inequalities for real numbers

Suppose a is a real number.

  1. 1.

    If a<0 then a is a negative number.

  2. 2.

    If a>0 then a is a positive number.

  3. 3.

    If a0 then a is a non-positive number.

  4. 4.

    If a0 then a is a non-negative number.

The first two inequalitiesMathworldPlanetmath are also called strict inequalities.
The second two inequalities are also called loose inequalities.


Suppose a and b are real numbers.

  1. 1.

    If a>b, then -a<-b. If a<b, then -a>-b.

  2. 2.

    If ab, then -a-b. If ab, then -a-b.

Lemma 1.

0<a  iff  -a<0.


If 0<a, then adding -a on both sides of the inequality gives -a=-a+0<-a+a=0.  This process can also be reversed. ∎

Lemma 2.

For any aR, either a=0 or 0<a2.


Suppose  a0, then by trichotomy, we have either  0<a  or  a<0, but not both.  If  0<a,  then  0=0a<aa=a2.  On the other hand, if  -(-a)=a<0, then  0<-a  by the previous lemma.  Then repeating the previous ,  0=0(-a)<(-a)(-a)=a2. ∎

Three direct consequences follow:

Corollary 1.


Corollary 2.

For any aR, 0<1+a2.

Corollary 3.

There is no real solution for x in the equation 1+x2=0.

Inequality for a converging sequence

Suppose a0,a1, is a sequence of real numbers converging to a real number a.

  1. 1.

    If ai<b or aib for some real number b for each i, then ab.

  2. 2.

    If ai>b or aib for some real number b for each i, then ab.

Title inequalities for real numbers
Canonical name InequalitiesForRealNumbers
Date of creation 2013-03-22 13:58:16
Last modified on 2013-03-22 13:58:16
Owner mathcam (2727)
Last modified by mathcam (2727)
Numerical id 12
Author mathcam (2727)
Entry type Definition
Classification msc 54C30
Classification msc 26-00
Classification msc 12D99
Related topic SummedNumeratorAndSummedDenominator
Defines strict inequality
Defines inequality