Introduction

We use a base-10 numbering system. What does this mean? Let’s take an example of counting to 20. In a tally system, you might make a mark for every number. You’d have 20 marks by the end of it. This is very inefficient. Our normal way of representing numbers is to count to 9, then represent 9+1 by resetting our count to zero and carrying over the 1.

Hopefully makes sense, we’ve been doing it like this since Kindergarten! This is more efficient, since instead of requiring one character for every thing we count, we can compress our representation of that number count to two digits.

Base-2 (Binary)

Hopefully everyone is familiar with the idea that computers use a binary number system. What does that mean? It means that every one of our digits can have only two possible values that we understand: 1 or 0. More accurately, we chould say that every digit has two voltage levels: a high voltage and a low voltage. Let’s use the number 5 as an example.

2² = 4 2⁰ = 1

4 + 1 = 5

Simple enough, so far!

This, by the way, is four bis, which we can call a nibble.

And this is 8 bits, which we call a byte. If you add 128 + 64 + 32 + 16 + 8 + 4 + 2 + 1, you get 255, which is the maximum unsigned value you can represent with a byte.

Converting From Decimal to Binary

One skill you will required to demonstrate in this course is the ability to do this conversion without a calculator.

There are several strategies to do this, but here’s one that I recommend:

Let’s start with an example: 78.

Use long division to divide 78 by 2.

78/2 = 39 with a remainder of 0.

The remainder becomes your least significant bit.

Next, divide your result by 2 again:

39/2 = 19, remainder = 1.

Again, your remainder gets added to the next least significant bit.

19/2 = 9, remainder = 1.

9/2 = 4, remainder = 1.

4/2 = 2, remainder = 0.

The last division of 2/2 = 1 with a remainder of 0. Remember to add the remainder to your binary number first. Since our final result is less than our radix, we can add this to the most significant bit of our binary number.

One word about style: It is good practice to divide your binary numbers into nibbles to make it easier to read. We will add a leading zero and write our answer like so:

0100 1110

Converting from Binary to Decimal

This should be easy if you can memorize the following:

For this course, we aren’t going to ask you to convert exceedingly long numbers, so this should work for any quiz/test questions. 0b is one common way to represent binary numbers, which we will use here.

Example: 0b10101100.

First, arrange this number in a way that’s easy to read:

(1 * 128) + (1 * 32) + (1 * 8) + (1 * 4) = 172.

Another Handy Resource

Base-8 (Octal)

Octal numbers have digits between 0 and 7, at which point they carry over. We use octal numbers with chmod, which we will discuss this week.

When being asked to do conversions with octal, the easiest approach is to start with binary and to group bits into groups of three.

0b10101100 becomes 10 101 100. Convert from binary to ‘decimal’ to get: 2 5 4

The way to correctly identify an octal number varies, but a leading zero seems to be most common. So our number is 0254.

To convert from octal to another format, convert each digit to binary, then group your binary digits into nibbles.

Base-16 (Hexadecimal)

Hexadecimal is used often to represent memory addresses. Hexadecimal numbers have digits between 0 and 15, at which point they carry over. Here’s how we represent these digits:

When being asked to do conversions with octal, the easiest approach is to start with binary and to group bits into groups of four.

Let’s convert from Hex into decimal. Here’s our Hex Number: 0xD6.

0x is the most common way of representing Hex numbers.

D = 13, which is 1101 in binary. 6 = 6, which is 0110 in binary.

0xD6 = 1101 0110. From here we can convert to decimal to get 214.

Summary

• Base-10 is the number system you are already familiar with.
• Base-2 is binary. Two possible values: 1 or 0.
• To convert decimal to binary, divide the decimal number by 2, and assign the remainder to the least signficant bit. (LSB). Continue until you have a result less than 2, and add this to your binary number.
• Base-8 is octal. Possible values: 0 - 7. Octal digits contain 3 bits.
• Base-16 is hexadecimal. Possible values: 0 - F. Hex digits contain 4 bits.

Bonus (DLC?) Content: Why Use Binary At All?

• Why Use Binary? - My Attempt
• Why Use Binary? - Computerphile

Calculating the Number of Bits for a Given Decimal Number

This is not a calculation that you will need to demonstrate on any tests in this course, but it is something that you will find useful in many practical situations.

Let’s say you are given a decimal number 65001. You want to decided what unsigned data type is required to store this number. Use this formula:

log₂65001 = 15.988 Take the ceiling of that number (round up to the next integer) to get the number of bits required for this number. (In this case, 16 bits are required.)

Using a calculator, it can be difficult to program a log₂ calculation, so in a pinch, you can use log₁₀:

log₁₀65001/log₁₀2 = 15.988