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The book says 22.00 but surely one can't go ahead and write this number correct to 2 decimal places as we don't know what they are. Unless it is standard practice to use zeros when are certain number of decimal places are required. What is the correct answer here?

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    $\begingroup$ "we don't know what they are": we know they are all zeroes. $\endgroup$ – Yves Daoust Oct 11 '16 at 10:55
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    $\begingroup$ If you are asked to write a whole number to k decimal places, then what could you do besides writing k zeros after the decimal point? Your book is correct. $\endgroup$ – StubbornAtom Oct 11 '16 at 10:57
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    $\begingroup$ if you had 22 pounds - how would you write that as pounds and pence? £22.00 is the answer - UK and US money is often written to 2 decimal places. In book keeping they would write £22.01 and £22.99 - but also £22.00 for a whole amount $\endgroup$ – Cato Oct 11 '16 at 11:04
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In the case of an integer such as 22 we do know what the decimal places are. There are an infinite number of zeroes. However it is common practice not to write any of them, so we tend to think of an integer as having no decimal places. So any integer $N$ to two decimal places is $N.00$.

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The correct answer is $22.00$. We know all infinitely many decimal places of the integer $22$ in its usual decimal representation: every one of them is $0$.

Say the decimal expansion of $22$ is $22.d_1d_2d_3\ldots\;$; by definition this means that

$$22=22+\frac{d_1}{10}+\frac{d_2}{10^2}+\frac{d_3}{10^3}+\ldots\;,$$

where each $d_k\in\{0,1,2,3,4,5,6,7,8,9\}$. And this is the case if and only if

$$\frac{d_1}{10}+\frac{d_2}{10^2}+\frac{d_3}{10^3}+\ldots=0\;,$$

which in turn is the case if and only if $d_k=0$ for $k=1,2,3,\ldots\;$. Thus, every decimal place in the (usual) decimal expansion of $22$ is $0$, and when we round to two places, we get $22.00$.

In fact $22$ does have another decimal expansion: $22=21.999\ldots\;$, because $0.999\ldots=1$. However, rounding this to two decimal places again results in $22.00$.

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  • $\begingroup$ Nice elementary explanation as usual. $\endgroup$ – StubbornAtom Oct 11 '16 at 11:00
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Perhaps you are confused with the use of so-called significant figures in other sciences. Here, the number $22$ sometimes means "a number in the range $[21.5,22.5)$ that we did not determine more precisely.

However, in mathematics, all numbers are exact. The number $22$ means exactly $22$, to any precision.

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