1.3. Real Numbers and their Decimal Expansions

In this section, we are going to study rational and irrational numbers from a different point of view. We will look at the decimal expansions of real numbers and see if we can use the expansions to distinguish between rationals and irrationals.

This part of the section introduces the concept of decimal expansions, which are ways of expressing numbers using decimal notation. Rational numbers can often be expressed as decimals, and this section sets the stage for understanding how these decimals can reveal whether a number is rational or irrational.

Think of decimal expansions like labels on jars in a kitchen. Rational numbers are like labeled jars (e.g., 1.5 or 0.33) that you can easily find, while irrational numbers are like jars without labels (e.g., 3.14159) where you have to guess what's inside based on a messy hint.

Let us take three examples: 3/10, 7/8, 1/7. Pay special attention to the remainders and see if you can find any pattern.

These examples illustrate how rational numbers can be expressed in decimal form. The process of long division shows that when you divide 3 by 10, 7 by 8, and 1 by 7, the results can be either terminating (like 0.875) or non-terminating with a repeating pattern (like 0.142857...). This highlights a key characteristic of rational numbers: their decimal forms either end after a finite number of digits or repeat indefinitely.

Imagine you are sharing pizza slices. If you have a pizza (represented by 3 out of 10 slices) that perfectly divides to 0.3 or 0.875, that's like a terminating decimal. But if you repeatedly cut a limited number of slices without finishing (like 1/7 yielding a repeating decimal), then you're stuck having the same types of slices over and over again.

The decimal expansion of rational numbers has only two choices: either they are terminating or non-terminating recurring.

Terminating decimals are those that come to an end (like 0.875 or 2.5), while non-terminating recurring decimals repeat a specific sequence of digits (like 0.333... which can be noted as 0.3 with a bar over it). Understanding this distinction is crucial because it helps us identify rational numbers just by looking at their decimal representation.

Think about using a stopwatch. If your race ends exactly at '2.5 seconds' (like a terminating decimal), it's clear you finished there. But if it keeps flashing '0.333...' without stopping, it's like running in circles, going on forever without truly finishing.

Thus, we see that the decimal expansion of rational numbers can be expressed in the form p/q, where p and q are integers and q ≠ 0. If we observe a decimal that is terminating or repeating, we can conclude that it is a rational number.

This part emphasizes the condition that determines a number's rationality based on its decimal form. If the decimal ends or repeats, we can claim the number is rational and can be expressed as a fraction of two integers. Conversely, this principle helps in identifying which numbers are irrational.

Consider a scenario where you’re collecting coins. Each time you can either take whole amounts (like $0.50,$0.75) or sometimes just loose changes that seem to go on forever (like $1.333... representing the constant addition of dimes in a jar). The former are your easy-to-count rational amounts, while the latter are endless and complex – like irrational amounts.

The decimal expansion of an irrational number is non-terminating non-recurring.

This describes that irrational numbers (like π or $\sqrt{2}$) cannot be expressed as a simple fraction, and their decimal representations neither terminate nor repeat. Their digits go on forever without falling into a predictable pattern. This differentiates them from rational numbers and further enriches our understanding of the real numbers as a whole.

Imagine drawing a spiral staircase. No matter how many turns you make, you notice that each step varies slightly and doesn't settle into a specific pattern or end. That's similar to how irrational numbers like $\sqrt{2}$ behave in decimal form—they go on forever without a fixed endpoint.

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