Normalise Binary

Normalised floating-point binary numbers are the binary equivalent of denary standard form. Very large or very small numbers have their digits shifted left or right so that they start immediately to the right of the binary point - this forms the mantissa - and the number of places it's been shifted to the right becomes the exponent (with a negative number indicating that the mantissa was shifted to the left). The only slight difference between normalised binary and denary standard form is that, in denary, the mantissa is greater than or equal to one and less than ten - i.e. there are digits to the left of the decimal point - whereas in binary, because we use two's complement form for negative numbers, the bit before the point is used as the sign, with all of the actual number coming after the point.

This page is a normalised binary calculator (using the two's complement method) - you can enter a denary number in the textbox below and the required mantissa and exponent will be displayed, along with an explanation of the process. If you aren't familiar with the use of binary for fractions and negative numbers, click here.

Note that the method used here is to normalise first and then use two's complement to convert negative numbers where necessary.  This is the method used in my textbook, the AQA Wikibook, the Cambridge International A Level textbook, etc.  Some books/courses apparently suggest finding the two's complement before normalising, which will give a different result.  There are also other methods for storing floating-point binary numbers, including IEEE 754.

Input a denary number:

Number of bits for the mantissa:


Number of bits for the exponent: