Exercises: Concept of natural numbers

In a tech shop, an employee is reviewing the inventory and measurements of some products. From the following list of values, state which ones represent natural numbers:

• a) $15$ smartphones.

• b) $1,5$ metres of charging cable.

• c) $40$ protective covers.

• d) $-3$ degrees in the cold storage room.

• e) $2/3$ of battery charge.

• f) $0$ errors in the log.

• g) $105$ wireless headphones.

A famous influencer has gained a total of three million four hundred and five thousand seven hundred and twenty followers on their account.

Write this quantity in figures using the decimal system and identify which digit is in the hundred thousands place.

Look at the position of the digits in the number $45.082$. State the place value of the digit $5$ and the digit $8$ in this number. Use the following diagram to help you and copy it into your notebook.

Express the number $70.432$ in expanded form using powers of $10$, breaking it down according to the place value of its digits.

Example: $625 = 6 \cdot 100 + 2 \cdot 10 + 5$

Identify which natural numbers correspond to the points A, B and C represented on the following number line:

Compare the following pairs of natural numbers using the symbols less than ($<$) or greater than ($>$):

1. a) $12.506$ ___ $12.605$
2. b) $9.999$ ___ $10.001$
3. c) $304.120$ ___ $304.099$
4. d) $1.000.500$ ___ $1.000.090$

The capacity of a football stadium is $74.682$ people. A sports newspaper wants to publish the story and decides to give an approximate figure. Round the number of attendees to the nearest thousand.

On the facade of an old theatre built in 1924, the date appears in Roman numerals.

Write the year of construction using the Roman numeral system. Remember the equivalences: $M=1000$, $D=500$, $C=100$, $L=50$, $X=10$, $V=5$, $I=1$.

Mateo is preparing a poster for Culture Week about the history of ancient Rome and needs to write the number $99$ in Roman numerals. To save space, he directly writes $IC$, arguing that since $C$ is worth $100$ and $I$ is worth $1$, placing it to the left simply subtracts it: $100 - 1 = 99$.

1. a) Explain which rule of the Roman numeral system Mateo has broken. From which specific symbols is the letter $I$ allowed to be subtracted?
2. b) Explain the correct procedure for writing numbers in the Roman system from their decomposition into tens and units, and write the number $99$ correctly step by step.

To organise transport for a school sports day, the management team notes down the registered students from three schools: $187$ from the first school, $312$ from the second, and $204$ from the third.

1. a) Before calculating the exact sum, estimate the total number of participants by rounding each of the three amounts to the nearest hundred.
2. b) Calculate the exact sum of the students and determine the difference between your estimate and the actual value.
3. c) If the transport company provides buses with $50$ seats each, would it have been safe to book the buses based solely on the estimate rounded to the hundreds? Explain what danger rounding to the nearest hundred entails compared to always rounding up in real-life logistical situations.

A famous YouTube channel currently has $14.508.392$ subscribers.

1. a) Write how this number is read using words.
2. b) State the value of the digit $5$ in this number according to its place value (units, tens, hundreds, etc.).
3. c) If the channel gains one million more subscribers, what will the ten millions digit be?

On the facade of an old theatre in your city, the opening date is engraved in Roman numerals: MDCCCXLIX.

1. a) Convert the opening year into the decimal number system.
2. b) If the theatre was renovated for the first time $75$ years later, write the year of that renovation in both decimal and Roman numerals.

Write the polynomial decomposition of the number of kilometres that light travels in one year (a light year), which is approximately $9.460.730.472.580$ km. To do so, use powers of base $10$.

Look at the following number line where several points are represented. Determine which natural numbers correspond to the letters $A, B$ and $C$, knowing that the distance between each main tick mark is constant.

In a tech shop, they are comparing the sales of three tablet models over the last year:

• Model Alpha: $234.090$ units.

• Model Beta: $243.009$ units.

• Model Gamma: $234.900$ units.

