实数

What is a Real Number?

Maybe you have already seen numbers like $1, 2, 3, 4, 100, 15.99, 4.29$ in daily life. In math classes, you might encounter fractions, other kinds of decimals, and radicals. Combined together, we already have a pretty diverse world of numbers: $5, -0.618, 3\sqrt{29}, \frac{4}{7}, -1$...

What do these numbers have in common? Yep, you guessed it—they are all real numbers!

The following illustration shows the further categorization of the real number system.

Understanding Real Numbers

Here are some common ways to understand what a real number is:

• A real number is any number that is not imaginary, meaning the square of it must be a positive real number.
• Every real number has an "order". This means if we take two real number $x$ and $y$, exactly one of these is true: $x<y, x=y,$ or $x>y$ (see "Compare Numbers on A Number Line" below).
• Every rational and irrational number is real. Any real number is either rational or irrational, but not both.

Number Line Representation

The mathematical definition of a real number is any value that can be found on the number line.

On the other hand, a number line is a line in which each point is a real number—a bit like a infinitely long ruler where each different point on it represent a different length.

Number Line: As Accurate As We Want

However, you can't "zoom in" on your ruler infinitely. For example, on your ruler, if the smallest distance between two ticks is $\frac{1}{16}$ inches, you cannot measure an object more accurately than $\frac{1}{16}$ inches, which is about $0.0625$ inches.

On a real line, you can zoom in as much you want to, and also have as many digits after the decimal point as you want to. The number $0.57721566490153286060...$, known as Euler-Mascheroni Constant, is one of the real numbers with infinitely many digits after the decimal point.

Comparing Numbers on A Number Line

When a number $A$ is on the right of another number $B$ on this number line, $A$ is larger than $B$. Take any two real numbers, $x$ and $y$, and treat them as points on the number line. These points follow the rule that states real numbers have an order where $x<y, x=y,$ or $x>y$.

The number line of real numbers is infinitely long, but does not include $+\infty$ and $-\infty$. Real numbers are represented in set theory by the symbol $\mathbb{R}$.

Real Number Properties

The following sets of numbers are all part of the real number system:

• Natural numbers $(\mathbb{N})$: Positive whole numbers.$$(1, 2, 3, ...)$$
• Whole Numbers $(\mathbb{W})$: Natural numbers and zero. $$(0, 1, 2, 3, ...)$$
• Integers $(\mathbb{Z})$: All whole numbers and negative numbers that are not decimals. $$(...,-3, -2, -1, 0, 1, 2, 3, ...)$$
• Rational Numbers $(\mathbb{Q})$: Fractions, integers, decimals that end, and repeating decimals. $$\text{ex: }\frac {1}{2}, 0.45, 0.3333333$$
• Irrational Numbers $(\mathbb{R\backslash Q})$: Decimals that go on forever. $$\text{ex: }\sqrt{3}, \pi, -\sqrt{11}$$

Real Number Properties and Operations

Real numbers have some really cool properties, like the reflexive, symmetric, and transitive properties. Also:

• The sum of two real numbers is always a real number.
• The difference of two real numbers is always a real number.
• The product of two real numbers is always a real number.
• The quotient of two real numbers is always a real number.

Really, any operation you do to two real numbers will give a real number. The only problem is that you can't take the square root of a negative real number.

(Video) What's a Real Number?

Here's a YouTube video by Don't Memorise that talks about the real number system.

$\large{\text{Summary}}$

Any number that can be put on a number line is a real number.

Integers like $-2$, rational numbers/decimals like $0.5$, and irrational numbers like $\sqrt{2}$ or $\pi$ can all be plotted on the number line, so they are real. Better yet, they can be ordered. That means if we plot any two real numbers on the number line, then we know which one is to the left.