Early computers designed to assist the collection of census data used fixed-point arithmetic, ie they represented all numbers with a fixed number of decimal positions before and after the decimal point similar to a modern cash register. This limits the range of numbers that can be handled: A machine with 12 digits before and 8 digits after the decimal point cannot store numbers larger than 999,999,999,999.99999999 and fractions smaller than 0.00000001.
Scientific applications often require a much larger number range than what can be accommodated in fixed-point arithmetic. Floating point arithmetic was invented to allow such scientific calculations.
A floating-point number consists of a mantissa and an exponent. The exponent indicates the power of ten, the mantissa the fraction of the power. The number 12,344.548, for example, is written in floating-point notation as 0.12344548 . 106. Its exponent thus has the value 6 and its mantissa the value 12344548.
The Zuse Z3 used 14 bits for the mantissa and 7 bits for the exponent. 7 binary digits convert to 27 = 128. This gave it a number range up to 10128.
The accuracy of floating-point calculations is determined by the number of digits in the mantissa:
0.42268.105 = 42,268.0
0.42268.10-5 = 0.00042268
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subtract: 42,267.999577
If this result is represented in floating-point notation it requires a mantissa with 11 digits: 0.42267999577.105. A computer with less than 11 mantissa digits will round the result to the nearest value. This can lead to erroneous results known as overflow (The true results is outside the range of the mantissa). The Z3 therefore had an overflow indicator to tell the user that the calculation exceeded the tolerated inaccuracy.