# If a and b are rational numbers satisfying

• Jamin2112
In summary, the conversation is about proving Proposition 1.15 which states that if a and b are real numbers satisfying a<b, then there are rational and irrational numbers between a and b. The professor suggests using the Archimedean property and the person attempting the solution starts by showing that (a+b)/2 can equal rational and irrational numbers. The conversation ends with a hint to try it with a=0.
Jamin2112

## Homework Statement

Prove Proposition 1.15.

Proposition 1.15. If a and b are real numbers satisfying a<b, then there are rational numbers and irrational numbers between a and b.

## Homework Equations

Professor said to use the Archimedean property

## The Attempt at a Solution

a < b ---> (a+a)/2 < (a+b)/2 ---> a < (a+b)/2

Similarly,

a < b ---> (a+b)/2 < (b+b)/2 ---> (a+b)/2 < b

Hence a < (a+b)/2 < b

Now I need some way to show that (a+b)/2 can equal rational and irrational numbers. Any suggestions?

Hi Jamin2112!

But you haven't used the Archimedean property.

Hint: try it with a = 0.

## What does it mean if a and b are rational numbers?

A rational number is a number that can be expressed as a ratio of two integers. This means that a and b are both whole numbers or fractions with integers as their numerator and denominator.

## Can a and b be any rational numbers?

Yes, a and b can be any rational numbers as long as they satisfy the given condition.

## What is the significance of a and b being rational numbers in this context?

In scientific and mathematical contexts, using rational numbers allows for more precise and accurate calculations and measurements.

## What happens if a and b are not rational numbers?

If a and b are not rational numbers, then they cannot satisfy the given condition. In this case, the statement may be false or invalid.

## How do I know if a and b satisfy the given condition?

To determine if a and b satisfy the given condition, you can plug in their values and see if the resulting equation or statement is true. You can also use logical reasoning and mathematical operations to prove or disprove the statement.

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