Understanding the Place Value and Value Comparison of Digits in Numbers

To understand how many times greater the value of 5 is in the number 2573 compared to its value in 6459, we need to carefully analyze the place value of each digit.

Place Value Analysis

Let's dive into the detailed place value of each 5 in the numbers 2573 and 6459.

Value of 5 in 2573

The digit 5 in the number 2573 is in the hundreds place. This means its value is:

5 × 100 500

We can break down the number 2573 as follows:

2000 (representing the thousands place) 500 (representing the hundreds place) 70 (representing the tens place) 3 (representing the units place)

Value of 5 in 6459

The digit 5 in the number 6459 is in the tens place. Therefore, its value is:

5 × 10 50

We can break down the number 6459 as follows:

6000 (representing the thousands place) 400 (representing the hundreds place) 50 (representing the tens place) 9 (representing the units place)

Comparing the Values

Now that we have determined the individual values of the digit 5 in both numbers, we can compare them.

Value of 5 in 2573: 500

Value of 5 in 6459: 50

To find out how many times greater 500 is compared to 50, we perform the division:

500 ÷ 50 10

This shows that the value of 5 in 2573 is 10 times greater than the value of 5 in 6459.

Note: The numbers mentioned assume a base-10 number system, which is the most common in everyday use. However, let's explore how this concept works in different bases.

Generalizing the Concept

The value ratio between digits in different positions and bases can be generalized. If both numbers are in base-10, the ratio between the values represented by two identical digits will be determined by the number of positions between them. For instance:

If the digits are adjacent in base-10, the ratio is 10^1. If there's one place between them, the ratio is 10^2. This pattern continues, so for n places between identical digits in base-10, the ratio is 10^n.

For instance, if the numbers in the original question were actually written in hexadecimal base-16, the first "5" would represent a value of 16 times the second one. The general formula for any base b is:

The ratio is b^n, where n is the number of places between the duplicate digits.

For example:

In a base-16 system, the ratio would be 16^n.

Thus, for any number base b, the value represented by identical digits at different positions can be calculated as b^n.