1024 (number) - Approximation To 1000

Approximation To 1000

See also: Binary prefix

The neat coincidence that 210 is nearly equal to 103 provides the basis of a technique of estimating larger powers of 2 in decimal notation. Using 210a+b ≈ 2b103a is fairly accurate for exponents up to about 100. For exponents up to 300, 3a continues to be a good estimate of the number of digits.

For example, 253 ≈ 8×1015. The actual value is closer to 9×1015.

In the case of larger exponents the relationship becomes increasingly more inaccurate with errors exceeding an order of magnitude for, for example:

$\begin{align} \frac{2^{1000}}{10^{300}} &= \exp \left( \ln \left( \frac{2^{1000}}{10^{300}} \right) \right) \\ &= \exp \left( \ln \left( 2^{1000}\right) - \ln\left(10^{300}\right)\right)\\ &\approx \exp\left(693.147-690.776\right)\\ &\approx \exp\left(2.372\right)\\ &\approx 10.72 \end{align}$

In measuring bytes, 1024 is often used in place of 1000 as the quotients of the units byte, kilobyte, megabyte, etc. In 1999, the IEC coined the term kibibyte for multiples of 1024, with kilobyte being used for multiples of 1000. As of 2011, this convention has not been widely adopted.

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