eight to base 10

(ii) 1101two to base 10

(iii) 5B9twelve to base 10

Solution (i) 7603eight to base 10

a. First Method: (Power Expansion)

Power Expansion entails multiplying out the number base using their base values.

• First, write the place values starting from the right-hand side.
• Write each digit under its place value.
• Multiply each digit by its base raised to the corresponding place value.
(i.e. baseplace value, Note: for this question, it will be 8place value)
• Add up the products. The answer will be the decimal number in base ten.

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)

Write down the base raised to the place value. The number we are converting is 7603eight ∴ it’s in base 8.

Multiply each digit by its base raised to the corresponding place value.

7603eight = (7 × 83) + (6 × 82) + (0 × 81) + (3 × 80)

= 3584 + 384 + 0 + 3

= 3,971ten

∴    7603eight = 3,971ten

b. Alternate Method – (Successive Multiplication)

7 × 8 = 56 + 6 = 62 × 8  = 496 + 0 = 496 × 8 = 3968 + 3 = 3,971ten

Solution (ii) 1101two to base 10

a. First Method: (Power Expansion)

Multiply each digit by its base raised to the corresponding place value

1101two = (1 × 23) + (1 × 22) + (0 × 21) + (1 × 20)

= 8 + 4 + 0 + 1

= 13ten

Alternate Method

1 × 2 = 2 + 1 = 3 × 2 = 6 + 0 = 6 × 2 = 12 + 1 = 13ten

Solution (iii) 5B9Twelve to base 10

a. First Method: (Power Expansion)

Multiply each digit by its base raised to the corresponding place value (Note: B =11)

5B9Twelve = (5 × 122) + (11 × 121) + (9 × 120)

= 720 + 132 + 9

= 861ten

Alternate Method

5 × 12 = 60 + 11 = 71 × 12  = 852 + 9 = 861ten

### Example 2

Convert 1101.101two to base ten

In the conversion of a decimal number to base 10, the whole number is treated as the examples above (i.e from the last digit the numbering is 0, 1, 2, 3, etc up to the first digit).

The decimal part on the other hand is numbered -1, -2, -3, etc immediately after the decimal point. The first number after the decimal point will be -1, the second number will be -2, etc.

Hence. We multiply the first number by the base raised to the power minus one (x-1) and so on.

Solution

1101.101two = (1 × 23) + (1 × 22) + (0 × 21) + (1 × 20) + (1 × 2-1) + (0 × 2-2) + (1 × 2-3)

= 8 + 4 + 0 + 1 + $$\frac {1}{2}$$ + 0 + $$\frac {1}{8}$$

= 8 + 4 + 0 + 1 + 0.5 + 0 + 0.125

=  13.625

1101.101two= 13.625ten

Alternate Method

Whole number part = 1 × 2 = 2 + 1 = 3 × 2 = 6 + 0 = 6 × 2 = 12 + 1 = 13

Fractional part = 1 × 2 = 2 + 0 = 2 × 2 = 4 + 1 = 5

The result for the whole number part is: 1101two = 13ten

The place value of the last column of the fractional part is 2-3

∴   0.101two= 5 × 2-3 = 5 × $$\frac {1}{8}$$ = 0.625

Summing up the results, then

1101.101two = 13 + 0.625

1101.101two = 13.625ten

### Example 3

Convert

(i) 4A3D.216 to base 10

(ii) 4B3.A612 to base 10

Solution (i) 4A3D.216 to base 10

Remember D = 13 from our Number Bases table

4A3D.216 = (4 × 163) + (A × 162) + (3 × 161) + (D × 160) + (2 × 16-1)

= (16,384) + (10 × 256) + (48) + (13 × 1) + (2 × 0.0625)

= 16,384 + 2,560 + 48 + 13 + 0.125

= 19,005.125

Solution (ii) 4B3.A612 to base 10

A = 10, B = 11

4B3.A612 = (4 × 122) + (B × 121) + (3 × 120) + (A × 12-1) + (6 × 12-2)

= (576) + (11 × 12) + (3) + (10 × 0.0833) + (6 × 0.00694)

= 576 + 132 + 3 + 0.833 + 0.04164

= 711.875

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