Chapter 1. Getting Started: The Real Number System FREE
When you withdraw R250 from an ATM in Polokwane, count 12 learners in a classroom, owe a friend R15, or measure the length of a soccer field at the FNB Stadium, you are using different kinds of numbers without thinking about it. South African shopkeepers price a loaf of bread at R16,50, engineers building the Gautrain calculate distances using values such as \(\sqrt{2}\), and a banker quotes inflation as 5,3 percent. Every one of these belongs to the real number system, the foundation on which all of Grade 10 Mathematics is built. This chapter sorts numbers into clear families, shows how they fit together, and gives you the language to describe them precisely.
1.1 The families of numbers
Numbers are grouped into sets that fit inside one another like nested boxes. Each set has a standard symbol.
- Natural numbers \(\mathbb{N} = \{1, 2, 3, 4, \dots\}\) are the counting numbers you use to count taxis or rands.
- Whole numbers \(\mathbb{N}_0 = \{0, 1, 2, 3, \dots\}\) add zero to the naturals.
- Integers \(\mathbb{Z} = \{\dots, -2, -1, 0, 1, 2, \dots\}\) include the negatives, useful for a R30 overdraft or 3 degrees below zero on a Sutherland winter morning.
- Rational numbers \(\mathbb{Q}\) are any numbers that can be written as a fraction \(\frac{a}{b}\) where \(a\) and \(b\) are integers and \(b \neq 0\). Examples are \(\frac{3}{4}\), \(-2\), \(0,75\) and \(0,\overline{3}\).
- Irrational numbers \(\mathbb{Q}'\) cannot be written as such a fraction. Their decimals never end and never repeat. Examples are \(\sqrt{2}\), \(\pi\) and \(\sqrt{7}\).
Together the rationals and irrationals make up the real numbers \(\mathbb{R}\). Notice the nesting: \(\mathbb{N} \subset \mathbb{N}_0 \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R}\), while the irrationals sit beside the rationals inside the reals.
1.2 Rational or irrational?
A number is rational if its decimal form either terminates (stops) or recurs (repeats a pattern forever). A number is irrational if its decimal goes on forever with no repeating pattern. A surd such as \(\sqrt{9} = 3\) is rational because the root is exact, but \(\sqrt{10}\) is irrational because no fraction equals it.
\(\sqrt{16} = 4\), which is a natural number, an integer and a rational. \(\frac{5}{8} = 0,625\) terminates, so it is rational. \(\pi = 3,14159\dots\) never terminates or repeats, so it is irrational. Final answer: rational, rational, irrational.
1.3 Converting recurring decimals to fractions
Because every recurring decimal is rational, it can be written as a fraction. The trick is to multiply by a power of 10 so the repeating part lines up, then subtract.
Let \(x = 0,454545\dots\) Since two digits repeat, multiply by 100: \(100x = 45,4545\dots\) Subtract: \(100x - x = 45\), so \(99x = 45\) and \(x = \frac{45}{99} = \frac{5}{11}\). Final answer: \(0,\overline{45} = \frac{5}{11}\).
1.4 Rounding and estimation
In real life we often round numbers. To round to a given decimal place, look at the next digit: if it is 5 or more, round up; otherwise round down. When a fuel price is R23,4567 per litre, rounding to two decimals gives R23,46. Always estimate first so you can check whether an answer is sensible. If \(\sqrt{50}\) is needed, note that \(7^2 = 49\), so \(\sqrt{50} \approx 7,1\), which a calculator confirms.
1.5 Simplifying surds
A surd is an irrational root such as \(\sqrt{12}\). To simplify, split off the largest perfect-square factor and take its root out: \(\sqrt{12} = \sqrt{4 \times 3} = \sqrt{4}\,\sqrt{3} = 2\sqrt{3}\). This uses the law \(\sqrt{a}\,\sqrt{b} = \sqrt{ab}\). Leaving the answer in surd form keeps it exact, which matters when a builder needs an accurate measurement rather than a rounded decimal.
\(72 = 36 \times 2\), so \(\sqrt{72} = \sqrt{36}\,\sqrt{2} = 6\sqrt{2}\). Final answer: \(6\sqrt{2}\).
1.6 The number line and interval notation
Every real number has exactly one place on the number line. We describe stretches of the line using interval notation. A round bracket excludes an endpoint and a square bracket includes it. For example, \([2, 5)\) means all real numbers from 2 (included) up to 5 (excluded). The same set in set-builder notation is \(\{x \in \mathbb{R} : 2 \le x < 5\}\). On the line we draw a closed dot at 2 and an open dot at 5. This notation describes ages that qualify for a discount or temperatures in a safe range, and you will use it throughout the year when solving inequalities.
Mastering these number families, conversions and notations gives you a precise vocabulary. Every later chapter, from exponents to trigonometry, assumes you can name a number correctly and place it on the line.