1/9/2023 0 Comments Real numbersAnd when we get into areas such as particle physics, the ability to measure unimaginably small differences in things like mass is absolutely crucial. When talking about global temperature variations over a number of years or decades, for example, even small fractions of a degree may be significant. In many areas of science and technology, however, we often need to be far more precise. And, let's face it, these predictions are often wrong anyway. For example, it is sufficient for the purposes of a public weather forecast to give the maximum and minimum temperatures expected on a particular day to the nearest degree. Of course, we often talk about these things in everyday life using integer values. ![]() Some examples that spring to mind include temperature, voltage, the pressure of a gas, or the density of a liquid. Real numbers are very useful when we want to measure things in the physical world that vary continuously (as opposed to discretely). ![]() We can therefore also say that the cardinality of the set of real numbers is greater than the cardinality of the set of natural numbers, despite both sets having an infinite number of members. They are said to be uncountably infinite. For now, it is enough to understand that we cannot count the real numbers in the same way that we can count the natural numbers or rational numbers. Clearly, not all infinities are equal! All we can say at this point is that there is a paradox here, but it is not one that we need to worry too much about unless, and until, we undertake the study of higher mathematics. At the same time, however, there must also be infinitely many real numbers between any two points on the number line, no matter how close together those points may be. It's fairly easy to see that there must be infinitely many real numbers on the number line. In the meantime, there are some other rather interesting things to consider. What is the first positive real number after zero? You could spend an awful lot of time thinking about that one! But we'll come back to the question of why rational numbers are countably infinite, whereas real numbers are not, later. Imagine, for example, that we want to start counting positive real numbers, starting with zero. Once you really start to think about this question, you begin to realise intuitively why we can't count the real numbers. So why can't we count the reals? Well, the key word here is continuum. But they can nevertheless be counted, and we'll see why in due course. how on Earth do you count rational numbers? I mean, they're fractions. Whole numbers, integers and rational numbers are all countably infinite! In fact, it's not only the natural numbers. For that reason, mathematicians tend to describe the set of natural numbers as being countably infinite. ![]() Nevertheless, we could continue to count. ![]() It is of course true that we would never actually finish counting, no matter how long we kept going. When we talk about the set of natural numbers, for example, we know that we can start at one (or zero, depending on how you choose to define the natural numbers) and count to. The word set implies that we are dealing with something that can be counted. Although we sometimes talk about the set of real numbers, mathematicians more often refer to it in terms of a field or continuum.
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