How To Convert To Base 10 From Other Bases

We will go over how to convert other bases to Base 10.

Example 1

Convert

(i) 7603 _eight to base 10

(ii) 1101 _two to base 10

(iii) 5B9 ₁₂ up to 10 base

Solutions (i) 7603 _eight to base 10

a. First Method: (Power Expansion)

Power Expansion entails multiplying out the base of the number by the base value.

• Then write down the place values beginning on the right-hand side.
• Write each digit under its place value.
• Multiply each number in the number of its basis raised to its corresponding place value.(i.e. baseplace value. Note that in this case the answer will be an 8place value)
• Add the two products. The result is the decimal number of base 10.

Many students find it difficult to find the place values. The easiest method is to start by going from the last digit until the first number. Every last Digit begins with a zero, the number that follows one, and it goes on. (0, 1 2, 3, etc)

Write down the number of the base increased to the place value. The number that we’re changing is 7603 _{eight, and} it’s in base 8.

Multiply each digit with its base, and then raise it to the equivalent place value.

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

= 3584 + 384 + 0 + 3

= 3,971_ten

7603_eight = 3,971_ten

b. Alternate Method – (Successive Multiplication)

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

Solutions (ii) 1101 _two to base 10

a. First Method: (Power Expansion)

Multiply each digit with its base and raise it to the appropriate place value

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

= 8 + 4 + 0 + 1

= 13_ten

Alternate Method

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

Solutions (iii) 5B9 ₁₂ in base 10.

a. First Method: (Power Expansion)

Multiply each number by the base of its value, and then raise it to the appropriate amount. ( Note: B = 11)

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

= 720 + 132 + 9

= 861_ten

Alternate Method

5 x 12 = 60 + 11 = 71 x 12 = 852 + 9 = 861 _ten

Example 2

Convert 1101.101 _twoto base Ten

When you convert the decimal number to base 10 the entire number is treated the same as in the above examples (i.e beginning with the last digit the numbering will be 0, 1, 2 3, etc. until the first number).

The decimal portion on the other hand , is identified as -1, -2, 3, etc, immediately after the decimal mark. The first number that follows at the point of decimal will be 1; while the next number will be -2, and so on.

Hence. The first number is multiplied by the number of the base raised to the power of 1 (x ^-1) and then on.

Solution

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

= 8 + 4 + 0 + 1 + 12 + 0 + 18

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

= 13.625

1101.101_two= 13.625_ten

Alternate Method

Whole number of parts is 1 2 + 1, 3, 2, = 6 + 0 = x 2 = 12 + 1 = 13.

Fractional portion = 1 2 x 2 = 2 + 2 = 4 + 1 = 5.

The final result for the entire number is two = 13 ten ₂ is 13 _Ten

The value of the column from the fractional portion is 2 ³

0.101_two= 5 x 2^-3 = 5 x 18 = 0.625

Compiling the results Then

1101.101_two = 13 + 0.625

1101.101_two = 13.625_ten

Example 3

Convert

(i) 4A3D.2 ₁₆ to base 10

(ii) 4B3.A6 ₁₂ to base 10

Solutions (i) 4A3D.2 ₁₆ to base 10

Remember the number 13 = D from our Table of Number Bases

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

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

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

= 19,005.125

Solutions (ii) 4B3.A6 ₁₂ to base 10

A = 10, B = 11

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

= (576) + (11 x 12) + (3) + (11 x 0.0833) + (6 x 0.00694)

= 576 + 132 + 3 + 0.9163 + 0.04164

= 711.96