Rounding Numbers

Rounding is a very easy and important concept in maths in general.

In general, if a number is 5 or above you round up. If it is 5 or below you round down.

Decimal Places (d.p)

To round by the number decimal places, all you need to do is count the number of places.

For example: "Give 56.6785 to 3 d.p.":

$56.6785 = \text{4 d.p.}$ so we need to look at the $(3+1)$ d.p. and see if it is 5 and above or below 5. $5$ is 5 or above so we need to ROUND UP.

$56.6785 \rightarrow 56.679 \text{ correct to 3 d.p.}$

There can be harder cases however:

"Give 89.997 to 2 d.p.":

Here the (2+1) d.p. is greater than or equal to 5 so we need to round up to (9+10) for the next number which would make the number 10 which can't happen. So what we do is change the 9 to a 0 and carry 1 to the next digit which is another 9, we do the same thing until we end up with 90. But since we are rounding to 2 d.p. we have to show 0's in their place:

$89.997 \rightarrow 90.00$

Significant figures (s.f.)

Significant figures are another way to round. You count significant figures as every value that is not a 0. So for example:

"Give 0.0345 to 2 s.f.":

$0.0345 \rightarrow 0.035$ because we don't count $0.0$ as significant figures (they just mean nothing!)

Estimating

You may be asked to estimate the value of an equation. To do this, round all the numbers to 1 or 2 s.f. For example:

"Estimate the value of $\frac{69.8+45.78 * 5}{27.8 - 5.4}$:"

$\dfrac{69.8+45.78 * 5}{27.8 - 5.4} \approx \dfrac{70 + 50 * 5}{30 - 5} \approx \dfrac{320}{25} = 12.8 \approx 13$

Bounds

Every measurement you make has a level (degree) of accuracy. This is because the equipment used to take the measurement is only so good. Bounds are the largest (upper bound) and smallest (lower bound) numbers that this measurement could be because of the level of accuracy.

For example "A cake is weighed and the result is 3.54 kg. However, the weighing machine was only accurate to 0.01kg, give the upper bound and lower bound:"

So the result is only accurate to 0.01kg therefore therefore the bounds will be:

$\text{Bound} = 3.54 \pm 0.005$

So in this example the Upper bound would be, 3.545 and the lower bound would be 3.535