1.3 Units  (Page 3/6)

Error margins

In a calculation that has many steps, it is best to leave the rounding off right until the end. For example, Jack and Jill walk to school. They walk 0,9 kilometers to get to school and it takes them 17 minutes. We can calculate their speed in the following two ways.

Method 1:

You will see that we get two different answers. In Method 1 no rounding was done, but in Method 2, the time was rounded to 2 decimal places. This made a big difference to the answer. The answer in Method 1 is more accurate because rounded numbers were not used in the calculation. Always only round off your final answer.

Scientific notation

In Science one often needs to work with very large or very small numbers. These can be written more easily in scientific notation, in the general form

where $d$ is a decimal number between 0 and 10 that is rounded off to a few decimal places. $e$  is known as the exponent and is an integer. If $e>0$ it represents how many times the decimal place in $d$ should be moved to the right. If $e<0$ , then it represents how many times the decimal place in $d$ should be moved to the left. For example $3,24\times {10}^3$ represents 3240 (the decimal moved three places to the right) and $3,24\times {10}^{-3}$ represents $0,00324$ (the decimal moved three places to the left).

If a number must be converted into scientific notation, we need to work out how many times the number must be multiplied or divided by 10 to make it into a number between 1 and 10 (i.e. the value of $e$ ) and what this number between 1 and 10 is (the value of $d$ ). We do this by counting the number of decimal places the decimal comma must move.

For example, write the speed of light in scientific notation, to two decimal places. The speed of light is 299 792 458 m $·$ s ${}^{-1}$ . First, find where the decimal comma must go for two decimal places (to find $d$ ) and then count how many places there are after the decimal comma to determine $e$ .

In this example, the decimal comma must go after the first 2, but since the number after the 9 is 7, $d=3,00$ . $e=8$ because there are 8 digits left after the decimal comma. So the speed of light in scientific notation, to two decimal places is 3,00 $\times$ 10 ${}^8$ m $·$ s ${}^{-1}$ .

Significant figures

In a number, each non-zero digit is a significant figure. Zeroes are only counted if they are between two non-zero digits 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. You estimate a number like this by removing significant figures from the number (starting from the 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 it would be 6,83.