Add Course Content For Mathematics Foundation Course of IITM

Author: simplearyan

.gitmodules

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+[submodule "themes/hextra"]
+	path = themes/hextra
+	url = https://github.com/imfing/hextra.git

content/iit-madras/Mathematics-1/Sets-and-Relations.md

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+---
+title: Sets and Relations
+date: 2025-08-09
+weight: 100
+description: "IIT Madras has launched the BS in `Data Science and Applications`. In this program, the course contents are delivered online and can be studied by anyone from anywhere, while the monthly quizzes and final semester exams will have to be attended in-person at designated centres."
+type: docs
+width: normal
+---

content/iit-madras/Mathematics-1/Week-1/Natural-Numbers-and-their-operations.md

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+---
+title: Natural Numbers and their operations
+date: 2025-08-09
+weight: 1
+image: https://sciencenotes.org/wp-content/uploads/2023/05/Natural-Numbers-Definition-and-Examples-1024x683.png
+emoji: 📃
+slug: "Natural Numbers and their operations"
+linkTitle: Natural Numbers and their operations🔥
+series_order: 1
+---
+
+Of course! Let's dive deep into the world of Natural Numbers and their operations. We'll start from the very basics and build up to the properties that govern them.
+
+{{< youtube WEC6jPWvoj8 >}}
+
+
+## Identifying Natural Numbers and Integers
+
+**Natural numbers** are positive whole numbers, used for counting: 1, 2, 3, 4, 5, ....[^1][^2]
+**Integers** include all positive and negative whole numbers and zero: ..., -3, -2, -1, 0, 1, 2, 3, ....[^3]
+
+![A diagram showing the relationship between Natural Numbers, Whole Numbers, and Integers](https://cdn1.byjus.com/wp-content/uploads/2020/10/Natural-Numbers-2.png)
+
+- Example images: Diagrams often show natural numbers on a number line starting at 1, with integers extending in both directions to negative numbers.[^4][^1]
+[^4]
+- Natural numbers set: $\mathbb{N} = \{1, 2, 3, 4, 5, \dots\}$.[^2][^1]
+- Integers set: $\mathbb{Z} = \{\dots, -2, -1, 0, 1, 2, \dots\}$.[^5][^3]
+
+
+## Distinguishing Between Notations: ($\mathbb{N}$), ($\mathbb{Z}$), ($\mathbb{Z}_n$)
+
+- **N** ($\mathbb{N}$): Natural numbers set ($1, 2, 3, ...$).[^6][^1]
+- **Z** ($\mathbb{Z}$): Integers set ($..., -2, -1, 0, 1, 2, ...$).[^6]
+- **ZN** ($\mathbb{Z}_n$): Modular integers—the set of possible remainders when dividing by $n$ ($0,1,\ldots, n-1$).
+
+![A graphic explaining what natural numbers are, with examples](https://sciencenotes.org/wp-content/uploads/2023/05/Natural-Numbers-Definition-and-Examples-1024x683.png)
+
+Example: Venn diagram notation can show relationships; N could fit inside Z on a diagram.[^7][^6]
+[^4]
+
+## Multiplication and Division as Addition/Subtraction
+
+- **Multiplication** is repeated addition:
+
+$$
+3 \times 4 = 3 + 3 + 3 + 3 = 12
+$$
+
+Imagine four groups, each with three objects—adding up all objects gives the same as multiplication.[^8]
+- **Division** is repeated subtraction:
+
+$$
+15 \div 5: 15 - 5 = 10, 10 - 5 = 5, 5 - 5 = 0
+$$
+
+The number of times 5 is subtracted until reaching zero is the quotient (here, 3).[^9][^8]
+
+Visual: Diagrams often show sets of objects grouped or repeatedly removed, or jumps on a number line illustrating subtraction.
+[^4]
+
+## Quotient, Remainder, and Modulus
+
+- **Quotient**: The result of division (number of times divisor "fits" in dividend).
+- **Remainder**: What’s left after dividing as far as possible.
+- **Modulus (mod)**: The remainder after division.
+
+Example: $14 \div 5 = 2$ remainder $4$, so $14$ mod $5 = 4$.
+
