1. REAL NUMBERS

In Class IX, you began your exploration of the world of real numbers and encountered irrational numbers. We continue our discussion on real numbers in this chapter. We begin with two very important properties of positive integers in Sections 1.2 and 1.3, namely the Euclid’s division algorithm and the Fundamental Theorem of Arithmetic.

This chunk introduces the concept of real numbers and builds a link to previous learning about irrational numbers. Students will recall that real numbers include both rational and irrational numbers. The section promises a deeper exploration of two significant mathematical properties: Euclid’s division algorithm (which relates to integer division) and the Fundamental Theorem of Arithmetic (which concerns the uniqueness of prime factorization).

Consider a baker who needs to divide 10 pies among 3 friends. In doing this, they may find that each friend can have 3 whole pies, and 1 pie remains. This is similar to Euclid's division algorithm, where the pie division results in a quotient (3) and a remainder (1).

Euclid’s division algorithm, as the name suggests, has to do with divisibility of integers. Stated simply, it says any positive integer a can be divided by another positive integer b in such a way that it leaves a remainder r that is smaller than b. Many of you probably recognise this as the usual long division process.

The algorithm states that for any two positive integers, 'a' and 'b', we can express 'a' in terms of 'b' as follows: a = b*q + r, where 'q' is the quotient (the result of division without decimals) and 'r' is the remainder (the leftover amount). The condition is that 'r' must be smaller than 'b'. This concept is foundational for understanding operations with integers.

Imagine you have 20 apples and want to divide them equally into bags that hold 7 apples each. You would fill 2 bags (making a total of 14 apples) and have 6 apples left over. Here, 20 is 'a', 7 is 'b', 2 is 'q', and 6 is 'r'.

The Fundamental Theorem of Arithmetic states that every composite number can be expressed as a product of primes in a unique way—this important fact is the Fundamental Theorem of Arithmetic.

This theorem emphasizes that every composite number (numbers greater than one that are not prime) can be uniquely factored into prime numbers. For instance, 12 can be factored as 2 × 2 × 3. While there may be multiple ways to express it by changing the order of factors (e.g., 3 × 2 × 2), the set of prime factors remains the same. This uniqueness is critical in various areas of mathematics, including number theory.

Think of building blocks where each block represents a prime number. You can use different arrangements to build a number like 12, but regardless of how you stack them, the specific blocks used (2, 2, and 3) determine what you've built. Just like unique addresses in a city, each composite number has its unique 'address' of prime factors.

We use the Fundamental Theorem of Arithmetic for two main applications. First, we use it to prove the irrationality of many of the numbers you studied in Class IX, such as 2, 3 and 5. Second, we apply this theorem to explore when exactly the decimal expansion of a rational number is terminating and when it is non-terminating repeating.

This chunk explains how the Fundamental Theorem of Arithmetic is not just theoretical but has practical applications. It aids in proving that certain numbers, like square roots of non-perfect squares (e.g., √2), are irrational. Additionally, it helps determine the nature of decimal expansions of rational numbers, distinguishing between terminating and repeating decimals based on the prime factors of their denominators.

If you think of a recipe, the unique combination of ingredients (the prime factors) can yield a dish (the composite number). Just as each recipe produces a specific dish, every rational number has a particular decimal representation based on its denominator. A dish with only ingredients allowing a full measure (like whole spices) might represent a terminating decimal.

--