13.3 Floating-point numbers, representation and manipulation

##### Why binary numbers are stored in normalised form
- To store the maximum range of numbers in the minimum number of bytes / bits
- Normalisation minimises the number of leading zeros/ones represented
- Maximising the number of significant bits // maximising the (potential)
- precision / accuracy of the number for the given number of bits
- … enables very large / small numbers to be stored with accuracy.
- Avoids the possibility of many numbers having multiple representations.
##### When underflow occurs in a binary floating-point system
- Following an arithmetic/logical operation
- … the result is too small to be precisely represented in the available system // When the number of bits is not enough / too small for the computer’s allocated word size / to represent the binary number

Example:
A student enters the following into an interpreter:
`OUTPUT(0.2 * 0.4)`
The student is surprised to see that the interpreter outputs the following:
`0.08000000000000002`
Explain why the interpreter outputs this value.

Answer:
- 0.2/0.4 cannot be represented exactly in binary / rounding error
- 0.2 has been represented by a value just greater than 0.2 // 0.4 has been represented by a value just greater than 0.4
- Therefore multiplying these two representations together increases the difference
- difference after the calculation is significant enough to be seen (given the number of positions after the decimal place)