+Visual: Number lines or division diagrams often show repeated subtraction steps, grouping leftover items as remainder.[^9]
+
+## Factors and Prime Factorization
+
+- **Factors**: Numbers that divide another exactly. Example: 12’s factors are 1, 2, 3, 4, 6, 12.
+- **Prime Factorization**: Breaking a number into prime numbers multiplied together.
+Example: 12 = 2 × 2 × 3 = $2^2 \times 3$.
+
+Visual: Factor trees are diagrams showing branches, each end-point is a prime number.[^1]
+[^4]
+
+***
+
+### Table: Sets and Concepts
+
+| Concept | Notation | Example | Image Reference |
+| :-- | :-- | :-- | :-- |
+| Natural numbers | $\mathbb{N}$ | 1,2,3,... | [^4] |
+| Integers | $\mathbb{Z}$ | ..., -1,0,1 | [^5] |
+| Modular integers | $\mathbb{Z}_n$ | 0,1,...,n-1 | [^4] |
+| Factors of 12 | n/a | 1,2,3,4,6,12 | [^4] |
+| Prime factorization | n/a | 12 = $2^2\times3$ | [^4] |
+
+---
+
+## Exercise Questions 🤯
+
+
+{{< border >}}
+
+### 1) How many natural numbers are there in the given list?
+
+**List:** 22, -17, 47, -2000, 0, 1, 43, 1729, 6174, -63, 100, 32, -9.
+
+**Solution:**
+Natural numbers ($\mathbb{N}$) are positive whole numbers: $\{1, 2, 3, ...\}$, and do not include 0 or negatives.[^1][^2]
+Filtering the list:
+
+- **22** — Yes
+- -17 — No
+- **47** — Yes
+- -2000 — No
+- 0 — No
+- **1** — Yes
+- **43** — Yes
+- **1729** — Yes
+- **6174** — Yes
+- -63 — No
+- **100** — Yes
+- **32** — Yes
+- -9 — No
+
+There are **8 natural numbers**: 22, 47, 1, 43, 1729, 6174, 100, 32.[^2]
+[^3]
+
+{{< /border >}}
+
+{{< border >}}
+
+### 2) If we distribute 3 pens to each student and 4 pens remain, what is the value of M?
+
+**Solution:**
+Given: Total pens = 70, pens per student = 3, pens left over = 4, number of students = $M$.
+
+Use Division Algorithm:
+
+$$
+\text{Total} = (\text{Pens per student} \times M) + \text{Remainder}
+$$
+
+So:
+
+$$
+70 = (3 \times M) + 4 \implies 66 = 3M \implies M = \frac{66}{3} = 22
+$$
+
+Thus, **M = 22**.[^4]
+
+{{< /border >}}
+
+{{< border >}}
+
+### 3) Which of the following will be the sixteenth prime number?
+
+**Solution:**
+List out the first sixteen primes:[^5]
+2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, **53**
+
+Therefore, the **sixteenth prime number is 53**.
+
+{{< /border >}}
+
+{{< border >}}
+
+### 4) If z mod 31 and z mod 74. Which of the following is (are) possible values for z?
+
+**Solution:**
+Possible remainders from modulus operations must be between 0 and one less than the divisor:[^4]
+
+- For $z \mod 31$: remainder must be in \$\$
+- For $z \mod 74$: remainder must be in \$\$
+
+Test options:
+
+- $z = 25$: Valid for both
+- $z = 34$: Invalid for mod 31
+- $z = 58$: Invalid for mod 31
+- $z = 67$: Invalid for mod 31
+
+Thus, **25** is the only possible value.
+
+{{< /border >}}
+
+{{< border >}}
+
+### 5) If Sheetal's birthday is on the Pth of October and Karthik's is on the Qth of November... find the value of P + 20.
+
+**Solution:**
+P and Q are perfect squares, October/November birthdays, and Sheetal is 10 days before Karthik:
+
+Perfect squares up to 31: 1, 4, 9, 16, 25
+Let $P = 25$ (25th October).
+
+If Karthik is born 10 days later:
+Days left in October after 25th = 31-25 = 6; Karthik's birthday is 4th November, which is also a perfect square.
+
+Thus:
+P = 25, Q = 4
+$P + 20 = 25 + 20 = 45$
+
+{{< /border >}}
+
+<div style="text-align: center">⁂</div>
+
+[^1]: https://byjus.com/maths/natural-numbers/
+
+[^2]: https://www.cuemath.com/numbers/natural-numbers/
+
+[^3]: https://www.pinterest.com/pin/633387438370444/
+
+[^4]: https://www.vedantu.com/maths/division-is-a-process-of-repeated-subtraction
+
+[^5]: https://study.com/academy/lesson/classification-of-numbers.html
+