Computer Science

Topic 4 - Representation of Numbers as Bit Patterns

What you need to know:-

- - - - Apply binary arithmetic techniques.
        Explain the representation of positive and negative integers in a fixed-length store using both two’s complement, and sign and magnitude
        representation.
        Describe the nature and uses of floating point form.
        State the advantages and disadvantages of representing numbers in integer and floating point forms.
        Convert between real number and floating point form.
        Describe truncation and rounding, and explain their effect upon accuracy.

Why do we need to know about binary?

Binary to Denary Conversion - POSITIVE WHOLE NUMBERS ONLY

This is the first in a series of computer science videos about the binary number system which is fundamental to the operation of a digital electronic computer. It covers the need for binary and details of how to convert positive whole numbers in base 10 into 8 bit binary, and vice versa.

Examples - Converting Decimal Integers into Binary

Examples - Converting Binary into Decimal

Binary to Denary Conversion - POSITIVE & NEGATIVE Whole Numbers using 2's complement

This video shows the 2's complement method. There are several ways to use this method of conversion. However, I find the third method (from 11.40 in video), is probably the easiest one to use.

Alternative conversion of POSITIVE & NEGATIVE Whole Numbers, using SIGN & MAGNITUDE

Fixed Point Binary

Up to this point, we have only represented whole numbers. Fixed point binary is how we start to represent decimal numbers, both positive and negative. However, there are some numbers which cannot be stored accurately using this method.

This video covers the representation of real numbers in binary using a fixed size, fixed point, register. It explains with examples how to convert both positive and negative denary numbers to and from fixed point binary format. It also covers the advantages and limitations of processors that make use of fixed point registers.

Floating Point Binary

This method allows us to represent numbers accurately in a given number of bits.

This video covers the representation of real numbers using floating point binary notation. It begins with a description of standard scientific form in base 10 (as used by scientists and engineers to denote very large or very small values) because floating point binary notation is similar in principle to standard form. It then explains with examples how to convert both positive and negative floating point binary numbers into denary.

Floating Point Rules.docx

Floating Point Binary - Precision

This video elaborates on the representation of real numbers using floating point binary notation. It explains how the relative allocation of bits between the mantissa and the exponent in a fixed size register can impact on the range and precision of the values that can be represented. It also draws a distinction between precision and accuracy, and points out some of the limitations of floating point binary. The video also mentions the importance of an awareness among programmers of how floating point binary works, including a demonstration of the kind of error that can occur when using single and double data types in a simple Visual Basic.NET program.

Normalising Floating Point Binary

This video covers the conversion of real numbers, both positive and negative, from denary into normalised floating point binary. It also covers the objectives of normalisation, explaining how to recognise un-normalised floating point binary representations of positive and negative numbers and then how to normalise them.