Number Sets: Natural to Real

The first four sets nest inside each other like boxes: every natural number is also a whole number, every whole number is also an integer, and every integer is also rational, because any integer can be written over 1 (for example $-5=\frac{-5}{1}$). We write this as $\mathbb{N}\subset \mathbb{W}\subset \mathbb{Z}\subset \mathbb{Q}\subset \mathbb{R}$, where $\subset$ means "is inside." The irrational numbers are different: they sit beside the rational numbers, not inside them. A real number is either rational or irrational, never both. In this course the natural numbers start at 1; some mathematicians include 0, so check how a question defines them.

Learn the Number Sets to better understand math | jensenmath.ca

Every rational number has a decimal that either terminates (stops) or repeats a block of digits forever. For example, $\frac{3}{8}=0.375$ terminates, and $\frac{4}{11}=0.363636\dots =0.\overline{36}$ repeats (the bar marks the repeating block). An irrational number has a decimal that does neither: $\sqrt{2}=1.41421356\dots$ and $\pi =3.14159265\dots$ go on forever with no repeating block. Be careful with two traps. First, a calculator screen shows only about 10 digits, so it cannot tell you whether a decimal repeats; $\sqrt{2}$ on a screen looks like it stops, but it does not. Second, $3.14$ and $\frac{22}{7}$ are rational approximations of $\pi$, not $\pi$ itself.

Converting repeating decimals to fractions 1 | Linear equations | Algebra I | Khan Academy

There is no integer between 3 and 4, and no integer between $-1$ and 0. The integers have gaps. The rational numbers do not: between any two different rational numbers you can always find another one. The easiest way is to take their average (add them and divide by 2), which lands exactly halfway. Between $\frac{1}{2}$ and $\frac{3}{4}$ sits $\frac{5}{8}$. Then between $\frac{1}{2}$ and $\frac{5}{8}$ sits $\frac{9}{16}$, and you can keep going forever. This property is called density, and it means there are infinitely many rational numbers between any two different rational numbers, even very close ones like 0.1 and 0.11.

Infinity ($\infty$) is not a number you can reach; it describes a process that never ends, like counting 1, 2, 3, … with no last number. Patterns that go on forever can still settle down. Picture walking toward a wall 1 m away: first you walk half the distance, then half of what is left, then half again. Your total distance is $\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\cdots$. After each step you are closer to 1 m, and you never pass it. We say the running total approaches the limit 1. No running total with a finite number of terms ever equals 1, but the value of the endless sum is defined to be its limit, so $\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\cdots =1$ exactly. The same idea explains why $0.999\dots$ (nines forever) equals exactly 1: it is the limit of $0.9,0.99,0.999,\dots$, whose gaps to 1 are $0.1,0.01,0.001,\dots$. A quick check: since $\frac{1}{3}=0.333\dots$, multiplying both sides by 3 gives $1=0.999\dots$.