The set of natural numbers

In the decimal number system, the value of a digit depends on its position. In the number 7 475, state the value represented by the first "7" (starting from the left) and the value represented by the second "7". How many times greater is the first than the second?

The estimated population of a social network in 2023 reached $1,205,040,300$ active users.

1. a) Write how this number is read in words.
2. b) Give its polynomial expansion using powers of base $10$.
3. c) What is the place value of the digit $5$ in this number?

Consider the natural number $N = 47,082$.

1. a) If we swap the thousands digit with the tens digit, what is the difference between the original number and the new number obtained?
2. b) How many complete hundreds are there in total in the number $N$?
3. c) Write the natural number closest to $N$ that ends in three zeros.

During a group discussion about the characteristics of the set $\mathbb{N}$, three students defend the following statements:

• Cian: "Between $10$ and $20$ there are infinitely many natural numbers, because $\mathbb{N}$ is an infinite set and numbers never end".

• Aoife: "If we invent an operation between natural numbers defined as $a \odot b = a \times b - (a + b)$, the set $\mathbb{N}$ is closed under it because we are only multiplying, adding, and subtracting".

• Darragh: "The number $0$ should not be part of $\mathbb{N}$ because natural numbers are used to count real objects, and if you have $0$ objects you are not counting anything tangible".

1. a) Refute Cian's statement by explaining the fundamental difference between the fact that $\mathbb{N}$ is an infinite set and the property that $\mathbb{N}$ is a discrete set.
2. b) Prove that Aoife's statement is false using a specific numerical counterexample with two natural numbers that result in an answer that does not belong to $\mathbb{N}$, and explain which property of subtraction causes this.
3. c) Respond to Darragh's argument by explaining what role zero plays in the positional decimal number system and why the current curriculum includes it as the first element of $\mathbb{N}$.

To equip the art classroom, a school purchases $4$ boxes of poster paints at $12$ € each, $3$ packs of paintbrushes at $9$ € each, and a roll of continuous paper for $48$ €. In addition, the shop applies a special loyalty offer that discounts $5$ € for each box of poster paints purchased.

The combined expression representing the final cost in euros is:

1. a) Without calculating the exact value on paper or using a calculator, make a quick mental estimate of the result by rounding each price or operation to the nearest ten and briefly explain how you set it up.
2. b) Solve the combined expression step by step, respecting the order of operations.
3. c) Compare your previous estimate with the exact result: was it accurate? Justify whether your estimate was higher or lower than the actual value.

Consider the following numerical expression without parentheses: $3 + 5 \times 4 - 2$. If we calculate it directly respecting the natural order of operations, we get $3 + 20 - 2 = 21$.

1. a) Place a single pair of parentheses in the expression so that the final result equals $13$.
2. b) Place a single pair of parentheses in a different position from the previous one so that the result equals $30$.
3. c) Where would you place two pairs of parentheses simultaneously so that the final result is exactly $16$?
4. d) Explain why, regardless of where you place the parentheses, you will never be able to obtain a decimal number from this operation, relating it to the properties of the set $\mathbb{N}$.

A secondary school organizes a visit to the National Gallery of Ireland in Dublin for a group of $28$ students and $3$ teachers. To travel on the Irish Rail DART, the school checks the following ticket options for the same zone:

• Single ticket: $3$ € per journey.

• 10-journey pass (transferable pass that can be shared by multiple people): $18$ € per pass.

Both students and teachers must make the outbound and return journeys on the same day.

1. a) Model the situation by setting up a single arithmetic expression with combined operations to calculate the minimum total cost, knowing that the group will buy as many 10-journey passes as possible and will purchase single tickets only for the remaining individual journeys.
2. b) Calculate this minimum cost step by step by applying the order of operations.
3. c) If it were decided to split the total cost obtained equally among the $31$ people going on the trip, would the cost per person belong to the set $\mathbb{N}$ of natural numbers? Justify your answer without needing to perform long division with decimals.

In a mathematical logic challenge, you are given an expression that is missing parentheses for the result to be correct. Place the necessary parentheses so that the equality holds:

Once solved, explain step by step the order of operations you followed to arrive at the result of $23$.

Walking through your city centre, you see a plaque on an ancient monument indicating its year of construction in Roman numerals: MDCCCXCVII. Convert that date into our decimal number system.

Aoife has seen a commemorative plaque on the facade of the municipal library and wants to transcribe the year it was founded, 1999, into the Roman numeral system. In her notebook she has written: $MIM$, and she justifies it to her classmate by saying: "I put an $M$ ($1,000$) and from another $M$ I subtract $I$ placed to its left ($1,000 - 1 = 999$); that way I have $1,000 + 999 = 1,999$".

1. a) Identify the two mistakes made by Aoife and explain which rules of Roman numerals she has broken.
2. b) Write the number 1999 correctly in Roman numerals, breaking down step by step each place value (thousands, hundreds, tens, and units).

Aoife wants to arrange her collection of $36$ stickers into a perfect square on her album cover. How many stickers must she place along each side of the square so that none are left over? Draw a simple diagram of the solution.

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

In a logistics warehouse, shoe boxes are arranged to form perfect cubes. If a shelf has a square base of $7 \times 7$ boxes and they are stacked to reach the same height to form a cube:

1. a) How many boxes are there in total on the shelf? Express it as a power and calculate its value.
2. b) If $150$ boxes are sold, what is the maximum number of boxes that can still form a perfect square on the warehouse floor? How many boxes would be left over in that case?

The Pythagoreans studied polygonal numbers. Look at the following sequence of hexagonal numbers and answer:

1. Draw or describe what the fourth hexagonal number pattern ($H_4$) would look like.

2. How many dots will pattern $H_5$ have?

3. Knowing that the formula to calculate the hexagonal number in position $n$ is $H_n = 2n^2 - n$, work out the value for $H_{12}$.

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

A four-digit natural number meets the following conditions:

• It is a palindromic number (it reads the same forwards and backwards).

• The sum of its digits is $24$.

• The thousands digit is three times the hundreds digit.

Reasonably determine which number it is and represent it using its expanded form.

In a mathematics class, the teacher proposes calculating the sum of the first $50$ non-zero even natural numbers:

Two students propose different ways to solve it without needing to add the fifty terms one by one:

• Cian's method: He realizes that he can pair the extremes by adding the first to the last ($2 + 100 = 102$), the second to the second-to-last ($4 + 98 = 102$), and so on with all possible pairs.

• Aoife's method: She prefers to factor out the common factor $2$ from the entire expression: $2 \times (1 + 2 + 3 + \ldots + 50)$. Then, she applies the triangular formula she learned to sum the first $n$ natural numbers, $\frac{n \times (n + 1)}{2}$, calculating $2 \times \frac{50 \times 51}{2}$.

1. a) Apply both methods to a reduced version with only the first 4 even numbers ($2 + 4 + 6 + 8$) and verify that you reach the same solution.
2. b) Analyze and compare both methods for the original case of the $50$ terms: which of the two requires fewer calculation steps and carries a lower risk of error? Justify your answer by naming the properties of operations in $\mathbb{N}$ that underpin each strategy.
3. c) Calculate the final value of the sum $S$ using the method you defended in the previous section.