The Set of Natural Numbers
Yaritza sells passion fruit juice outside her school and writes down how many cups she sold each day: 0, 12, 25… Is there a natural number that is the largest? And is there one between 7 and 8? The set of natural numbers, $\mathbb{N}$, has its own rules, and it's worth getting them straight before we start operating with them seriously.
The set ℕ
ℕ inside the real numbers
When subtraction doesn't fit inside $\mathbb{N}$, the set is extended: the integers $\mathbb{Z}$ appear (with negatives), then the rational numbers $\mathbb{Q}$ (with fractions), and finally the real numbers $\mathbb{R}$. Each set contains the previous one:
$$\mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R}$$So $5$ is natural, an integer, rational, and real all at once, but $-3$ is an integer and is not natural.
Place value, reading, and rounding
Our system is decimal and positional: a digit is worth more or less depending on the place it occupies. In $7{,}475$ the first $7$ is in the thousands place and is worth $7{,}000$; the second is in the tens place and is worth $70$.
| Place | Millions | Hundred thousands | Ten thousands | Thousands | Hundreds | Tens | Ones |
|---|---|---|---|---|---|---|---|
| Power | $10^6$ | $10^5$ | $10^4$ | $10^3$ | $10^2$ | $10^1$ | $10^0$ |
Each position is worth 10 times the one to its right. That's why a number can be written as the sum of each digit times its power of $10$: this is the expanded form.
How to read a large number
Digits are grouped in threes, from right to left, and each group is read with its own name: ones, thousands, millions. The comma separates the groups.
Rounding a number
Rounding means replacing a number with a nearby one that ends in zeros. To round to a place (tens, hundreds, thousands…), look at the digit to its right:
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 addition, subtraction, multiplication, division, exponents, and parentheses, the order in which you operate is not up to you: there is a fixed hierarchy.
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 $23$.
Roman numerals
Before the decimal system, people in Rome wrote numbers with letters. You can still see them on clocks, on plaques on monuments, or on the chapters of a book.
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.
Numbers with special patterns
Some natural numbers hide a geometric pattern. You already know the perfect squares ($1, 4, 9, 16, 25, \ldots$): with that many dots (or boxes, or stickers) you can build a complete square, and the number of dots on each side is its square root. So $36$ dots form a $6 \times 6$ square because $6^2 = 36$.
Perfect cubes
If you stack layers of a square until the height equals the side, you build a cube. A perfect cube is a number of the form $n^3 = n \times n \times n$. For example, a $4 \times 4 \times 4$ cube of boxes has $4^3 = 4 \times 4 \times 4 = 64$ boxes. The first ones are $1, 8, 27, 64, 125, \ldots$
Hexagonal numbers
Another family is the hexagonal numbers, which the Pythagoreans studied: the dots are arranged to form hexagons that get bigger and bigger.
Palindromic numbers read the same forward and backward, like $4{,}774$. To find one that meets certain conditions, use symmetry: in a four-digit one, $abba$, you only need to find $a$ and $b$.
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?
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:
-
Draw or describe what the fourth hexagonal number pattern ($H_4$) would look like.
-
How many dots will pattern $H_5$ have?
-
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:
-
It is a palindromic number (it reads the same forwards and backwards).
-
The sum of its digits is $24$.
-
The thousands digit is three times the hundreds digit.
Reasonably determine which number it is and represent it using its expanded form.
Summary
Back to the beginning: there is no largest natural number (you can always add $1$) and between $7$ and $8$ there is no natural number.