A Propos .999999... equals 1?

Dear Alex,

A propos of your page on 0.999... and 1, another example:

Suppose your boss wishes to give you a bonus to enable you to purchase a computer, say for $2,000. (If you bought it through corporate channels, it would take six months, you would overpay by several hundred dollars, and wind up with last-years model.)

If he gives you a $2,000 bonus, by the time you have paid income tax on the extra income, you can no longer afford the purchase. If your tax rate on the additional income is, say, r, you are short by 2000·r. So your boss throws in an additional $2000·r dollars. But, you still have to pay tax on the additional money, so you are still not "whole." If he throws in an additonal $2000·r², you will still be short by $2000·r³. Evidently, it will require an infinite sum of bonuses to enable you to buy the computer at your boss's expense:

2000 + 2000·r + 2000·r² + 2000·r³ + ...

But you can also calculate the required amount by simple algebra:

Let B indicate the total bonus necessary to buy the computer and pay all the required additional income tax (at rate r). After the taxes are subtracted, you should be left with the inteded purchase price:

B - r·B = $2000,

so that

or B = $2000/(1-r).

(If your marginal tax rate is 50% -- not so hard for a two-income couple paying City, State, and Federal income taxes -- your boss has to pay out twice the price of the intended purchase.)

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