Floating point numbers are stored in scientific notation using base 2 not base 10.
There are a limited number of bits so they are stored to a certain number of significant binary figures.
There are various number of bytes (bits) used to store the numbers - the bits being split between the mantissa (the number) and the exponent (the power of 10 (being in the base of the storage - in binary, 10 equals 2 in decimal) by which the mantissa is multiplied to get the binary/decimal point back to where it should be), examples:
The numbers are stored normalised:
In decimal numbers the digit before the decimal point is non-zero, ie one of {1, 2, ..., 9}.
In binary numbers, the only non-zero digit is 1, so *every* floating point number in binary (except 0) has a 1 before the binary point; thus the initial 1 (before the binary point) is not stored (it is implicit).
The exponent is stored by adding an offset of 2^(bits of exponent - 1), eg with 8 bit exponents it is stored by adding 2^7 = 1000 0000
Zero is stored by having an exponent of zero (and mantissa of zero).
Example 10 (decimal):
10 (decimal) = 1010 in binary → 1.010 × 10^11 (all digits binary) which is stored in single precision as:
sign = 0
exponent = 1000 0000 + 0000 0011 = 1000 00011
mantissa = 010 0000 0000 0000 0000 0000 (the 1 before the binary point is explicit).
Example -0.75 (decimal):
-0.75 decimal = -0.11 in binary (0.75 = ½ + ¼) → 1.1 × 10^-1 (all digits binary) → single precision:
sign = 1
exponent = 1000 0000 + (-0000 0001) = 0111 1111
mantissa = 100 0000 0000 0000 0000 0000
Note 0.1 in decimal is a recurring binary fraction 0.1 (decimal) = 0.0001100110011... in binary which is one reason floating point numbers have rounding issues when dealing with decimal fractions.
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The numbers stored in a computer in floating-point notation are stored in scientific notation - but note that internally, they are stored in base-2, not in base-10.
It is the use of scientific notation.
1.230 X 10^3
You can use websites like Wolfram Alpha, Scientific Calculator, or OnlineConversion to look up scientific notation. Simply enter the number in scientific notation and these websites will provide you with the equivalent decimal representation or vice versa.
Scientific notation produces convenient numbers when working with very small or very large quantities.
1.024 x 102On calculators and computers, this is usually represented as: 1.024E2