What is a Floating Point Number?
A basic data type, but what is it?
A simple question. this is a number which can store a decimal point. But why don’t we use this for every type of number? Read on to find out!
Difficulty: Beginner | Easy | Normal | Challenging
Prerequisites:
- None, although negative numbers are stored as Two’s Complement which is not covered here (guide HERE)
Terminology
Data types: A representation of the tyoe of data that can be processed, for example Integer or String
Exponent: The section of a decimal place after the decimal place
Floating Point: A number without a fixed number of digits before and after the decimal point
Integer: A number that has no fractional part, that is no digits after the decimal point
Mantissa: The section of a Floating point number before the decimal place
Precision: How precise or accurate something is
Real numbers: Another name for Floating Point Numbers
Floating point numbers:
Why they are required:
Compared to Floating Point numbers Integers are precise and there can never be any rounding errors. However, Integer division typically means 1 / 2 = 1 which may not be suitable for all uses being coded.
A simple definition:
A Floating Point number usually has a decimal point. This means that 0, 3.14, 6.5, and -125.5 are Floating Point numbers.
Since Floating Point numbers represent a wide variety of numbers their precision varies.
Storing Integer Numbers
Integer numbers can be stored by just manipulating bit positions. One possible way of doing this is shown in the image below:
We can only store (2 to the power of n) — 1 numbers, but this is a simple way to store Integer numbers.
Storing Floating Point
Floating Point numbers can’t be stored exactly like Integer numbers are. The issue is there is a decimal place — so what is the first number we store.
0.1
0.01
0.001
0.0001
We would have to define this for each number we stored, and then we would be restricted to that decision…so we would not be able to change from that choice.
So clearly this isn’t the way that we store Floating Point numbers. We split a Floating Point number into sign, exponent and mantissa as in the following diagram showing 23 bits for the mantissa and 8 bits for the exponent:
The above image shows an exponent (in Denary) of 1, with a mantissa of 1 — that is 1.1
Now in a real example this would be stored as Two's complement and even the mantissa can be offset by 127, but this basic example shows how it might be solved.
Precision
Single precision
Single precision Floating Point numbers are 32-bit. That means that 2,147,483,647 is the largest number can be stored in 32 bits.
That is, 2³¹ − 1 = 2,147,483,647
(remember: -1 because of the sign bit)
The smallest number that can be stored is the negative of the largest number, that is -2,147,483,647
Double precision
Double precision Floating Point numbers are 64-bit. That means that 9,223,372,036,854,775,807 is the largest number that can be stored in 64 bits.
That is, ²³¹ − 1 = 9,223,372,036,854,775,807
(remember: -1 because of the sign bit)
The smallest number that can be stored is the negative of the largest number, that is -9,223,372,036,854,775,807
Issues
Overflow
Floating Point overflow occurs when an attempt is made to store a number that is larger than can be adequately stored by the model chosen. This is known as a floating Point overflow.
Memory usage of Floating Point Numbers
The memory usage of Floating Point numbers depends on the precision precision chosen for the implementation.
Conclusion:
Floating Point numbers are used in the real application of computing. This involves sign, exponent and mantissa as different parts of the number to store the number at the precision you desire.
Extend your knowledge
- Two’s complement is shown in this article, but Wikipedia have a nice article (HERE)
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