Counting Four-Digit Numbers: From 1000 to 9999

Introduction:
Four-digit numbers are a common concept in mathematics and can be particularly useful when dealing with numeric ranges, such as in the case from 1000 to 9999. This article will explain various methods to count the total number of four-digit integers within this range.

1. The Simple Method: Subtract the Limits

The easiest way to find the number of four-digit numbers between 1000 and 9999 is to use the formula:

(Total Upper;Limit - Lower;Limit 1)

Applying the formula:

(Total 9999 - 1000 1 9000)

Hence, there are 9000 four-digit numbers in the given range.

2. Understanding the Formula

The formula can be broken down to explain its logic. By starting from the lower limit (1000) and ending at the upper limit (9999) and adding 1, we account for all the numbers in the range, including the first and the last number.

3. Distinct Digits Condition

Now, let's consider the scenario where we need to find the number of four-digit numbers with all distinct digits. The first digit (thousands place) can be any number from 1 to 9 (9 choices). The second digit (hundreds place) can be any number from 0 to 9 except the one used in the first place (9 choices). Following this pattern, the third digit (tens place) has 8 choices and the last digit (units place) has 7 choices. Multiplying these choices, we get the total combinations:

(9 times 9 times 8 times 7 4536)

This reveals that there are 4536 four-digit numbers with distinct digits.

4. Another Method Using Sequences

To confirm the count, we can use the arithmetic sequence formula. The first term (a1) is 1000, and the common difference (d) is 1. The last term (an) is 9999. Using the formula for the nth term of an arithmetic sequence:

(a_n a_1 (n - 1)d)

Solving for n:

(9999 1000 (n - 1) times 1)

(n - 1 8999)

(n 9000)

This confirms that there are 9000 four-digit numbers.

5. Applying the Formula for Any Range

To find the number of four-digit numbers between any two limits, say 1000 and n (where n 9999), we use the formula:

(n - 1000 1 n - 999)

For example, if n 9999:

(9999 - 1000 1 9000)

This confirms our previous findings.

Conclusion

Counting four-digit numbers is an essential skill in understanding numeric ranges and can be approached using various methods, including simple subtraction, arithmetic sequences, and distinct digit conditions. The total number of four-digit numbers from 1000 to 9999 is 9000, as confirmed by multiple methods.