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Mathematics LibreTexts

6.5: More x -mals

  • Page ID
    9860
  • \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

    \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)

    It should come as no surprise that we can use this reasoning about division in the “Dots & Boxes” model in other bases as well.

    The following picture shows that working in base 5,

    $$1432_{five} \div 13_{five} = 110_{five} R2_{five},\; \text{meaning}\; 1432_{five} = 110_{five} \cdot 13_{five} + 2_{five} \ldotp$$

    base5diva-768x280.png

    Think / Pair / Share

    Carefully explain the connection between the picture and the equation shown above.

    • Show in the picture where you see \(1432_{five}\) from the equation.
    • Where do you see \(13_{five}\)?
    • Where do you see \(110_{five}\) and \(2_{five}\)?

    Example: \(1432_{five} \div 13_{five}\)

    Here’s where we left off the division, with a remainder of 2:

    base5divb-768x185.png

    Now we can unexplode one of those two remaining dots. Then we’re able to make another group of \(13_{five}\).

    base5divc-768x178.png

    Once again, there are two dots left over, not in any group. So let’s unexplode one of them.

    base5divd-768x181.png

    And we still have two dots left over. Why not do it again?

    base5dive-768x169.png

    It seems like we’re going to be doing the same thing forever:

    • Start with two dots in some box.
    • Unexplode one one of the dots, so you have one dot in your original box and five in the box to the right.
    • Form a group of \(3_{five}\). That uses the one dot in your original box and three dots in the box to the right.
    • So you have two dots left in a box.
    • Unexplode one of the dots, so you have one dot in your original box and five in the box to the right.
    • This feels familiar…

    We conclude:

    $$1432_{five} \div 13_{five} = 110.111 \ldots_{five} = 110. \bar{1}_{five} \ldotp$$

    Think / Pair / Share

    The equation

    $$1432_{five} \div 13_{five} = 110. \bar{1}_{five} \ldotp$$

    is a statement in base five. What is it saying in base ten?

    “\(1432_{five}\)” is the number

    $$1 \cdot 125 + 4 \cdot 25 + 3 \cdot 5 + 2 \cdot 1 = 242_{ten} \ldotp$$

    • What is \(13_{five}\) in base 10? Be sure to explain your answer.
    • What is \(110. \bar{1}_{five}\) in base 10? Explain how you got your answer.
    • Translate the equation above to a statement in base ten and check that it is correct.

    Problem 2

    1. Draw pictures to compute \(8 \div 3\) in a base ten system, and show the answer is \(2. \bar{6}\).
    2. Draw the pictures to compute \(8_{nine} \div 3_{nine}\) in a base 9 system, and write the answer as a decimal. (Or is it a “nonimal”?)

    Problem 3

    1. Draw the pictures to compute \(1 \div 11\) in a base ten system, and show the answer is \(0. \overline{09}\).
    2. Draw the base 3 pictures to compute \(1_{three} \div 11_{three}\), and write the answer as a decimal (“trimal”?) number.
    3. Draw the base four pictures to compute \(1_{four} \div 11_{four}\), and write the answer as a decimal (“quadimal”?) number.
    4. Draw the base six pictures to compute \(1_{six} \div 11_{six}\), and write the answer as a decimal (“heximal”?) number.
    5. Describe any patterns you notice in the computations above. Do you have a conjecture of a general rule? Can you prove your general rule is true?

    Problem 4

    Remember that the fraction \(\frac{2}{5}\) represents the division problem \(2 \div 5\). (This is all written in base ten.)

    1. What is the decimal expansion (in base ten) of the fraction \(\frac{2}{5}\)?
    2. Rewrite the base-ten fraction \(\frac{2}{5}\) as a base four division problem. Then find the decimal expansion for that fraction in base four.
    3. Rewrite the base-ten fraction \(\frac{2}{5}\) as a base five division problem. Then find the decimal expansion for that fraction in base five.
    4. Rewrite the base-ten fraction \(\frac{2}{5}\) as a base seven division problem. Then find the decimal expansion for that fraction in base seven.
    5. Barry said that in base fifteen, the division problem looks like $$2_{fifteen} \div 5_{fifteen},$$and the decimal representation would be \(0.6_{fifteen}\). Check Barry’s answer. Is he right?

    Problem 5

    Expand each of the following as a “decimal” number in the base given. (The fraction is given in base ten.)

    $$\begin{split} (a)\; \frac{1}{9}\; \text{in base 10} \quad \qquad &(b)\; \frac{1}{2}\; \text{in base 3} \\ (c)\; \frac{1}{3}\; \text{in base 4} \quad \qquad &(d)\; \frac{1}{4}\; \text{in base 5} \\ (e)\; \frac{1}{5}\; \text{in base 6} \quad \qquad &(f)\; \frac{1}{6}\; \text{in base 7} \\ (g)\; \frac{1}{7}\; \text{in base 8} \quad \qquad &(h)\; \frac{1}{8}\; \text{in base 9} \end{split}$$

    Do you notice any patterns? Any conjectures?

    Problem 6 (Challenge)

    What fraction has decimal expansion \(0. \bar{3}_{seven}\)? How do you know you are right?