How to Change Number Bases Part 2

In the previous post, we have learned how to change numbers form one base to other. In this post, we are going to discuss more examples of number bases particularly the two number systems used in computers: the binary and the hexadecimal system.

The Binary Number System

The binary number system has base 2 and only uses 1 and 0 as digits. The binary number 1101 in expanded form is

1 \times 2^3 + 1 \times 2^2 + 0 \times 2^1 + 1 \times 2^0 or 

8 + 4 + 0 + 1 = 13 in base 10. If we want to convert base 10 numbers to base 2, then we look for the highest multiple of the powers of 2 and subtract from the number each time until the difference is 0.

Example: Convert 74 to base 2.

The highest number that can be subtracted from 74 is 2^6 = 64. So, just like in the previous post, we can create a table with the largest number as 2^6 as shown below. Since, we can subtract 2^6 from 74, we write 1 under 2^6.

number bases 2-1

Subtracting, we have 74 - 2^6 = 10. We cannot subtract 2^5 and 2^4 from 10, so we write 0 under these numbers.

number bases 2-2

We know that 10 = 2^3 + 2^1, so we write 1 under them and write 0 in the remaining cells. The final answer is shown in the next table.

number bases 2-3

From the table, this means that 74_{10} = 1001010_2.

Exercises:

1.) Convert 213 to binary.
2.) Convert 100110 to decimal

The Hexadecimal Number System

The hexadecimal number system or the base 16 number system uses 0, 1, 2, all the way up to 15 as digits.

In base 10, when we talk about digit, we are taking about one number. One thing that probably comes to your mind is how do we deal with digits containing two digits in base 16? For example, in the hexadecimal number, is 14 a 1-digit number or a 2-digit number? To solve this problem, letters were used to denote digits from 10 to 15: A = 10, B = 11, C = 12, D = 13, E = 14 and F = 15. That means that the expanded form of E20A is

14 \times 16^3 + 2 \times 16^2 + 0 \times 16^1 + 10 \times 16^0

or 57344 + 512 + 0 + 10 = 58866 in base 10.

Just like in the previous examples, in converting base 10 numbers to base 16 numbers, we need to look for the highest multiples of powers of 16 and subtract each time until the difference is 0.

Example: Convert 12452_{10} to base 16.

We know that 16^4 = 65536, so we can only subtract multiples of 16^3 = 4096 from 12452. Obviously, 12452 divided by 4096 will give an integral quotient of 3, so we write 3 under 16^3 in the table below.

number bases 2-4

Now, 12452 - 3(16^3) = 12452 - 12288 = 164. Clearly, we cannot subtract 16^2 from 164, so we put 0 under it.

number bases 2-5

Next, there are ten 16^1 in 164, so we write A under it. Next, 164 – 10(16) = 4. So, we write 4 under 16^0.

number bases 2-6

Exercise:

Convert 18B2 to base 10.
Convert 15000 to base 16.

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One thought on “How to Change Number Bases Part 2

  1. Hi,
    Nice rundown. I hope you also will tell about the shortcut of 16 [i.e. 10base16] being 2**4, and how simple it is to convert between binary and hex, four digits1 digit.

    While you are on it, you could do the same for octal, not that I am sure there are still computers around that use it (I remember such…)

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