Absolute Value and the Real Line

The Absolute Value of a Real Number

Definition: If $a$ is a real number, then we define the absolute value of the number $a$ denoted $\mid a\mid$ or $\mathrm {abs} (a)$ as: $\mid a\mid =\left\{{\begin{matrix}a&\mathrm {if\,a>0,} \\0&\mathrm {if\,a=0,} \\-a&\mathrm {if\,a<0.} \end{matrix}}\right.$.

For example, suppose we want to find the absolute value of $5$. Well since $5>0$, we note that $\mid 5\mid =5$. If we wanted to find the absolute value of $-5$ then since $-5<0$ we note that $\mid -5\mid =-(-5)=5$. We will now look at some important properties of the absolute values of real numbers utilizing The Order Properties of Real Numbers.

• Theorem 1: If $a$ is a real number then $\mid a\mid =\mid -a\mid$.

• Proof: We will split this proof up into three cases.

• Case 1: Suppose that $a>0$. Then $-a<0$. Therefore by the definition of the absolute value of a number, $\mid a\mid =a$, and $\mid -a\mid =-(-a)=a$, and so $\mid a\mid =\mid -a\mid$.

• Case 2: Now suppose that $a=0$. Therefore $-a=0$ and clearly $\mid a\mid =0$ and $\mid -a\mid =0$, and so $\mid a\mid =\mid -a\mid$.

• Case 3: Lastly suppose that $a<0$. Then $-a>0$. We obtain that $\mid a\mid =-a$ and $\mid -a\mid =-a$, and so $\mid a\mid =\mid -a\mid$.

• In all three cases we get that $\mid a\mid =\mid a\mid$. $\blacksquare$

• Theorem 2: If $a$ and $b$ are real numbers then $\mid ab\mid =\mid a\mid \mid b\mid$.

• Proof: We will split this proof up into three cases.

• Case 1: Suppose that $a=0$ or $b=0$ or both $a,b=0$. Then $a\cdot b=0$, and so $\mid ab\mid =0$. Similarly $\mid a\mid \mid b\mid$ is either $0\cdot \mid b\mid$ or $\mid a\mid \cdot 0$ or $0\cdot 0$, all of which equal $0$, so $\mid ab\mid =\mid a\mid \mid b\mid$.

• Case 2: Suppose that $a,b>0$. Then $ab>0$ and so $\mid ab\mid =ab$ and $\mid a\mid \mid b\mid =ab$, so $\mid ab\mid =\mid a\mid \mid b\mid$.

• Case 3: Suppose that one of $a>0$ and $b<0$. Then $ab<0$. So $\mid ab\mid =-ab$, and $\mid a\mid \mid b\mid =a\cdot -b=-ab$, so $\mid ab\mid =\mid a\mid \mid b\mid$.

• Case 4: Suppose that $a,b<0$. Then $ab>0$ and so $\mid ab\mid =ab$ and $\mid a\mid \mid b\mid =-a\cdot -b=(-1)(-1)ab=ab$. So $\mid ab\mid =\mid a\mid \mid b\mid$.

• In all four cases we get that $\mid ab\mid =\mid a\mid \mid b\mid$. $\blacksquare$

• Theorem 3: If $a$ is a real number then $\mid a\mid ^{2}=a^{2}$.

• Proof: We know that $a^{2}>0$ and there by applying Theorem 2 we get that $a^{2}=\mid a^{2}\mid =\mid a\cdot a\mid =\mid a\mid \mid a\mid =\mid a\mid ^{2}$. $\blacksquare$

• Theorem 4: If $c\geq 0$ then $\mid a\mid \leq c$ if and only if $-c\leq a\leq c$.

• Proof: $\Rightarrow$ If $\mid a\mid \leq c$ then we have that both $a\leq c$ and $-a\leq c$ or rather $a\geq -c$ which is equivalent to saying that $-c\leq a\leq c$.

• $\Leftarrow$ Suppose that $-c\leq a\leq c$. Then $a\leq c$ and $-c\leq a\Leftrightarrow c\geq -a$ so then $\mid a\mid \leq c$. $\blacksquare$

• Theorem 5: If $a$ is a real number then $-\mid a\mid \leq a\leq \mid a\mid$.

The Real Line

The real line

In mathematics, the real line, or real number line is the line whose points are the real numbers. That is, the real line is the set R of all real numbers, viewed as a geometric space, namely the Euclidean space of dimension one. It can be thought of as a vector space (or affine space), a metric space, a topological space, a measure space, or a linear continuum.