1. a) Order the models from highest to lowest sales using the symbol $>$.
2. b) If the company's goal was to sell a quarter of a million units of the most successful model, how many units were they short of reaching that figure?

The budget for building a new sports centre is $4.376.512$ €.

1. a) Approximate this budget by rounding to the nearest hundred thousand.
2. b) Approximate by truncating to the nearest ten thousand.
3. c) Which of the two previous approximations is closer to the actual price? Justify your answer by calculating the difference in both cases.

A group of archaeologists has found an ancient code that uses an additive number system with the following symbols:

• △ represents $100$ units.

• □ represents $10$ units.

• | represents $1$ unit.

1. a) Write the natural number represented by the combination: △△△□□□□|||||||
2. b) If we add $45$ units to that number, how would the result be written using these symbols?

Using only the digits $2, 0, 5, 8$ and $9$, without repeating any:

1. a) Write the largest possible even number.
2. b) Write the smallest possible odd number (bearing in mind that a number does not usually start with the digit $0$).
3. c) Calculate the difference between both numbers.

A four-digit natural number is a secret code that you must discover using the following clues:

1. Rounding it to the hundreds gives $4.800$, but truncating it to the hundreds results in $4.700$.

2. The tens digit is an odd number strictly greater than $6$.

3. The number is a palindrome (it reads the same from left to right as from right to left).

1. a) Determine this secret number with reasoned steps, justifying step by step why you rule out the other options.
2. b) Write the obtained number using its polynomial decomposition with powers of $10$.

According to municipal register data (INE), three neighbouring villages in a region have the following populations: Robledillo has $8.472$ inhabitants, El Pinar has $12.835$, and Valdesoto has $9.604$ residents. For an institutional campaign, two ways of simplifying the figures are being considered:

1. a) Calculate the estimated total population of the region if you first round the number of inhabitants of each village to the nearest thousand.
2. b) Calculate the estimated total population if, instead of rounding, you truncate the number of inhabitants of each village to the thousands.
3. c) Find the exact actual population by adding the three original figures. Compare the difference with the actual value in each case and explain why rounding usually provides an overall approximation much closer to reality than truncation.

The security PIN of Lucía's new mobile phone is a six-digit number with the following characteristics:

• It is a palindrome (it reads the same forwards as backwards).

• The sum of all its digits is $28$.

• The hundred thousands digit is three times the hundreds digit.

• The number does not contain any digit repeated more than twice.

• The value of the tens digit is an even prime number.

Determine Lucía's secret number and write its expanded form using powers of $10$.

In a space mission, a probe has detected an asteroid located at a distance of $135$ billion metres from Earth.

1. a) Express this quantity in kilometres using a single power of base $10$.
2. b) If the probe travels at a constant speed of $10.000$ km/h, calculate how many hours it will take to reach the asteroid. Express the final result in standard form.

An old computer system uses a light code to represent numbers in binary system. The system has $6$ light bulbs. When a bulb is on, it represents a $1$, and when it is off, it represents a $0$. The bulbs are read from right to left with power of $2$ values ($2^0, 2^1, 2^2...$).

If we want to represent the number of students in a secondary school that has $5$ groups of 1st ESO, $4$ groups of 2nd ESO and each group has exactly $25$ students:

1. a) What is the total number of students in decimal?
2. b) What would be the bulb code (on/off) needed to represent that number? Are $6$ bulbs enough to represent this number? Justify your answer.

An archaeologist finds a Roman inscription indicating the year of construction of a temple. The inscription is $\overline{IV}CDLXIX$ (remember that a horizontal bar over a symbol multiplies its value by $1000$). However, they suspect there is an error and that the temple was actually built $144$ years later.

1. a) Convert the year on the inscription into the decimal system.
2. b) Calculate the actual year of construction in the decimal system.
3. c) Write the resulting actual year in Roman numerals again.

We have the natural number $7.402$.

1. a) How does its value change if we swap the units digit with the thousands digit? Calculate the difference between both numbers.
2. b) If we add a zero between the hundreds digit and the tens digit of the original number, by how much does the initial number increase?

