The concept of a decimal number

In a biology laboratory, a student is measuring the thickness of different plant leaves. Write in figures the following decimal numbers obtained in their measurements:

1. a) Twelve units and eight tenths.
2. b) Seven units and twenty-five hundredths.
3. c) Zero units and one hundred and four thousandths.
4. d) Five units and nine ten-thousandths.

At the Olympic Games, swimming times are measured with great precision. A swimmer took $48.052$ seconds to complete an event. State the place value of each digit in this number:

• The $4$ is in the:

• The $8$ is in the:

• The $0$ is in the:

• The $5$ is in the:

• The $2$ is in the:

Darragh is ordering numbers for an exam and states the following: "The number $0.19$ is greater than $0.8$ because $19$ is a much larger number than $8$. Furthermore, $0.8$ and $0.80$ represent different quantities because the second one has more decimal places". a) Find the two reasoning errors Darragh makes and explain, using the place value of the digits (tenths and hundredths), why his statements are incorrect. b) Write the correct comparison between $0.19$ and $0.8$ using the appropriate symbol ($>$ or $<$) after matching their number of decimal places.

Look at the following grid, which represents one unit divided into one hundred equal parts. State which decimal number is represented by the orange-shaded part and how this number is read.

Identify which decimal numbers are represented by the letters $A$, $B$ and $C$ on the following number line:

Draw a number line in your notebook and plot the following decimal numbers. For the negative numbers, use a different color from the one for the positive ones.

$A = 0.4$

$B = -0.25$

$C = 0.85$

$D = -0.7$

It may help you to draw the line from $-1$ to $1$, with $0$ in the center, and divide each unit into 20 equal parts, so that each mark is worth $0.05$. For example, if each unit takes up 10 cm of your notebook, the marks will be 0.5 cm apart. Write each point's letter and value next to it.

You can do it on paper and send a photo with your answer.

Order the following decimal numbers from smallest to largest:

In a Formula 1 championship, the qualifying lap times of four drivers around the circuit were as follows:

• Driver A: $1:24.058\text{ s}$

• Driver B: $1:24.5\text{ s}$

• Driver C: $1:24.06\text{ s}$

• Driver D: $1:24.0578\text{ s}$

Order the drivers from fastest to slowest using the symbols $>$ or $<$ as appropriate.

Order the following decimal numbers from least to greatest, justifying your answer by expanding at least four of their decimal digits:

• $A = 2.3\hat{5}$

• $B = 2,\widehat{35}$

• $C = 2.35$

• $D = 2.35\hat{3}$

• $E = 2.355$

• $F = 2.353$

Once ordered, find a terminating decimal number and a pure recurring decimal number that lie between the two middle values of your list.

Classify the following decimal numbers, stating whether they are terminating, pure recurring or mixed recurring:

1. a) $4.25$
2. b) $7,\widehat{3}$
3. c) $1.02\widehat{5}$
4. d) $12.444...$
5. e) $0.123$
6. f) $5.6\widehat{12}$

Without dividing the terms, state what type of decimal number each of the following fractions will produce (terminating or recurring). Justify your answer by analysing the prime factors of the denominator:

1. a) $\dfrac{7}{40}$
2. b) $\dfrac{5}{12}$
3. c) $\dfrac{11}{21}$
4. d) $\dfrac{3}{250}$

Look at the following figure, where a square of area $1\text{ u}^2$ has been divided into sections.

1. a) Express the part shaded in orange as a decimal.
2. b) Express the part left blank as a decimal.
3. c) What type of decimal numbers have you obtained (terminating or recurring)?

Classify the following decimal numbers in the attached table (Terminating, Pure recurring, Mixed recurring or Irrational):

• $3.141592...$ (Number $\pi$)

• $2.15$

• $7.444...$

• $0.1232323...$

• $\sqrt{2}$

• $5.0616161...$

We want to place the irrational number $\sqrt{7}$ on the number line, whose exact value has infinite non-repeating decimal places. a) Without using a calculator, estimate between which two consecutive integers $\sqrt{7}$ lies based on the nearest perfect squares. Next, estimate with reasoning between which two consecutive tenths ($2.5$ and $2.6$, or $2.6$ and $2.7$, etc.) it must be placed, checking by hand the square of those tenths. b) Knowing that the approximate value of this number is $\sqrt{7} = 2.645751...$, check whether your estimation by tenths was correct. c) Based on the density property of decimals, propose a terminating decimal and an eventually recurring decimal that lie between $2.64$ and $\sqrt{7}$.