Just like the set of real numbers, the real line is usually denoted by the symbol R (or alternatively, $\mathbb {R}$, the letter “R” in blackboard bold). However, it is sometimes denoted R¹ in order to emphasize its role as the first Euclidean space.

This article focuses on the aspects of R as a geometric space in topology, geometry, and real analysis. The real numbers also play an important role in algebra as a field, but in this context R is rarely referred to as a line. For more information on R in all of its guises, see real number.

As a linear continuum

The order on the number line

Each set on the real number line has a supremum.

The real line is a linear continuum under the standard < ordering. Specifically, the real line is linearly ordered by <, and this ordering is dense and has the least-upper-bound property.

In addition to the above properties, the real line has no maximum or minimum element. It also has a countable dense subset, namely the set of rational numbers. It is a theorem that any linear continuum with a countable dense subset and no maximum or minimum element is order-isomorphic to the real line.

The real line also satisfies the countable chain condition: every collection of mutually disjoint, nonempty open intervals in R is countable. In order theory, the famous Suslin problem asks whether every linear continuum satisfying the countable chain condition that has no maximum or minimum element is necessarily order-isomorphic to R. This statement has been shown to be independent of the standard axiomatic system of set theory known as ZFC.

As a metric space

The metric on the real line is absolute difference.

An ε-ball around a number a

The real line forms a metric space, with the distance function given by absolute difference:

$d(x,y)=|x-y|.$

The metric tensor is clearly the 1-dimensional Euclidean metric. Since the n-dimensional Euclidean metric can be represented in matrix form as the n-by-n identity matrix, the metric on the real line is simply the 1-by-1 identity matrix, i.e. 1.

If p ∈ R and ε > 0, then the ε-ball in R centered at p is simply the open interval (p − ε, p + ε).

This real line has several important properties as a metric space:

• The real line is a complete metric space, in the sense that any Cauchy sequence of points converges.
• The real line is path-connected and is one of the simplest examples of a geodesic metric space.
• The Hausdorff dimension of the real line is equal to one.

As a topological space

The real line can be compactified by adding a point at infinity.

The real line carries a standard topology, which can be introduced in two different, equivalent ways. First, since the real numbers are totally ordered, they carry an order topology. Second, the real numbers inherit a metric topology from the metric defined above. The order topology and metric topology on R are the same. As a topological space, the real line is homeomorphic to the open interval (0, 1).

The real line is trivially a topological manifold of dimension 1. Up to homeomorphism, it is one of only two different connected 1-manifolds without boundary, the other being the circle. It also has a standard differentiable structure on it, making it a differentiable manifold. (Up to diffeomorphism, there is only one differentiable structure that the topological space supports.)

The real line is a locally compact space and a paracompact space, as well as second-countable and normal. It is also path-connected, and is therefore connected as well, though it can be disconnected by removing any one point. The real line is also contractible, and as such all of its homotopy groups and reduced homology groups are zero.

As a locally compact space, the real line can be compactified in several different ways. The one-point compactification of R is a circle (namely, the real projective line), and the extra point can be thought of as an unsigned infinity. Alternatively, the real line has two ends, and the resulting end compactification is the extended real line [−∞, +∞]. There is also the Stone–Čech compactification of the real line, which involves adding an infinite number of additional points.

In some contexts, it is helpful to place other topologies on the set of real numbers, such as the lower limit topology or the Zariski topology. For the real numbers, the latter is the same as the finite complement topology.

As a vector space

The bijection between points on the real line and vectors

The real line is a vector space over the field R of real numbers (that is, over itself) of dimension 1. It has the usual multiplication as an inner product, making it a Euclidean vector space. The norm defined by this inner product is simply the absolute value.

As a measure space

The real line carries a canonical measure, namely the Lebesgue measure. This measure can be defined as the completion of a Borel measure defined on R, where the measure of any interval is the length of the interval.

Lebesgue measure on the real line is one of the simplest examples of a Haar measure on a locally compact group.

In real algebras

The real line is a one-dimensional subspace of a real algebra A where R ⊂ A. For example, in the complex plane z = x + iy, the subspace {z : y = 0} is a real line. Similarly, the algebra of quaternions

q = w + x i + y j + z k

has a real line in the subspace {q : x = y = z = 0 }.

When the real algebra is a direct sum $A=R\oplus V,$ then a conjugation on A is introduced by the mapping $v\mapsto -v$ of subspace V. In this way the real line consists of the fixed points of the conjugation.

Licensing

Content obtained and/or adapted from:

• The Absolute Value of a Real Number, mathonline.wikidot.com under a CC BY-SA license
• Real line, Wikipedia under a CC BY-SA license