Absolute Value and the Real LineMATH 464/506, Real Analysis

J. Robert Buchanan

Department of Mathematics

Summer 2007

J. Robert Buchanan Absolute Value and the Real Line

Absolute Value

Definition

The absolute value of a real number a, denoted by |a|, isdefined by

|a| =

a if a > 0,0 if a = 0,−a if a < 0.

Theorem1 |ab| = |a||b| for all a, b ∈ R.2 |a|2 = a2 for all a ∈ R.3 If c ≥ 0, then |a| ≤ c if and only if −c ≤ a ≤ c.4 −|a| ≤ a ≤ |a| for all a ∈ R.

Proof.

J. Robert Buchanan Absolute Value and the Real Line

Absolute Value

Definition

The absolute value of a real number a, denoted by |a|, isdefined by

|a| =

a if a > 0,0 if a = 0,−a if a < 0.

Theorem1 |ab| = |a||b| for all a, b ∈ R.2 |a|2 = a2 for all a ∈ R.3 If c ≥ 0, then |a| ≤ c if and only if −c ≤ a ≤ c.4 −|a| ≤ a ≤ |a| for all a ∈ R.

Proof.

J. Robert Buchanan Absolute Value and the Real Line

Absolute Value

Definition

The absolute value of a real number a, denoted by |a|, isdefined by

|a| =

a if a > 0,0 if a = 0,−a if a < 0.

Theorem1 |ab| = |a||b| for all a, b ∈ R.2 |a|2 = a2 for all a ∈ R.3 If c ≥ 0, then |a| ≤ c if and only if −c ≤ a ≤ c.4 −|a| ≤ a ≤ |a| for all a ∈ R.

Proof.

J. Robert Buchanan Absolute Value and the Real Line

Triangle Inequality

Theorem

If a, b ∈ R, then |a + b| ≤ |a| + |b|.

Proof.

Corollary

If a, b ∈ R, then1 ||a| − |b|| ≤ |a − b|,2 |a − b| ≤ |a| + |b|.

Corollary

If a1, a2, . . . , an are any real numbers, then

|a1 + a2 + · · · + an| ≤ |a1| + |a2| + · · · + |an|.

J. Robert Buchanan Absolute Value and the Real Line

Triangle Inequality

Theorem

If a, b ∈ R, then |a + b| ≤ |a| + |b|.

Proof.

Corollary

If a, b ∈ R, then1 ||a| − |b|| ≤ |a − b|,2 |a − b| ≤ |a| + |b|.

Corollary

If a1, a2, . . . , an are any real numbers, then

|a1 + a2 + · · · + an| ≤ |a1| + |a2| + · · · + |an|.

J. Robert Buchanan Absolute Value and the Real Line

Triangle Inequality

Theorem

If a, b ∈ R, then |a + b| ≤ |a| + |b|.

Proof.

Corollary

If a, b ∈ R, then1 ||a| − |b|| ≤ |a − b|,2 |a − b| ≤ |a| + |b|.

Corollary

If a1, a2, . . . , an are any real numbers, then

|a1 + a2 + · · · + an| ≤ |a1| + |a2| + · · · + |an|.

J. Robert Buchanan Absolute Value and the Real Line

Triangle Inequality

Theorem

If a, b ∈ R, then |a + b| ≤ |a| + |b|.

Proof.

Corollary

If a, b ∈ R, then1 ||a| − |b|| ≤ |a − b|,2 |a − b| ≤ |a| + |b|.

Corollary

If a1, a2, . . . , an are any real numbers, then

|a1 + a2 + · · · + an| ≤ |a1| + |a2| + · · · + |an|.

J. Robert Buchanan Absolute Value and the Real Line

Real Line

Remarks:Geometrically we may regard |a| as the distance along thenumber line from 0 to a.The distance between a and b in R is |a − b|.

Definition

Let a ∈ R and ǫ > 0. The ǫ-neighborhood of a is the set

Vǫ(a) = {x ∈ R : |x − a| < ǫ}.

Remark: x ∈ Vǫ(a) means x satisfies the following equivalentinequalities:

−ǫ < x − a < ǫ

a − ǫ < x < a + ǫ

J. Robert Buchanan Absolute Value and the Real Line

Real Line

Remarks:Geometrically we may regard |a| as the distance along thenumber line from 0 to a.The distance between a and b in R is |a − b|.

Definition

Let a ∈ R and ǫ > 0. The ǫ-neighborhood of a is the set

Vǫ(a) = {x ∈ R : |x − a| < ǫ}.

Remark: x ∈ Vǫ(a) means x satisfies the following equivalentinequalities:

−ǫ < x − a < ǫ

a − ǫ < x < a + ǫ

J. Robert Buchanan Absolute Value and the Real Line

Real Line

Remarks:Geometrically we may regard |a| as the distance along thenumber line from 0 to a.The distance between a and b in R is |a − b|.

Definition

Let a ∈ R and ǫ > 0. The ǫ-neighborhood of a is the set

Vǫ(a) = {x ∈ R : |x − a| < ǫ}.

Remark: x ∈ Vǫ(a) means x satisfies the following equivalentinequalities:

−ǫ < x − a < ǫ

a − ǫ < x < a + ǫ

J. Robert Buchanan Absolute Value and the Real Line

Result

Theorem

Let a ∈ R. If x belongs to the neighborhood Vǫ(a) for everyǫ > 0, then x = a.

Proof.

J. Robert Buchanan Absolute Value and the Real Line

Result

Theorem

Let a ∈ R. If x belongs to the neighborhood Vǫ(a) for everyǫ > 0, then x = a.

Proof.

J. Robert Buchanan Absolute Value and the Real Line

Homework

Read Section 2.2.

Page 34: 1, 2, 14 , 15

Boxed problems should be written up separately and submittedfor grading at class time on Friday.

J. Robert Buchanan Absolute Value and the Real Line