# What are the biggest and smallest represent-able numbers with single precision floating points?

I am trying to understand the limits of the floating point representation.

On a 32-bit computer with 7 bits for the exponent and 24 bits for the mantissa, I want to know the biggest and smallest numbers.

My calculation:

Base 2

Biggest positive number = $$+ 1 \times 2^{127}$$

Smallest positive number = $$+ 2^{-24} \times 2^{-127}$$

Biggest negative number = $$- 2^{-24} \times 2^{-127}$$

Smallest negative number = $$-1 \times 2^{127}$$

Decimal

Biggest positive number = $$+1 \times 10^{38}$$

Smallest positive number= $$+ 10^{-7} \times 10^{-38}$$

Biggest negative number= $$- 10^{-7} \times 10^{-38}$$

Smallest negative number = $$-1 \times 10^{38}$$

Is this a correct calculation?

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More to the point: in inexact arithmetic, there exist representable numbers $\eta$ such that $x+\eta=x$ for any $x$, and for $x=1$, machine $\epsilon$ is the largest such $\eta$. – J. M. Jul 17 '11 at 15:42