Exercises: Concept of natural numbers

In a tech store, 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) $1.5$ meters 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: $625 = 6 \times 100 + 2 \times 10 + 5$

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 soccer 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 theater built in 1924, the date appears in Roman numerals.

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

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 theater 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 theater was renovated for the first time $75$ years later, write the year of that renovation in both decimal and Roman numerals.

Write the expanded form of the number of kilometers that light travels in one year (a light year), which is approximately $9,460,730,472,580$ km. To do this, 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 center is $4,376,512$ pesos.

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.

Lucía's new cell phone PIN is a six-digit number with the following features:

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

• The sum of all its digits is $24$.

• The hundred thousands digit is triple the hundreds digit.

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

• The tens digit is an even prime number.

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

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

1. a) Express this quantity in kilometers 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 scientific notation.

An old computer system uses a light code to represent numbers in the 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 year of secondary, $4$ groups of 2nd year of secondary 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.