Why Is the Distributive Property Key in Simplifying Algebraic Expressions?

Click For Summary

Discussion Overview

The discussion revolves around the justification of the distributive property and other axioms of real numbers in the context of simplifying algebraic expressions. Participants explore the application of these properties to specific algebraic statements and the importance of assumptions in abstract algebra.

Discussion Character

  • Homework-related
  • Technical explanation
  • Conceptual clarification

Main Points Raised

  • Some participants assert that the distributive property justifies the simplifications in the given algebraic expressions.
  • Others express uncertainty about the completeness of their answers and the significance of the assumptions regarding the variable x.
  • One participant notes that the assumptions are necessary in abstract algebra to clarify notation and ensure consistency in operations.
  • There is a discussion about the appropriate axioms needed for the first expression, with some suggesting the axiom of equality and others pointing out that it does not fit the structure required for the distributive property.
  • Participants consider the associative property of multiplication as a potential justification for the first expression.

Areas of Agreement / Disagreement

Participants generally agree on the correctness of the answers for two of the expressions but disagree on the justification for the first expression, with multiple competing views on which axiom applies.

Contextual Notes

There is a noted ambiguity regarding the use of different multiplication operators in the expressions, which could lead to confusion about their interpretation. Additionally, the discussion highlights the need for clarity in the assumptions made in algebraic contexts.

bergausstein
Messages
191
Reaction score
0
in the following exercises, assume that x stands for an unknown real number, and assume that $x^2=x\times x$. which of the properties of real numbers justifies each of the following statement?

a. $(2x)x=2x^2$
b. $(x+3)x=x^2+3x$
c. $4(x+3)=4x+4\times 3$

my answers
a. distributive property
b. distributive property
c. distributive property.

i just don't know if my answers are complete. and I'm also bothered why this part " assume that x stands for an unknown real number, and assume that $x^2=x\times x$." is important.

thanks!
 
Last edited:
Mathematics news on Phys.org
bergausstein said:
in the following exercises, assume that x stands for an unknown real number, and assume that $x^2=x\times x$. which of the properties of real numbers justifies each of the following statement?

a. $(2x)x=2x^2$
b. $(x+3)x=x^2+3x$
c. $4(x+3)=4x+4*3$

my answers
a. distributive property
b. distributive property
c. distributive property.

i just don't know if my answers are complete. and I'm also bothered why this part " assume that x stands for an unknown real number, and assume that $x^2=x\times x$." is important.

thanks!

Hi bergausstein!

The extra assumptions are needed, since in abstract algebra you can't really assume anything. In principle you're limited to exactly what the axioms give you. Anything else needs to be specified. These assumptions are matters of notation, so that you know that squaring a number is in all respects the same as multiplying that number by itself.
Actually, the extra assumptions in this case are so standard, that I consider it a bit of overkill to mention them.

Your answers to (b) and (c) are correct. However, for (a) you will need a different axiom.

Btw, is there a reason you used a different multiplication operator in (c)?
Luckily there is only 1 multiplication operator in the field of the real numbers, but otherwise that would be ambiguous.
 
what axiom do i need for a? let me guess, is it an axiom of equality?
 
bergausstein said:
what axiom do i need for a? let me guess, is it an axiom of equality?

Axiom of equality? That's not really one of the axioms of the real numbers, although you are implicitly using the axioms that belong to an equivalence relation.

What I mean is that the distributive property is a(b+c)=ab+ac.
But if I look at (a) I have (2x)x = 2(xx).
Those do not look like they have the same structure...
 
I like Serena said:
Axiom of equality? That's not really one of the axioms of the real numbers, although you are implicitly using the axioms that belong to an equivalence relation.

What I mean is that the distributive property is a(b+c)=ab+ac.
But if I look at (a) I have (2x)x = 2(xx).
Those do not look like they have the same structure...

we can use associative property of multiplication (ab)b = a(bb) am i correct?
 
bergausstein said:
we can use associative property of multiplication (ab)b = a(bb) am i correct?

Yep!
 

Similar threads

MHB Inequality of positive real numbers
  • · Replies 1 ·
Replies
1
Views
1K
MHB Find a,b,c,d for Max $a+b-c+d$
  • · Replies 1 ·
Replies
1
Views
1K
High School Commutative & Associative property of negative numbers
  • · Replies 4 ·
Replies
4
Views
3K
MHB What is the Sum of x, y, and z in a Non-Negative Real Number System?
  • · Replies 4 ·
Replies
4
Views
2K
MHB Which procedure takes the minimum time to solve modulus functions?
  • · Replies 5 ·
Replies
5
Views
1K
Undergrad Algebraic property of real numbers
  • · Replies 30 ·
2
Replies
30
Views
3K
MHB Using Properties of Real Numbers: Justifying Equalities
  • · Replies 2 ·
Replies
2
Views
2K
MHB Real Roots of Cubic Equation: $x^3+a^3x^2+b^3x+c^3=0$
  • · Replies 1 ·
Replies
1
Views
2K
MHB Can All Roots of a Quartic Polynomial Be Real?
  • · Replies 1 ·
Replies
1
Views
2K
MHB How to Simplify Rational Expressions with Variables
  • · Replies 4 ·
Replies
4
Views
2K