15.1: Searching for Digits
The first 3 digits after the decimal for the decimal expansion of $\frac37$ have been calculated. Find the next 4 digits.
Let’s think about infinite decimals.
The first 3 digits after the decimal for the decimal expansion of $\frac37$ have been calculated. Find the next 4 digits.
Your teacher will give your group a set of cards. Each card will have a calculations side and an explanation side.
The cards show Noah’s work calculating the fraction representation of $0.4\overline{85}$. Arrange these in order to see how he figured out that $0.4\overline{85} = \frac{481}{990}$ without needing a calculator.
Use Noah’s method to calculate the fraction representation of:
Use this technique to find fractional representations for $0.\overline{3}$ and $0.\overline{9}$.
Label all of the tick marks. Plot $\sqrt{2}$ on all three number lines. Make sure to add arrows from the second to the third number lines.
Label all of the tick marks on the number lines. Use a calculator to get a very accurate approximation of $\pi$ and plot that number on all three number lines.
Not every number is rational. Earlier we tried to find a fraction whose square is equal to 2. That turns out to be impossible, although we can get pretty close (try squaring $\frac75$). Since there is no fraction equal to $\sqrt{2}$ it is not a rational number, which is why we call it an irrational number. Another well-known irrational number is $\pi$.
Any number, rational or irrational, has a decimal expansion. Sometimes it goes on forever. For example, the rational number $\frac{2}{11}$ has the decimal expansion $0.181818 . . . $ with the 18s repeating forever. Every rational number has a decimal expansion that either stops at some point or ends up in a repeating pattern like $\frac2{11}$. Irrational numbers also have infinite decimal expansions, but they don't end up in a repeating pattern. From the decimal point of view we can see that rational numbers are pretty special. Most numbers are irrational, even though the numbers we use on a daily basis are more frequently rational.