Q14. - Geometry

Question

For the following figure,

 

a. Could you list all the numbers between 1 and 2?

b. Is there a point on the number line for every number between 1 and 2?

c. Is there any limit to the number of points between $S$ and $T$?

Step-by-Step Solution

Answer

a. There are infinite numbers between 1 and 2.

b. A point on the number line for every number between 1 and 2 is possible.

c. All the numbers between 1 and 2 are moving towards 2, thus numbers between 1 and 2 cannot be greater than 2.

Part a. Step 1. Check the figure.

Consider the number line.

Part a. Step 2. Step description.

From the figure, there are infinite numbers between two numbers and on a number line every point is paired with a number and every number is paired with a point.

Part a. Step 3. Step description.

Since between any two number there are infinite rational and irrational numbers so there are infinite numbers 1 and 2.

 

Thus, there are infinite numbers between 1 and 2.

Part b. Step 1. Check the figure.

Consider the number line.

Part b. Step 2. Step description.

From the figure, there are infinite numbers between two numbers and on a number line every point is paired with a number and every number is paired with a point.

Part b. Step 3. Step description.

Since between any two number there are infinite rational and irrational numbers so a point on the number line for every number between 1 and 2 is possible.

 

Thus, a point on the number line for every number between 1 and 2 is possible.

Part c. Step 1. Check the figure.

Consider the number line.

Part c. Step 2. Step description.

From the figure, because between any two numbers there are infinite rational and irrational numbers and the numbers are in increasing order.

Part c. Step 3. Step description.

Since between any two numbers there are infinite rational and irrational numbers and the numbers are in increasing order.so all the numbers between 1 and 2 are moving towards 2, thus numbers between 1 and 2 cannot be greater than 2.

 

Thus, all the numbers between 1 and 2 are moving towards 2, thus numbers between 1 and 2 cannot be greater than 2.