Welcome to Week 1 of Mathematics-1 for Data Science!
This course will delve into the fundamental building blocks of mathematics essential for data science. We'll begin with some familiar concepts and gradually progress to more advanced topics.
Numbers for Counting: Natural Numbers
We use numbers primarily for counting objects. For instance, if we have 7 balls and 7 pencils, we need to know that both sets contain the same number of items: 7. This number represents a quantity common to both sets.
We're all familiar with numbers like 1, 2, 3, and 4. When we see multiple objects, we can count them. However, there's one crucial number of Indian origin that deserves special mention: 0 (zero).
Zero is essential because it allows us to represent the absence of something to count. Without zero, our place value system, which we use to manipulate numbers, wouldn't function correctly.
These numbers, starting with 0, are often called natural numbers. There's some variation in how different books define them. Some may only include 1, 2, 3, and so on. We'll use the symbol N with a double line across it (ℕ) to represent the set of natural numbers.
Expanding our Number System: Integers
Sometimes, people might exclude 0 from the set of natural numbers. To emphasize its inclusion, we might write N₀. Regardless, whenever we talk about natural numbers, we always include 0.
But what can we do with natural numbers? We can add, subtract, multiply, and divide them. These are the basic arithmetic operations you learned in school.
However, from a mathematical perspective, an interesting question arises: When we perform operations on natural numbers, do we always get another natural number?
- Addition: If we add two natural numbers, do we always get another natural number? (Yes)
- Subtraction: What happens with subtraction? If we subtract a larger number from a smaller one (e.g., 6 - 5), we go below 0. Subtracting things you don't have doesn't make sense.
Therefore, to perform subtraction meaningfully, we need to expand the scope of our numbers. This is where negative numbers come in: -1, -2, -3, and so on.
We now have both positive numbers (0, 1, 2, and so on) and negative numbers. These are technically non-negative because 0 is neither positive nor negative. With the addition of negative numbers, we have a set called integers, denoted by the symbol Z with a bar across it (ℤ).
Natural numbers start at 0 and go forward (0, 1, 2, 3, ...). Integers, on the other hand, extend infinitely in both directions, from negative infinity to positive infinity.
Visualizing Integers: The Number Line
Thinking of integers as a sequence is helpful. On the left, we have very small numbers, and on the right, we have very large numbers. This sequence is often called the number line. As you move from left to right, the numbers increase. This arrangement helps us understand integers.
Operations on Integers: Multiplication and Division
Let's explore multiplication and division with integers.
- Multiplication: When we say 7 times 4, we're essentially taking 7 objects and making 4 copies of each. If we want to know the total number of objects, we add the copies together: 7 from the first group, 7 from the second group, and so on. This repeated addition is what multiplication represents.
- Division: Division can be visualized as repeated subtraction. You keep subtracting the number you're dividing by until you reach 0. If you have a remainder (e.g., dividing 19 by 5), it means you can't distribute the objects evenly.
Prime Numbers and Factorization
As we delve deeper into numbers, we'll encounter prime numbers. A prime number has exactly two factors: 1 and itself. The smallest prime number is 2. Examples include 2, 3, 5, 7, and so on. There are clever ways to find prime numbers, like the Sieve of Eratosthenes.
One of the essential properties of numbers is that any integer can be uniquely factored into a product of prime numbers. For instance, 12 can be expressed as 2 x 2 x 3, where 2 and 3 are its prime factors.
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