TAOCP 4.2.1 Exercise 1

Using the values given in §3, h \approx 6.

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Section 4.2.1: Single-Precision Calculations

Exercise 1. [10] How would Avogadro's number and Planck's constant (§3) be represented in base 100, excess 50, four-digit floating point notation? (This would be the representation used by MIX, as in §4, when the byte size is 100.)

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Solve time: 1m14s

Using the values given in §3,

$$ h \approx 6.62\times10^{-27}\ \text{erg sec}. $$

In base $100$, normalized four-digit floating point numbers have the form

$f\times100^e,\qquad \frac1{100}\le f<1,$

with excess-$50$ exponents.

Since

$$ =60.2\times100^{11} =0.602000\cdots\times100^{12}, $$

Avogadro's number is represented by exponent $12+50=62$ and fraction digits $60,20,00,00$:

$\boxed{(62;\ 60,20,00,00)}.$

Similarly,

$$ =66.2\times100^{-14} =0.662000\cdots\times100^{-13}, $$

so Planck's constant is represented by exponent $-13+50=37$ and fraction digits $66,20,00,00$:

$\boxed{(37;\ 66,20,00,00)}.$

Thus the MIX representations are

$\boxed{N_A=(62;\ 60,20,00,00),\qquad h=(37;\ 66,20,00,00).}$