Exercises: Concept

In the decimal number 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 using figures and 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 on which several points are represented. State which natural number corresponds to each labelled letter.

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 collected by a player is expressed by the following polynomial expansion:

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

Write the number of resources in the base-10 positional system and state the place value of the digit $2$ within this number (thousands, ten thousands, hundred thousands, etc.).

Natural numbers can be represented using rectangular arrays. For example, the number $12$ can be represented as an array of $3 \times 4$ or $2 \times 6$.

1. a) Draw all possible rectangular arrays (excluding single-row arrays) for the number $18$.
2. b) What relationship do the dimensions of these rectangles have with the factors of the number $18$?

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

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 number 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 $10$.

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 gives a perfect square ($C_n$).

In the figure you can see how the dots of $T_4$ and $T_5$ are arranged.

Knowing that the formula for the $n$th triangular number is $T_n = \frac{n \times (n + 1)}{2}$, prove graphically and numerically that the sum of the fourth triangular number ($T_4$) and the fifth triangular number ($T_5$) equals the fifth square number ($C_5$), that is, the number of dots in a $5 \times 5$ square.

1. a) Make the drawing: arrange the dots of $T_5$ and $T_4$ so that, together, they form a square of $5 \times 5$ dots (use two different colors).

1. b) Calculate $T_4$, $T_5$ and $C_5 = 5^2$ using the formulas and check that $T_4 + T_5 = C_5$.

An art collector wants to arrange his gallery so that his $144$ paintings form a perfect square. However, he decides to buy more paintings so that the gallery becomes 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 length of the original square?

A circular botanical garden is to be designed. The budget allows buying enough seeds to cover an area of $31,400$ square meters.

1. a) If we use the approximate value of $\pi \approx 3.14$, what will be the radius of the garden?
2. b) The municipality decides that the garden must be surrounded by a square fence whose side length is exactly equal to the diameter of the circular garden. How many meters 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?