Binary & Hexadecimal Explained Simply

Binary & Hexadecimal Explained Simply

If you have just started learning Computer Science, Binary can look strange at first. A page full of 1s and 0s does not immediately feel like a number system. It can look more like a secret code.

Many students ask the same questions: Why do computers use Binary? Why do we need Hexadecimal? Why can't computers just use the normal numbers we use every day?

Those are good questions. In this FutureLogic lesson, we are going to answer them properly. We will not just memorise methods. We will understand why they work.

By the end of this lesson, you will be able to:

Why students find Binary difficult

Most students do not struggle with Binary because it is too hard. They struggle because they are often asked to convert numbers before anyone has explained why Binary exists.

That is the wrong order. If you start with a table of place values, Binary feels like something to memorise. If you start with the computer hardware, Binary begins to make sense.

Why do computers use Binary?

Imagine a simple light switch. It has only two reliable positions:

• OFF
• ON

Now imagine shrinking that switch until billions of them can fit inside a tiny computer chip. Those tiny electronic switches are called transistors.

A transistor can be thought of as having two reliable electrical states. Engineers represent those two states using the digits 0 and 1.

This is why Binary is a base-2 number system. It only uses two digits: 0 and 1. Your revision notes summarise this clearly: electronic circuits have two states, ON and OFF, represented by 1 and 0.

What are Binary place values?

In Denary, the number system we normally use, each column is worth ten times more as you move left. That is why we have ones, tens, hundreds and thousands.

Binary works in a similar way, but because it is base-2, each column is worth two times more as you move left.

The key place values for an 8-bit binary number are:

These are the values you should write above every 8-bit binary number before converting. Your revision notes give exactly the same exam warning: do not try to do the place values in your head.

Binary to Denary: how to convert

To convert Binary to Denary, look at each column. If the bit is 1, include that place value. If the bit is 0, ignore that place value.

Worked example: Convert 01010101 to Denary

So:

The small ₂ means Binary. The small ₁₀ means Denary. You do not always need to write those small base numbers in your exam answer, but they are useful while learning.

Denary to Binary: how to convert

Converting Denary to Binary works in the opposite direction. Start with the largest place value on the left and ask one question each time:

If it fits, write 1 and subtract it. If it does not fit, write 0 and move to the next place value.

Worked example: Convert 45 to Binary

Reading the bits from top to bottom gives:

The leading zeros matter. In an 8-bit answer, 00101101 is better than just writing 101101, because it shows the complete 8-bit pattern.

What is Hexadecimal?

Hexadecimal is a base-16 number system. That means it uses sixteen symbols instead of two or ten.

The symbols are:

The letters represent the values from 10 to 15.

Why is Hexadecimal used?

Computers use Binary, but long Binary numbers are difficult for humans to read. Hexadecimal makes Binary shorter and easier to work with.

The important fact is this:

Four bits are called a nibble. This makes conversion between Binary and Hexadecimal very neat.

Binary to Hexadecimal: how to convert

To convert Binary to Hexadecimal, split the binary number into groups of four bits from the right. Then convert each group into one hexadecimal digit.

Worked example: Convert 11111111 to Hexadecimal

So:

This is why Hexadecimal is useful. Eight binary digits become just two hexadecimal digits.

Worked example: Convert 10100111 to Hexadecimal

So:

Binary Addition

Binary addition works column by column, just like Denary addition. The difference is that Binary can only use 0 and 1, so when a column becomes too large, you carry into the next column.

What is Overflow?

Overflow happens when the result of a binary addition is too large to fit into the number of bits available.

For example, in 8 bits:

The true answer needs 9 bits. But if only 8 bits are available, the extra bit cannot be stored. The stored result becomes:

That is overflow. The result is wrong because the answer did not fit into the available number of bits.

What is Two's Complement?

Two's complement is a method used to represent negative numbers in Binary.

The method has two steps, and the order matters:

1. Invert all the bits.
2. Add 1.

Worked example: Find the two's complement of 00101101

So the two's complement version is:

Common Exam Mistakes

Examiner Tips

• Show your working. Method marks can still be awarded even if the final answer is wrong.
• Write place values above numbers. It takes seconds and prevents common errors.
• Check carries carefully. A missed carry can ruin the answer.
• Practise conversions regularly. Speed and accuracy come from repetition.

Quick Quiz

1. Convert 10101010 into Denary.
2. Convert 45 into 8-bit Binary. Show your working.
3. What is hexadecimal A in Binary?
4. What causes overflow, and how can you detect it?
5. Convert 00110011 to its two's complement.

Quick Quiz Answers

1. 10101010 = 128 + 32 + 8 + 2 = 170
2. 45 = 32 + 8 + 4 + 1 = 00101101
3. Hexadecimal A = 1010
4. Overflow occurs when the result is too large for the available bits. It can be detected when there is a carry out of the most significant bit.
5. Invert: 11001100. Add 1: 11001101

Final Summary