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I have a statement that says:

In relation with the number $-\frac{23}{7}$, what of the following statements are correct ?

$I)$ When writing it with three significant digits we obtain an approximation by default.

$II)$ When truncating to the thousandth we obtain an approximation by excess.

$III)$ The difference between, truncate it to the unit and write it with a one significant digit It is zero.

My development was:

$I)$ first, the fraction in decimal is equal to $-3.28571429$, and with 3 significant digits is $-3.28$ Also, when we talk about significant figures, there are 2 types of approximation, by excess and by default. It is by excess when the approximate number is greater than the original. It is by default when the approximate number is less than the original.

$ -3.28 $ is greater than $ -3.28571429 $, therefore it is false.

$II)$ When truncating to the thousandth, I get:$ -3.285$ that is greather than the original value, therefore is true

$III)$ If i truncate it to the unit: $-3$ and if i write it with one significant digit is $-3$, so $-3 + 3 = 0$, therefore is true

so, i answered $II)$ and $III)$ are correct, but here the problem, since all were correct. What am I wrong about?

ESCM
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1 Answers1

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There is a difference between making a number significant and truncating a number. Significant means a rounded number (instead of just dropping the $n$-th decimal you turn the previous decimal to the "integer" closed to the previous decimal part: with "integer" I mean considering it as an integer within decimals), truncated means dropped digits (just throwing them away without rounding them).

What you did in $II)$ is correct: you truncated $−3.28571429...$ by dropping $...571429$.

In $I)$ you should round $-3.28571429...$ at the number $8$, which means that $...8571429$ should become $...9$. In "integer" style: $8,571429...$ lies closest to $9$ (minus sign doesn't change anything due to symmetry around $0$ in the real numbers). This means that $-3.28571429...$ becomes $-3.29$ and we know that $-3.29<-3.28571429...$, so it is indeed an approximation by default.

In $III)$ you happen to have a correct answer, since both truncating and making the number significant yield the same result.

In conclusion: truncating is different from making a number significant.

The Phenotype
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  • so, you aproximate to $-3.29$ , because you have: $-3,285$, then when does $5 $ come after $ 8 $, does it aproximate to the next integer? – ESCM May 03 '18 at 20:17
  • And If the number would have been $-3,284$, would it have been default? ($-3.28$) – ESCM May 03 '18 at 20:18
  • @Mattiu When rounding, the numbers $0.0, 0.1, 0.2, 0.3, 0.4$ approximate to $0$ and $0.5, 0.6, 0.7, 0.8, 0.9$ approximate to $1$. In general a number $0.5\leq x <1.5$ approximates to $1$ and we clearly have that $8.5< 8.571429<9.5$, so this approximates to $9$. The number $-3,284$ would indeed round to $-3,28$ and be default since $-3,28>-3.28571429...$. – The Phenotype May 03 '18 at 23:41
  • Many many thanks – ESCM May 04 '18 at 00:31