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Station  8A

Question 1

Up to now our machines have consisted of a row of boxes extending infinitely far to the left. Why not have boxes extending infinitely far to the right as well?

Let’s go back to working with a 1101 \leftarrow 10 machine and see what such boxes could mean in that machine.
(Then it should become clear what they mean in other machines as well.)

To keep the left and right boxes visibly clear, we’ll separate them with a point. (Society calls this point, for a 1101 \leftarrow 10 machine at least, a decimal point.)
I8S8A - Image005

So, what does it mean to have dots in the right boxes? What are the values of dots in those boxes?

Since this is a 1101 \leftarrow 10 machine, we do know that ten dots in any one box explode to make one dot one place to the left. So ten dots in the box just to the right of the decimal point are equivalent to one dot in the 11 s box. Each dot in that box must be worth one-tenth.
I8S8A - Image008

We have:
I8S8A - Image009

In the same way, ten dots in the next box over are worth one-tenth. And so each dot in that next box must be worth one-hundredth.
I8S8A - Image010

We have:
I8S8A - Image011

And ten one-thousands make a hundredth, and ten ten-thousands make a thousandth, and so on.
I8S8A - Image012

We see that the boxes to the left of the decimal point represent place values as given by the powers of ten, and the boxes to the right of the decimal point place values given by the reciprocals of the powers of ten.

We have just discovered decimals!

When people write 0.30.3, for example, they mean the value of placing three dots in the first box after the decimal point.
I8S8A - Image014

We see that 0.30.3 equals three tenths: 0.3=3100.3 = \dfrac{3}{10}.

Seven dots in the third box after the decimal point is seven thousandths: 0.007=710000.007 = \dfrac{7}{1000}.
I8S8A - Image018

Comment: Some people might leave off the beginning zero and just write .007=71000.007 = \dfrac{7}{1000}. It’s just a matter of personal taste.

Some people read 0.60.6 out loud as “point six” and others read it out loud as “six tenths.” Which is more helpful for understanding what the number really is?

Question 2

There is a possible source of confusion with a decimal such as 0.310.31. This is technically three tenths and one hundredth: 0.31=310+11000.31 = \dfrac{3}{10} + \dfrac{1}{100}.
I8S8A - Image023

But some people read 0.310.31 out loud as “thirty-one hundredths,” which looks like this.
I8S8A - Image025

Are these the same thing?

Well, yes! With three explosions we see that thirty-one hundredths becomes three tenths and one hundredth.

Comment: You can also show that 310+1100\dfrac{3}{10} + \dfrac{1}{100} and 31100\dfrac{31}{100} are the same with the arithmetic of adding fractions. We have

310+1100=30100+1100=31100\dfrac{3}{10} + \dfrac{1}{100} = \dfrac{30}{100} + \dfrac{1}{100} = \dfrac{31}{100}.
(Do you see that this is really the result of performing three unexplosions in a picture of 310+1100\dfrac{3}{10} + \dfrac{1}{100}?)

A teacher asked his students to each draw a 1101 \leftarrow 10 machine picture of the fraction 3191000\dfrac{319}{1000}.

JinJin drew:
I8S8A - Image032

Subra drew:
I8S8A - Image033

The teacher marked both students as correct. Are each of these solutions indeed valid? Explain your thinking. (By the way, the teacher doesn’t mind if students just write numbers instead of drawing dots.)

Question 3

Multiple choice!

The decimal 0.230.23 equals:

(A) 2310\dfrac {23}{10}

(B) 23100\dfrac {23}{100}

(C) 231000\dfrac {23}{1000}

(D) 2310000\dfrac {23}{10000}

Question 4

The decimal 0.04090.0409 equals:

(A) 409100\dfrac {409}{100}

(B) 4091000\dfrac {409}{1000}

(C) 40910000\dfrac {409}{10000}

(D) 409100000\dfrac {409}{100000}

Question 5


Some decimals give fractions that simplify further.

For example,

0.5=510=120.5 = \dfrac{5}{10} = \dfrac{1}{2}


0.04=4100=1250.04 = \dfrac{4}{100} = \dfrac{1}{25}.

