The first and third numbers are between 0 and 1 while the second and fourth numbers are between 1 and 2. Two of them are rational and can be graphed exactly.
Graph:
Least to greatest:
{0.1, 4/5, sqrt(2), sqrt(3) }
In this set of numbers, we have been given both rational and irrational values. Let's consider these types of numbers individually first.
The Rational Numbers
The numbers 45 and 0.1 are both rational, they can be written in the form ab. The number 45 is already in this form and we can write
0.1 as 1/9.
Now let's graph these on a number line. To graph the first number, we can draw a number line from 0 to 1 and divide the segment into five equal spaces.
To graph 19, we instead divide the same segment into nine equal parts.
The Irrational Numbers
The other numbers are both irrational, they cannot be written as a fraction. We will need to approximate their places on the number line. If we use a calculator to find their square roots, we have:
sqrt(2)=1.41421... sqrt(3)=1.73205...
Notice that the first square root is somewhat greater than 1.4 and the second one is slightly greater than 1.7. By dividing the segment between 1 and 2 into ten equal parts, we can relatively accurately graph these numbers. Let's add them to our existing number line.
Least to Greatest
Now that we have graphed each number on the number line, we can write them in order from least to greatest. Lesser numbers are found on the left and greater numbers are on the right. This gives us the following order.
{0.1, 45, sqrt(2), sqrt(3) }
Extra
How do you rewrite 0.1 as a fraction?
The expression 0.1 is the same thing as 0.11111... where 1 repeats indefinetely. Let's call the fraction that equals 0.11111... for x so we get the equation
x=0.11111...
By multiplying the equation by 10 we create a new equivalent equation.
Now, by subtracting both sides by x, we get 9x on the left-hand-side while we on the right-hand-side can get rid of the repeating decimals. Finally we divide both sides by 9 which isolates x.
