1.02 Rounding

Natural numbers and real numbers

Natural numbers are positive whole numbers that we use to count. e.g. $0,1,2,3,...,21,...1045,...$0,1,2,3,...,21,...1045,...

Real numbers are all rational and irrational numbers combined. e.g. $7,0.62,-76.5,1\frac{5}{91},\sqrt{2},\pi$7,0.62,−76.5,1591​,√2,π

An example of numbers that are not real are square roots of negative numbers, because we cannot calculate a square root of a negative number. e.g. $\sqrt{-1},\sqrt{-81}.$√−1,√−81.

Another example of an expression that is not real is when you divide by zero. e.g. $\frac{11}{0}$110​. This is not real because you cannot divide a quantity by zero.

 A good way to check if a number or calculation is real is to put it in your scientific calculator and press the equals sign. If it comes up with an error, then it is not real. (Make sure your calculator is not in complex numbers mode). 

 

Worked Example

Example 1

 

Classify the following numbers as natural and/or real: 

a) $102$102     b) $\frac{1}{4}$14​     c) $-65$−65     d) $\sqrt{100}$√100     e) $\sqrt{-62}$√−62    f)$\sqrt{50}$√50

a) This is a positive whole number, so it is a natural number.

b) This is a fraction, so it is not a natural number, but it is a real number.

c) This is a negative number, so it is not a natural number, but it is a real number.

d) $\sqrt{100}=10$√100=10 which is a whole positive number. Therefore this is a natural number and a real number. 

e) This is not a real number since we cannot square root a negative number. 

f) $\sqrt{50}=7.0710678...$√50=7.0710678..., so it is not a natural number, but it is a real number.

 

Rounding 

Rounding is a way to simplify our numbers, to help with estimating. With whole numbers, we might round to the nearest ten, or hundred. With decimals, we might round to the nearest unit, tenth or hundredth.

Rounding $13$13 to the nearest ten is the same as asking if $13$13 is closer to $10$10, or closer to $20$20. As we know that $15$15 is halfway between $10$10 and $20$20, and $13$13 is less than halfway, $13$13 must be closer to $10$10.

Rounding $175$175 to the nearest hundred is the same as asking if $175$175 is closer to $100$100 or $200$200.  As we know that $150$150 is halfway between $100$100 and $200$200, and $175$175 is greater than $150$150, then $175$175 is closer to $200$200 than $100$100.

It's this notion of "halfway-ness" that leads us to the following rule for rounding.

 

Practice questions

Question 1

Question 2

Question 3