The set of natural numbers
A Metro ticket costs $\$5$, and you can buy 1, 2, 3 tickets... But is there a natural number that is the largest? And one that lies between $7$ and $8$? Natural numbers have their own rules, and it's worth getting them straight before operating with them.
Natural numbers and place value
Place value, reading, and decomposition
Our system is decimal and positional: the value of a digit depends on the place it occupies. Each place is worth $10$ times the one to its right.
| Millions | Thousands | Ones |
|---|---|---|
| H T O | H T O | H T O |
(H = hundreds, T = tens, O = ones within each group of three digits.)
To read a number, you split it into groups of three digits with commas and read each group with its name. For example, $3{,}052{,}400$ is read "three million fifty-two thousand four hundred".
The expanded form writes the number as a sum of each digit times its power of $10$:
$$47{,}082 = 4\times10^4 + 7\times10^3 + 0\times10^2 + 8\times10 + 2$$The place value of a digit is what it is worth because of where it sits: in $47{,}082$, the $7$ is worth $7{,}000$ and the $8$ is worth $80$.
Whole hundreds and rounding
To find out how many whole hundreds are in a number, count how many times $100$ fits: it is the quotient of dividing by $100$, that is, the number left after removing the last two digits. In $47{,}082$ there are $470$ whole hundreds, with $82$ ones left over.
To round to a ten, hundred, or thousand, look at the digit that follows the place you are rounding to:
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If it is $5$ or more, the digit in that place goes up by one.
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If it is less than $5$, it stays the same.
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The digits to the right become zeros.
The closest number ending in zeros is exactly the rounding to that place.
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)…
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. a) Write how this number is read in words. b) Give its polynomial expansion…
The estimated population of a social network in 2023 reached $1,205,040,300$ active users.
- a) Write how this number is read in words.
- b) Give its polynomial expansion using powers of base $10$.
- c) What is the place value of the digit $5$ in this number?
Consider the natural number $N = 47,082$. a) If we swap the thousands digit with the tens digit, what is the difference between the original number and the new number…
Consider the natural number $N = 47,082$.
- a) If we swap the thousands digit with the tens digit, what is the difference between the original number and the new number obtained?
- b) How many complete hundreds are there in total in the number $N$?
- c) Write the natural number closest to $N$ that ends in three zeros.
Order of operations
When an expression mixes several operations, the order is not free.
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…
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:
$$5 + 3 \times 8 - 6^2 : 4 + 2 = 23$$Once solved, explain step by step the order of operations you followed to arrive at the result of $23$.
Perfect squares and cubes
A perfect square is a natural number that can be arranged in a complete square of dots, with none left over. It is the result of multiplying a natural number by itself: $n^2$.
A perfect cube is $n^3$: the pieces form a complete cube.
The first perfect squares: $1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144\ldots$ The first cubes: $1, 8, 27, 64, 125\ldots$
Lucía 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…
Lucía 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…
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:
- a) How many boxes are there in total on the shelf? Express it as a power and calculate its value.
- 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?
Numbers with special patterns
Some numbers hide a pattern. Besides perfect squares, there are hexagonal numbers, which are arranged in ever larger hexagons, and palindromic numbers, which read the same forward and backward.
A four-digit palindrome has the form $abba$ (the first digit equal to the last, and the second equal to the third). This helps you set up conditions on its digits, such as their sum or the relationship between two of them.
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…
The Pythagoreans studied polygonal numbers. Look at the following sequence of hexagonal numbers and answer:
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Draw or describe what the fourth hexagonal number pattern ($H_4$) would look like.
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How many dots will pattern $H_5$ have?
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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…
A four-digit natural number meets the following conditions:
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It is a palindromic number (it reads the same forwards and backwards).
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The sum of its digits is $24$.
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The thousands digit is three times the hundreds digit.
Reasonably determine which number it is and represent it using its expanded form.
Roman numerals
They are written with seven letters: $I=1$, $V=5$, $X=10$, $L=50$, $C=100$, $D=500$, $M=1{,}000$. A smaller letter to the right of a larger one is added ($VII=7$); to the left, it is subtracted ($XC=90$). No letter is repeated more than three times in a row ($4$ is $IV$) and the value of a letter does not depend on its position, which is why there is no Roman zero.
For example, $MCMXCIV = 1{,}000 + 900 + 90 + 4 = 1{,}994$.
Walking through your city center, you see a plaque on an ancient monument indicating its year of construction in Roman numerals: MDCCCXCVII. Convert that date into our…
Walking through your city center, 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.
Summary
Go back to the beginning: there is no largest natural number, since you can always add $1$, and between $7$ and $8$ there is no natural number.