How many times has the number 1 been repeated from 1 to 199

The 1 is considered the simplest and most fundamental numeral in mathematics. As we count up from 1 to 199, we may intuitively assume that 1 appears just once at the beginning. However, let us explore how many times this seemingly straightforward number is within this range.

To determine the frequency of the number 1, we must carefully analyze each number from 1 to 199. Let us break it down into different sections to make our analysis more manageable.

From 1 to 9, the digit 1 appears only once, in its natural position as the starting number. So, there are 9 numbers in this range that contain the digit 1.

In the next range, from 10 to 19, the digit 1 appears an additional 10 times. This is because the number 1 is the last digit of all the numbers in this range (11, 12, 13, …, 19). Therefore, the count increases by one each time, resulting in 10 occurrences of the digit 1.

From 20 to 99, there are two positions where the digit 1 can appear: as the tens digit or as the ones digit. The tens digit can be occupied by any digit from 2 to 9, resulting in 8 possibilities. Each of these possibilities yields 10 occurrences of the digit 1, as it can take on any value from 0 to 9 as the ones digit. So, there are a total of 8 x 10 = 80 occurrences of the digit 1 in this range.

Moving on to the range from 100 to 199, we need to consider the hundreds, tens, and ones digits. The hundreds digit can only be filled with the number 1, so it appears once in each hundred (100, 101, 102, …, 199). Therefore, it appears 100 times.

The tens and ones digits now have the same possibilities as the previous range, from 0 to 9, giving us 10 occurrences of the digit 1 per ten numbers. Since there are ten tens within the hundreds range, we have an additional 10 x 10 = 100 occurrences of the digit 1 in this section.

To consolidate our findings, we can sum up the occurrences in each range: 9 + 10 + 80 + 100 + 100 = 299. Thus, the digit 1 appears a total of 299 times from 1 to 199.

Surprisingly, the number 1 is repeated nearly 300 times in this relatively small range. This underlines its significance as a basic building block in the world of numbers.

In conclusion, when counting from 1 to 199, the number 1 appears 299 times. It shows up as the starting number, as the last digit of numbers in the teens, as the tens digit in numbers from 20 to 99, and as the hundreds digit in numbers from 100 to 199. This demonstrates the crucial role played by the number 1 within the numerical sequence and its significance in shaping the foundations of mathematics.