Exercises: Concept

In the decimal numeral system, the value of a digit depends on its position. Consider the number of followers of a famous streamer, which is exactly $452\,807$. Write its polynomial expansion using powers of $10$.

Write the following numbers in figures and using powers of $10$:

1. a) Eight million, three hundred thousand units.
2. b) Thirty-two billion units.
3. c) One hundred thousand.

Look at the following number line where several points have been plotted. Indicate which natural number corresponds to each of the labeled letters.

Roman numerals are still used to mark centuries or on the facades of old buildings. Translate the following numbers written by an architect into our decimal system:

1. a) $XIX$
2. b) $CDXLIV$
3. c) $MCMLXXXVIII$

In computing, the binary system (base $2$) is used to store information. Knowing that only the digits $0$ and $1$ are used in this system, translate the binary number $10110_{2}$ into our decimal system.

The ancient Greeks studied numbers according to the geometric shapes they could form. Look at the following triangular numbers and answer: what is the fifth triangular number in the sequence?

State whether the following statements are True or False and briefly justify your answer:

1. a) In the number $77\,777$, all digits have the same value.
2. b) The decimal numeral system is a positional system.
3. c) The set of natural numbers has a last element.

Order the following natural numbers from smallest to largest using the symbols $<$ or $>$ as appropriate:

In a strategy video game, the number of wood resources accumulated by a player is expressed by the following polynomial expansion:

$4 \cdot 10^6 + 2 \cdot 10^5 + 0 \cdot 10^4 + 8 \cdot 10^3 + 5 \cdot 10^2 + 1 \cdot 10^1 + 9 \cdot 10^0$

Write the number of resources in the Decimal Numeral System and indicate what value the digit $2$ represents within this number (thousands, tens of thousands, hundreds of thousands, etc.).

Look at the following inscription on the facade of an old theater in your city, which indicates the year it was finished: MDCCCLXXXIV.

1. a) Translate this number into the Decimal Numeral System.
2. b) If the theater was renovated $140$ years after its opening, in what year did the renovation take place? Write the result in Roman numerals as well.

The Mayan numeral system is positional and uses base $20$ levels. In the first level (the bottom one), the symbols are multiplied by $20^0 = 1$, and in the second level (the top one), they are multiplied by $20^1 = 20$.

Knowing that a dot represents $1$ unit and a line represents $5$ units, identify what decimal number the following Mayan diagram represents:

Triangular numbers are formed by arranging dots in the shape of an equilateral triangle. The first triangular number is $1$, the second is $3$, and the third is $6$.

1. a) Draw the dots corresponding to the fourth and fifth triangular numbers.
2. b) Use the formula $T_n = \frac{n \cdot (n + 1)}{2}$ to calculate the triangular number in position $n = 12$.

In the binary system (base $2$), only the digits $0$ and $1$ are used. This system is the basic language of computers.

1. a) Express the decimal number $25$ in the binary system using the method of successive divisions by $2$.
2. b) What decimal number does the binary number $11011_2$ represent?

An astronomer is studying the distance between two stars and notes in his notebook that they are $720\,000\,000$ miles apart.

1. a) Express this amount using scientific notation.
2. b) If a spaceship travels at a constant speed of $10^4$ mph, how many hours will it take to cover that distance? Express the result as a power of $10$.

Natural numbers can be represented by rectangular shapes. For example, the number $12$ can be represented as a rectangle of $3 \times 4$ or $2 \times 6$.

1. a) Draw all the possible rectangular shapes (excluding the single-row one) for the number $18$.
2. b) What is the relationship between the dimensions of these rectangles and the factors of the number $18$?

State, with reasons, whether the following statements about the set of natural numbers are true or false:

1. a) Between any two natural numbers, there is always another natural number.
2. b) The decimal numeral system is positional because the value of a digit depends on its position.
3. c) Any natural number can be expressed as a sum of distinct powers of base $10$ (without repeating any).
4. d) The Roman numeral VV is a valid way to write the decimal number $10$.

In a strategy video game, an ancient civilization uses a base 12 positional numeral system. The symbols for the values ten and eleven are $\alpha$ and $\beta$ respectively. A player finds a chest that requires entering a decimal code to open, but the clue is written in the civilization's system: $2\beta\alpha_ {12}$.

Determine the value of the code in our decimal system by performing the polynomial expansion of the number.

Triangular numbers ($T_n$) can be visualized as dots forming an equilateral triangle. A fascinating property is that the sum of two consecutive triangular numbers always results in a perfect square ($C_n$).

Knowing that the formula to find the nth triangular number is $T_n = \frac{n \cdot (n + 1)}{2}$, show graphically and by numerical calculation that the sum of the fourth triangular number ($T_4$) and the fifth triangular number ($T_5$) is equal to the fifth square number ($C_5$).

A popular short video social network stores approximately $4,5 \cdot 10^6$ new clips every day. Each video takes up, on average, a space of $1,2 \cdot 10^7$ bytes.

Calculate, expressing the result in scientific notation:

1. a) The total space in bytes that the platform needs to store the videos of a single day.
2. b) The total space it will need after a leap year ($366$ days).

In computing, the binary system (base 2) is fundamental. A "byte" is made up of 8 "bits" (each bit is a 0 or a 1). The maximum number that can be represented with one byte is $11111111_2$.

1. a) Convert this number to the decimal system.
2. b) If a 16-bit processor can represent numbers up to $2^{16} - 1$, express this maximum value in scientific notation to two decimal places.

An art collector wants to organize his gallery so that his $144$ paintings form a perfect square. However, he decides to buy more paintings so that the gallery is a rectangle where the length is twice the width.

What is the minimum number of additional paintings he must buy to form this rectangle if he wants the width to be greater than the side of the original square?

Apply the order of operations and the properties of powers to find the value of the following "magic" expression used in a social media challenge:

Show all the intermediate steps of the work.

A "light-year" is the distance that light travels in one year. Knowing that the speed of light is approximately $300.000$ mi/s, calculate:

1. a) How many miles light travels in one hour (express the result in scientific notation).
2. b) If the closest star to the Sun, Proxima Centauri, is $4,22$ light-years away, how many miles separate us from it? (Use a year of $365$ days and express the result in scientific notation).

A circular botanical garden is to be designed. The budget allows buying enough seeds to cover an area of $31.400$ square feet.

1. a) If we use the approximate value of $\pi \approx 3,14$, what will the radius of the garden be?
2. b) The city council decides that the garden must be surrounded by a square fence whose side is exactly equal to the diameter of the circular garden. How many feet of fencing are needed?
3. c) What is the area of the square plot of land that lies outside the circle but inside the fence?