In simple English, what does it mean to be transcendental?

Solution 1:

We will play a game. Suppose you have some number $x$. You start with $x$ and then you can add, subtract, multiply, or divide by any integer, except zero. You can also multiply by $x$. You can do these things as many times as you want. If the total becomes zero, you win.

For example, suppose $x$ is $\frac23$. Multiply by $3$, then subtract $2$. The result is zero. You win!

Suppose $x$ is $\sqrt[3] 7$. Multiply by $x$, then by $x$ again, then subtract $7$. You win!

Suppose $x$ is $\sqrt2 +\sqrt3$. Here it's not easy to see how to win. But it turns out that if you multiply by $x$, subtract 10, multiply by $x$ twice, and add $1$, then you win. (This is not supposed to be obvious; you can try it with your calculator.)

But if you start with $x=\pi$, you cannot win. There is no way to get from $\pi$ to $0$ if you add, subtract, multiply, or divide by integers, or multiply by $\pi$, no matter how many steps you take. (This is also not supposed to be obvious. It is a very tricky thing!)

Numbers like $\sqrt 2+ \sqrt 3$ from which you can win are called algebraic. Numbers like $\pi$ with which you can't win are called transcendental.

Why is this interesting? Each algebraic number is related arithmetically to the integers, and the winning moves in the game show you how so. The path to zero might be long and complicated, but each step is simple and there is a path. But transcendental numbers are fundamentally different: they are not arithmetically related to the integers via simple steps.

Solution 2:

$\sqrt2$ satisfies the equation: $$x^2-2=0$$ Similarly, $\sqrt[\Large3]3$ satisfies the equation: $$x^3-3=0$$ Numbers like this, that satisfy polynomial equations, are called algebraic numbers. (Specifically, the coefficients of these polynomials need to be integers.)

Another algebraic number is $\frac12$, since it satisfies: $$2x-1=0$$ In fact, all rational numbers are algebraic. But, as the first two examples show, not every algebraic number is rational.

Now, it's not obvious, but if you add up or multiply together two algebraic numbers, you get another algebraic number. For example, $\sqrt2+\sqrt[\Large3]3$ satisfies the equation: $$x^6-6x^4-6x^3+12x^2-36x+1=0$$

(In case you're wondering, complex numbers can also be algebraic. In fact, it's not hard to show that a complex number is algebraic if and only if its real and imaginary parts are algebraic.)

A real (or complex) number that's not algebraic is called transcendental. In 1873, the number $e\approx2.71828$ was proven transcendental. In 1882, $\pi\approx3.14159$ was, too. It is unknown if $e+\pi$ is transcendental. In fact, we're not even sure if it's irrational! Same goes for similar numbers such as $\pi^\pi$ and $e\pi$. ($e^\pi$, however, is transcendental.)

Solution 3:

Among the real numbers, some are integer.

Others are rational, i.e. they are solutions of a linear equation such as

$$px=q$$ where $p,q$ are integer. The numbers not falling in this scheme are called irrational.

A rather obvious generalization of this principle are numbers that are solutions of a polynomial equation such as

$$px^3+qx^2+rx+s=0$$ where $p,q,r,s$ are integer (any other degree can do). These numbers are called algebraic, which is the converse of transcendental.

The algebraic numbers enjoy a special property: even though there is an infinity of them, they can be numbered (they are said to be countable). By contrast, the transcendental numbers cannot, there is a "larger" infinity of them.

You easily understand that all integers are rational and all rationals are algebraic.

Among the functions of the real variable, some are polynomials.

A rational fraction is the quotient of two polynomials, i.e. a function $y=\dfrac{Q(x)}{P(x)}$, that verifies an equation like

$$P(x)y=Q(x).$$

More generally, an algebraic function $y=f(x)$ is such that it can be expressed as the root of a polynomial with coefficients that are themselves polynomials in $x$:

$$P(x)y^3+Q(x)y^2+R(x)y+S(x)=0.$$

A function that is not algebraic is called transcendental.

Looking closer, one can observe that algebraic items are defined from equations that use a finite number of additions and multiplications. Transcendental items require "stronger" tools (such as an infinite number of terms).

Solution 4:

The only thing cryptic I see in the quoted definition of "transcendental number" is that you haven't first defined what an algebraic number is. An algebraic number is a number that is a root of a polynomial with rational coefficients. That is equivalent to saying it's a root of a polynomial with integer coefficients. Thus the roots of $$ \frac 5 8 x^3 - \frac{21}2 x^2 + \frac{17}{12} x + 19 = 0 $$ are algebraic numbers. The common denominator of these coefficients is $24$, and multiplying both sides by that we get $$ 15x^3 - 252 x^2 + 34 x + 456 = 0 $$ and that equation has the same roots but has integer coefficients.

Rational numbers are algebraic numbers. For example $\dfrac{17}{12}$ is a root of $$ x - \frac {17}{12} = 0 $$ or of $$ 12x - 17 = 0. $$

The function $x\mapsto \sqrt[3] x = f(x)$ is an algebraic function by the given definition, since it satisfies the polynomial equation $$ f(x)^3 - x = u^3 - x = 0 $$ in the variable $u=f(x)$. In other words, it is not only polynomial functions that satisfy polynomial equations.