Conversely, if a fraction can be rewritten to have a denominator that is a power of ten, then we can easily write it as a decimal.

For example,

35=610\dfrac{3}{5} = \dfrac{6}{10} and so 35=0.6\dfrac{3}{5} = 0.6


1320=13×520×5=65100=0.65\dfrac{13}{20} = \dfrac{13 \times 5}{20 \times 5} = \dfrac{65}{100} = 0.65.

What fractions (in simplest terms) do the following decimals represent?

0.050.05, 0.20.2, 0.80.8, 0.0040.004

Question 6

Write each of the following fractions as a decimal.

25\dfrac {2} {5}, 125\dfrac {1} {25}, 120\dfrac {1} {20}, 1200\dfrac {1} {200}, 22500\dfrac {2} {2500}

Question 7


The decimal 0.0500.050 equals

(A)50100\dfrac {50} {100}

(B)120\dfrac {1} {20}

(C)1200\dfrac {1} {200}

(D) None of these?

Question 8

The decimal 0.0002080.000208 equals

(A) 52250\dfrac {52} {250}

(B) 522500\dfrac {52} {2500}

(C) 5225000\dfrac {52} {25000}

(D) 52250000\dfrac {52} {250000}

Question 9

Write each of the following fractions as decimals.

720\dfrac {7} {20}, 1625\dfrac {16} {25}, 301500\dfrac {301} {500}, 1750\dfrac {17} {50}, 34\dfrac {3} {4}

Question 10


What fraction does the decimal 2.32.3 represent?

Question 11

What fraction does 17.0417.04 represent?

Question 12

What fraction does 1003.10031003.1003 represent?

Question 13

Let’s explore the question: Do 0.190.19 and 0.1900.190 represent the same number or different numbers?

Here are two dots and boxes pictures for the decimal 0.190.19:
I8S8A - Image069

Here are two dots and boxes picture for the decimal 0.1900.190
I8S8A - Image070

(A) Explain how one “unexplosion” establishes that the first picture of 0.190.19 is equivalent to the second picture of 0.190.19.

(B) Explain how several unexplosions establishes that the first picture of 0.1900.190 is equivalent to the second picture of 0.1900.190.

(C) Explain how explosions and unexplosions in fact establish that all four pictures are equivalent to each other.

(D) In conclusion then: Does 0.1900.190 represent the same number as 0.190.19?

Question 14

To a mathematician, the expressions 0.190.19 and 0.1900.190 represent exactly the same numeric quantity.

But you may have noticed in science class that scientists will often write down what seems likes
unnecessary zeros when recording measurements. This is because scientists want to impart more information to the reader than just a numeric value.

For example, suppose a botanist measures the length of a stalk. By writing the measurement as 0.1900.190 meters in her paper, the scientist is saying to the reader that she measured the length of the stalk to the nearest one thousandth of a meter and that she got 11 tenth, 99 hundredths, and 00 thousandths of a meter. Thus we are being told that the true length of the stalk is somewhere in the range of 0.18950.1895 and 0.19050.1905 meters.

If she wrote in her paper, instead, just 0.190.19 meters, then we would have to assume she measured the length of the stalk only to the nearest hundredth of a meter and so its true length lies somewhere between 0.1850.185 and 0.1950.195 meters.

Question 15


How does 123412 \dfrac {3} {4}, for example, appear as a decimal?

Well, 1234=12+3412 \dfrac {3} {4} = 12 + \dfrac {3} {4} and we can certainly write the fractional part as a decimal. (The non-fractional part is already in the 1101 \leftarrow 10 machine format!)

We have

1234=12+7510012 \dfrac {3} {4} = 12 + \dfrac {75} {100}

and so we see

1234=12.7512 \dfrac {3} {4} = 12.75.

Write each of the following numbers in decimal notation.

(A) 53105 \dfrac {3} {10}

(B) 7157 \dfrac {1} {5}

(C) 131213 \dfrac {1} {2}

(D) 106320106 \dfrac {3} {20}

(E) 7825\dfrac {78} {25}

(F) 94\dfrac {9} {4}

(G) 13140\dfrac {131} {40}

You can either play with some of the optional stations below or go to the next island!

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