A famous YouTuber currently has twelve million three hundred thousand views on their channel. Their goal is to reach two billion views by the end of the year.

1. a) Write both quantities in figures.
2. b) How many views are they short of reaching their goal? Express the difference in expanded form.
3. c) If on average they get one million views every three days, will they reach their goal in the $300$ days remaining in the year?

State whether the following statements about the set of natural numbers ($\mathbb{N}$) are true or false, giving an example or counterexample in each case:

• 1) The place value of a digit in the decimal system is independent of its position if all the digits of the number are equal.

• 2) When multiplying a natural number by $10$, its number of digits always increases by one.

• 3) The smallest natural number is zero.

• 4) In the Roman numeral system, the value of a symbol is always added to the previous one.

Marta wants to organize her collection of $144$ video games on a shelf so that they are arranged in perfect squares.

1. a) If she wants to make the largest possible square, how many games will its side have? Will she have any games left over?

1. b) If she decides to use all the games and split them into two shelves that each form an identical perfect square, is it possible? Justify your answer by finding the side length of those two squares or by explaining why it cannot be done.

In a logic challenge, you are asked to find out how many three-digit natural numbers have a sum of digits exactly equal to $4$. Two classmates suggest different strategies:

• Method A (Sequential exploration): Check the three-digit numbers starting at $100$, sum the digits of each one mentally, and note down only those that add up to $4$ until reaching $999$.

• Method B (Ordered positional decomposition): Represent the number as $c \cdot 100 + d \cdot 10 + u$, where the hundreds digit $c$ can only be $1, 2, 3$, or $4$, and list the possible combinations of digits in an orderly way by fixing the hundreds from greatest to least.

1. a) Compare the efficiency of both methods: why is Method A not an appropriate mathematical strategy for this problem?
2. b) Apply Method B: list all the triplets of digits $(c, d, u)$ in an orderly manner and deduce the complete list of numbers that meet the condition.
3. c) Without checking the numbers you obtained one by one, justify using place value why none of them can end in a units digit greater than $3$ if the hundreds digit is at least $1$.

A science outreach magazine makes several mistakes in an article about numeral systems:

"The Roman system is more perfect than the decimal system because it does without zero, an empty and unnecessary symbol. In the decimal system, if we remove the zero from $405$, it is confused with $45$, which proves it is a defective system. Furthermore, Roman rules are so simple that to write any number it is enough to subtract the smaller symbol from the larger one; for example, $1.999$ is simply written as $MIM$."

As a mathematics student, write a well-founded response:

1. a) Identify the two mistakes regarding Roman rules made when writing $1.999$ as $MIM$. Decompose $1.999$ according to the place value of its digits and write its correct Roman representation.
2. b) Refute the claim that zero is a flaw: explain why the decimal system needs zero to be positional and what fundamental limitation the Roman system has by lacking it.
3. c) Justify with a calculation example why the decimal system historically facilitated arithmetic operations (such as adding or multiplying medium-sized quantities) compared to the Roman system.

Topic challenge. How do natural numbers describe the reality of your surroundings? Imagine you are a demographic analyst and you need to create a profile of your province.

1. Data collection: Go to the official INE website (ine.es) and find the population of your municipality and four neighbouring municipalities.

2. Spreadsheet processing: Create a table in Calc or Excel. Order the five municipalities from highest to lowest population using the signs $>$ and $<$. Add columns to decompose each value according to its place value (ones, tens, hundreds, thousands...) and round each figure to the nearest ten thousand.

3. Analysis and final product: Prepare a brief report (one page) in which you calculate the exact difference in population between the most populated and the least populated municipality. Argue why the set of natural numbers ($\mathbb{N}$) must be used to census people rather than any other number set.

Assessment criteria:

• Rigorous data extraction from the INE and clean tabulation in a spreadsheet.

• Mastery of key concepts of $\mathbb{N}$: ordering, place value, and rounding.

• Clear exposition and reasoned justification in the report's conclusions.