0.2 Error margins

Error margins - grade 11

We have seen that numbers are either rational or irrational and we have see how to round-off numbers. However, in a calculation that has many steps, it is best to leave the rounding off right until the end.

For example, if you were asked to write $3\sqrt{3}+\sqrt{12}$ as a decimal number correct to two decimal places, there are two ways of doing this.

Method 1

Method 2

In the example we see that Method 1 gives 8,66 as an answer while Method 2 gives 8,65 as an answer. The answer of Method 1 is more accurate because the expression was simplified as much as possible before the answer was rounded-off.

In general, it is best to simplify any expression as much as possible, before using your calculator to work out the answer in decimal notation.

Significant figures - extension

In a number, each non-zero digit is a significant figure. Zeroes are only counted if they are between two non-zerodigits or are at the end of the decimal part. For example, the number 2000 has 1 significant figure (the 2), but $2000,0$ has 5 significant figures. Estimating a number works by removing significant figures from your number (starting fromthe right) until you have the desired number of significant figures, rounding as you go. For example $6,827$ has 4 significant figures, but if you wish to write it to 3 significant figures it would mean removing the 7 and rounding up, so itwould be $6,83$ . It is important to know when to estimate a number and when not to. It is usually good practise to only estimate numbers when it is absolutelynecessary, and to instead use symbols to represent certain irrational numbers (such as $\pi$ ); approximating them only at the very end of a calculation. If it is necessary to approximate a number in the middle of a calculation, then it isoften good enough to approximate to a few decimal places.

End of chapter exercises

1. Calculate:
   1. $\sqrt{16}-\sqrt{72}$ to three decimal places
   2. $\sqrt{25}+\sqrt{2}$ to one decimal place
   3. $\sqrt{48}-\sqrt{3}$ to two decimal places
   4. $\sqrt{64}+\sqrt{18}-\sqrt{12}$ to two decimal places
   5. $\sqrt{4}+\sqrt{20}-\sqrt{18}$ to six decimal places
   6. $\sqrt{3}+\sqrt{5}-\sqrt{6}$ to one decimal place
2. Calculate:
   1. $\sqrt{x-2}$ , if $x=\mathrm{3,3}$ . Write the answer to four decimal places.
   2. $\sqrt{4+x}$ , if $x=\mathrm{1,423}$ . Write the answer to two decimal places.
   3. $\sqrt{x+3}+\sqrt{x}$ , if $x=\mathrm{5,7}$ . Write the answer to eight decimal places.
   4. $\sqrt{\mathrm{2x}-5}+\frac{1}{2}\sqrt{x+1}$ , if $x=\mathrm{4,91}$ . Write the answer to five decimal places.
   5. $\sqrt{(\mathrm{3x}-1)+(\mathrm{4x}+3)}-\sqrt{x+5}$ , if $x=\mathrm{3,6}$ . Write the answer to six decimal places.
   6. f) $\sqrt{(\mathrm{2x}+5)-(x-1)+(\mathrm{5x}+2)}+\frac{1}{4}\sqrt{4+x}$ , if $x=\mathrm{1,09}$ . Write the answer to one decimal place