Before we focus on the Binary number system, we need to understand number bases. This topic looks at the meaning of number bases and exercises using binary, octal, denary and hexadecimal (oh, and a sprinkling of dozenal as well). We are learning ...- About number bases.
So that we can ...- Represent numbers in different number bases
- Binary - Denary - Hexadecimal - Convert from one number base to another
- Binary <−> Denary - Understand how computers use different numbers bases
- Hexadecimal is used to represent binary - Use of binary and hexadecimal in representation of colours, MAC addresses, assembly language and machine code.
What's all this?
Before we start, let's look carefully at how humans and computers count. Task 1.1 Reading and questionsWhere we learn about the different ways that humans, computers and cartoon mice count Reading and notesRead this passage and make notes in your notebook / folder.
In your notebooksAnswer the following questions in full sentences.- Look at the image of
**Mickey Mouse**- how many fingers (and thumbs) does he have? - How many different values could he represent using his fingers (ignore zero)?
- What base would he count in?
- What values would their be in his number system?
- What would the base be called? (think geometric shapes ...)
This seems like an odd question doesn't it? Surely it's one thousand, one hundred and one! Yes, but only in Base 10! All the values in a number do is tell you how many of each place value there are in the value of the number. In Base 10, it's units, tens, hundreds, thousands etc - each place value is worth 10 times the previous one (Base 10, see?) This is what we are used to.Watch the presentation (go full screen to make it easier to read) ...
Thinking in any other number base than 10 is really hard - we naturally try to convert the number into base 10 to make it make sense to us - if we were the same as Mickey and had only 8 fingers, we would struggle to think in anything other than Base 8 (octal, if you didn't already know) and we'd count very differently.Task 2.1 Practice converting different bases to denaryWhere we learn how to convert values in different number bases to denary On the worksheetWhen you are ready, attempt the worksheet called Number conversion. Print out the completed worksheet single sided and hand it in or stick it in your notebooks for assessment.Image of Mickey Mouse from http://pngimg.com/
For your examination, you will need to be confident converting binary numbers to and from denary. It's common to convert to and from a byte - 8 bits. Consider the example below.2 makes it easy to simplify this table (thankfully) by removing the 'Place', 'Factor' and 'Worth' rows as in the example below. There is only ever one lot of each place value in binary unlike other bases - this actually makes binary the easiest base to work with!each place is worth twice the previous, slot in the binary number and add up the places which have a '1' in them.Task 3.1 Binary to denary conversionWhere we learn how to convert binary numbers to their denary value In your notebooks / on paperAttempt to answer the following in your notebooks. Show
all you working. The first one has been done for you - make sure that you look back over the examples if you are struggling, or ask for help!- Convert 01100011
_{2}to denary.
- Convert 00000011
_{2}to denary. - Convert 11000011
_{2}to denary. - Convert 01001111
_{2}to denary. - Convert 11110000
_{2}to denary. - Convert 00000000
_{2}to denary. - Convert 00110011
_{2}to denary. - Convert 11111111
_{2}to denary. - Convert 01010101
_{2}to denary. - Convert 10101010
_{2}to denary.
Task 3.2 Denary to binary conversionWhere we learn how to convert denary numbers into their binary value This is a little more complicated. Your teacher will show you a method before you begin the exercise. In your notebooks / on paperAttempt to answer the following questions in your notebook, showing
all your working. The first one has been done for you. If you are struggling, ask for help!- Convert 187
_{10}to binary.
- Convert 12
_{10}to binary. - Convert 15
_{10}to binary. - Convert 32
_{10}to binary. - Convert 45
_{10}to binary. - Convert 79
_{10}to binary. - Convert 0
_{10}to binary. - Convert 102
_{10}to binary. - Convert 156
_{10}to binary. - Convert 255
_{10}to binary.
We nearly always write binary numbers in groups of eight bits called one byte. Here is the problem ... - Computers work in binary
- Humans work in denary
- Binary numbers are considerably longer than their equivalent denary number
- Binary numbers are hard to remember
- Humans make mistakes
Here are some solutions ... - Break the binary number down into smaller parts
- Represent each part in another number base
It therefore means that we can only use bases which neatly represent the place values of binary to do this. So ...We could group the bits in 2's and use Base 4 but this only makes the number half the length - can we do better?We could group the bits in 3's and use Base 8 but 8 bits doesn't divide into 3s neatly ...4's and use Base 16 - this seems to work OK, except that each group of 4 bits has a range of 0 to 15 and we know that we can't use 2 digits to represent a single digit in a number, so what do we do???Yeah - but why invent new ones when we can use some we already have? Task 4.1 Binary / Hexadecimal conversionWhere we learn how to convert between binary and hexadecimal Your teacher will show you the method for converting binary to hexadecimal and vice versa. In your notebooks / on paperWhen you are ready, answer the following questions in your notebook, showing all your working. The first one has been done for you. Ask for help if you are struggling. - Convert 11010100
_{2}to hexadecimal. - Convert 10111100
_{2}to hexadecimal. - Convert 00000000
_{2}to hexadecimal. - Convert 11111111
_{2}to hexadecimal. - Convert 01010101
_{2}to hexadecimal.
In your notebooks / on paperThere is no mystery about hexadecimal to binary conversion - it's just the opposite to binary to hex. Try the following exercises in your book. The first one has been done for you. - Convert 5C
_{16}to binary. - Convert 14
_{16}to binary. - Convert F9
_{16}to binary. - Convert 0A
_{16}to binary. - Convert 10
_{16}to binary.
Research the use of hexadecimal numbers in ... - Hyper Text Markup Language (HTML);
- Media Access Control (MAC) addresses;
- Assembly programming language.
dozenal.Find some binary jokes to share with the class next lesson. How about trying to pronounce hexadecimal numbers? ReferencesImages of Mickey Mouse from http://pngimg.com/ FAQQ : What other bases are commonly used?A : The only number bases commonly used by computer scientists are Base 2 (binary), Base 10 (denary) and Base 16 (Hexadecimal). Base 8 (octal) used to be used in the early days of computing machines because they used 6-bit bytes which split neatly into two octal nybbles.Q : Do you need to know the prefixes like 0x, 0d, and 0b?A : Absolutely not for the examination, but definitely if you want to be a programmer!Q : How do I convert numbers quickly in an exam?A : Err, practise, practise, practise. Play the 'binary game' from the assessment task, play 2048, learn how to pronounce hexadecimal numbers. |