Estimation is a very underrated tool in math and science. It is very useful for us to see and gain an understanding of what the number actually means in nature.
In short, if a digit of a number is greater than or equal to 5 we round the number up whilst if a digit of a number is less than 5 we round down.
We are able to round integers to a given power of 10. We look for the digit of the number by 1 power of 10 less than the power of 10 we are rounding to and round up or down depending on that digit. For example we can round 73 to the nearest 100 or round 780 to the nearest 100:
$$ \color{red}{7}3 \Rightarrow 100 $$
$$ 7 \color{red}{8}0 \Rightarrow 800 $$
Decimal places are the number of digit places there are after the decimal place. On the other hand significant figures are the number of digits after non-zero digits in a number. Like with other forms of rounding, we look for the digit after the level of accuracy we are rounding to. For example, let’s round 0.00567 to 2 decimal places and 2 significant figures:
$$ 0.00\color{red}{5}67 \Rightarrow 0.01 $$ $$ 0.0056\color{red}{7} \Rightarrow 0.0057 $$
If we have rounded a number by a certain amount it means we have a range of results the number could actually be. The upper bound is the highest number value the number can be, the lower bound the lowest. As an example, assume the number 2.45 is accurate to 2 decimal places, find the upper and lower bound:
$$ 2.45 \Rightarrow 2.455 $$ $$ 2.45 \Rightarrow 2.445 $$
For both the upper and lower bound we use the number 5 because it is the maximum value if you were to round down (the actual maximum would be .49 recurring recurring but we assume that equals .5) and the maximum value if you were to round up.
More coming soon…