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:

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.

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 $\pi$)

• $2.15$

• $7.444...$

• $0.1232323...$

• $\sqrt{2}$

• $5.0616161...$

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

Three siblings measure their height. Lucía is $1.54$ meters tall and Marcos is $1.55$ meters tall. Find a decimal number that lies between Lucía's height and Marcos's height.

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

$A = 0.4$

$B = -0.25$

$C = 0.85$

$D = -0.7$

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.

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.