The Set of Natural Numbers

Plot the points corresponding to the natural numbers $A = 2$, $B = 7$ and $C = 11$ on the following number line.

State whether the following statements about the set of natural numbers ($\mathbb{N}$) are true or false, justifying your answer with an example or a counterexample:

1. a) The set $\mathbb{N}$ is a finite set because we can always write the largest possible number.
2. b) Between any two natural numbers, for example $4$ and $5$, there is always another natural number.
3. c) Every natural number has a successor and, except for the first one, they all have a predecessor.
4. d) The decimal numeral system is positional because the value of a digit depends on its position.

In the decimal numeral system, the value of a digit depends on its position. In the number 7,475, indicate what value the first "7" (starting from the left) represents and what value the second "7" represents. How many times larger is the first compared to the second?

A famous streamer has reached the figure of $2.045.703$ subscribers on his channel. Perform the polynomial expansion of this number using powers of base $10$.

The estimated population of a social network in the year 2023 reached the figure of $1.205.040.300$ active users.

1. a) Write how this number is read in English.
2. b) Perform 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 closest natural number to $N$ that ends in three zeros.

A four-digit natural number meets the following conditions:

• It is a palindromic number (it reads the same from left to right as from right to left).

• The sum of its digits is $22$.

• The thousands digit is three times the hundreds digit.

Determine, with reasons, what number it is and represent it using its polynomial expansion.

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 true:

Once solved, explain step by step the order of operations you followed to reach the result of $25$.

A courier company organizes its deliveries into three types of boxes. The shipping cost is calculated with the following expression, where $x$ is the number of small boxes, $y$ the number of medium boxes and $z$ the number of large boxes:

If a customer sends $15$ small boxes, $15$ medium boxes and $15$ large boxes:

1. Calculate the total cost of the operation.

2. Apply the distributive property to extract the common factor and rewrite the expression so that it is easier to calculate for the same number of boxes of each type.

3. If the price of each box goes up by $\$2$, how would the common factor change?

Lucy wants to organize her collection of $36$ stickers to form a perfect square on the cover of her album. How many stickers must she put on each side of the square so that none are left over? Draw a simple diagram of the solution.

In a logistics warehouse, shoe boxes are organized 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 the 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?

Imagine a bacterium that reproduces by binary fission (divides into two) every $20$ minutes. If we start with a single bacterium at 8:00 a.m.:

1. How many bacteria will there be at 10:00 a.m.? Express it as a power.

2. Represent the situation using a diagram or table that relates the elapsed time to the number of bacteria.

3. If the container where the bacteria are is completely full when there are $2^{18}$ bacteria, at what time will this happen?

A hummingbird's heart beats approximately $1200$ times per minute.

1. How many times will it beat in a full day? Express the result in scientific notation.

2. If the hummingbird lives an average of $5$ years, how many heartbeats will it have in its entire life? (Consider a year as $365$ days).

3. Use the order of operations to write a single expression that calculates the total heartbeats in those $5$ years.

The ancient Babylonians used a sexagesimal system (base 60) to measure time, a system we still use today. If a basketball game lasts $1$ hour, $22$ minutes and $15$ seconds, express the total duration of the game solely in seconds.

A content creator has recorded three videos for his channel. The first lasts $1$ h $15$ min $40$ s, the second lasts $45$ min $25$ s and the third $1$ h $5$ s.

1. a) What is the total duration of the three videos expressed in single-unit form (seconds)?
2. b) If he wants the playlist to last exactly $4$ hours, how much footage does he have left to record? Express the result in mixed-unit form (hours, minutes and seconds).

At an archaeological site in ancient Mesopotamia, a tablet with a grain inventory has been found. The Mesopotamian system is sexagesimal (base $60$) and positional. The symbols used are the vertical wedge ($\curlyvee$) which is worth $1$ and the horizontal wedge ($\prec$) which is worth $10$.

Translate the number represented on the following tablet into our decimal system, bearing in mind that the space between the groups of symbols indicates a change in the order of units:

Write the complete arithmetic expression and the final result in the decimal numeral system.

Walking through the center of your city, you see a plaque on an ancient monument that indicates the year of its construction in Roman numerals: MDCCCXCVII. Translate that date into our decimal numeral system.

An archaeologist has discovered an inscription at the entrance of an ancient temple that indicates the year of its construction in Roman numerals: MMCDLIV.

1. a) Translate the year into the Decimal Numeral System.
2. b) If the temple was restored $136$ years after its construction, write the year of the restoration in Roman numerals as well.

The language used by computers is the binary system (base 2), which only uses the digits $0$ and $1$. If the binary number $101$ is equivalent to the number $5$ in our decimal system, and $110$ is equivalent to $6$, what decimal number is the binary number $1000$ equivalent to?

In computing, numbers are usually represented in the binary system. A storage device indicates that it has the following sectors occupied, represented by the binary number $101101_2$.

1. a) What number does it represent in our decimal system?
2. b) Write the decimal number $75$ in binary code.

A data server uses LED bulbs to show error codes in binary. Each bulb that is on represents a $1$ and each bulb that is off represents a $0$. The system reads the bulbs from right to left, where the first bulb on the right represents $2^0$, the second $2^1$, the third $2^2$, and so on.

If the server shows the following status, determine the decimal number of the error code and write its polynomial expansion in powers of base $10$.

The Pythagoreans represented numbers using dots to form geometric shapes. Look at the following sequence of triangular numbers and, without needing to draw all the dots, indicate how many dots the fifth number in the sequence ($T_5$) would have.

Triangular numbers are those that can be represented by dots forming an equilateral triangle. Look at the following sequence:

1. a) How many dots will the sixth triangular number ($T_6$) have?
2. b) Draw the triangular number $T_4$ in your notebook and check if the sum of two consecutive triangular numbers $T_3 + T_4$ results in a perfect square number.

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

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

2. How many dots will the shape $H_5$ have?

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