Insert three rational numbers (in fraction form) between the decimal numbers $0,\hat{7}$ and $0.7\hat{8}$.

To solve it, follow these steps:

1. Find the generating fractions of both numbers.

2. Convert these fractions to a common denominator.

3. Find three fractions that lie between them and check that their decimal expressions satisfy the insertion condition.

Represent the following numbers approximately on the number line and determine exactly which of them is the greatest:

• $X = 1.4\hat{6}$

• $Y = \sqrt{2}$ (use the approximation $1.4142...$)

• $Z = \frac{11}{7}$

Since $\sqrt{2}$ is an irrational number, it has no generating fraction. Therefore:

• For $X$, calculate its generating fraction and compare it with $Z$ (for example, by reducing them to a common denominator).

• For $Y$, use its decimal approximation and compare it with the other two.

Justify the position of each number on the line by doing the necessary divisions with at least three decimal places. Use the line in the figure, where each small mark is worth $0.01$.

You can do it on paper and send a photo with your answer.

During a class discussion about the density property of decimals, Cian claims: "Between $0,\widehat{2}$ and $0.23$ no other decimal number can fit because, writing $0,\widehat{2} = 0.222...$, after the $2$ in the hundredths only $3$ can follow, so there is no gap in between". a) Prove that Cian's reasoning is false: explain how the decimal numeral system guarantees the existence of infinitely many intermediate values and find three numbers of different types (one terminating, one pure recurring, and one eventually recurring) that lie strictly between $0,\widehat{2}$ and $0.23$. b) Construct an irrational number within that same interval showing its first eight decimal places. Explain precisely the formation rule you used to ensure that its digits never form a repeating pattern and that it cannot be expressed as a fraction.

In a biology laboratory, the size of a microorganism was measured, giving a value of $0.04586$ mm. Due to a software error, the microscope shows this value approximated in two different ways:

• Approximation X: truncated to the thousandths.

• Approximation Y: rounded to the thousandths.

Calculate the relative error made in each of the two approximations. Which of the two estimates is more accurate for the scientific study? Justify your answer by comparing the errors obtained.

Four sprinters have recorded times in seconds with very close decimal expansions: $A = 12,\widehat{34}\text{ s}$, $B = 12.3\widehat{4}\text{ s}$, $C = 12.3434\text{ s}$, and $D = 12,\widehat{343}\text{ s}$. To order them from least to greatest time, two championship judges propose different approaches: Judge 1 proposes: "Let's round all the times to the nearest hundredth; that way it will be faster and simpler to order them". Judge 2 proposes: "Let's expand each number to its fifth decimal place and compare column by column from left to right". a) Apply Judge 1's proposal. Is it possible to break the tie and rank the four athletes with their method? Explain in detail what happens and why this procedure fails in this case. b) Apply Judge 2's proposal, write out the expansion to five decimal places of each time, and order the athletes from least to greatest using the symbol $<$. Justify why five decimal places are enough in this case and why, in general, comparing digit by digit up to the first differing digit never fails. c) If a fifth athlete records an irrational time whose first digits are $12.343443444...\text{ s}$, between which two athletes from the previous list would she rank? Justify it by comparing the exact decimal place in which it differs from its neighbours.

Three siblings measure their height. Aoife is $1.54$ metres tall and Cian is $1.55$ metres tall. Find a decimal number that lies between Aoife's height and Cian's height.

You are asked to construct four different decimal numbers that lie strictly between $3.4$ and $3.5$ (that is, greater than $3.4$ and less than $3.5$). Each one must satisfy one of the following conditions: a) A terminating decimal that has exactly three decimal places. b) A pure recurring decimal (remember that the repetend must start immediately after the decimal point). c) An eventually recurring decimal (with non-recurring and recurring parts). d) An irrational number. For this last one, write its first seven decimal places and explain what rule or pattern you followed so that no block of digits ever repeats.

Met Éireann has recorded the accumulated precipitation in litres per square metre ($\text{l/m}^2$) during a day of storms at four stations in the province of Munster: Station A: $24,\widehat{7}\text{ l/m}^2$; Station B: $24.75\text{ l/m}^2$; Station C: $24.7\widehat{5}\text{ l/m}^2$; Station D: $24,\widehat{75}\text{ l/m}^2$. a) Write out the expansion of each of the four records with at least four decimal places and order them from greatest to least precipitation using the symbol $>$. b) To verify the sensors, the technical team needs to input a test value that is a terminating decimal and lies strictly between the two rainiest records on the list. Propose a number that satisfies this requirement and justify why it belongs to that interval.