Expressing Smallest and Largest n-digit Number in Terms of Powers of 10

You know what makes a number large: it’s more number of digits and greater digits in each place value. Based on that, if you want to have the largest two-digit number for example, you’d simply take a two-digit number and fill both the place values with 9’s because the digit 9 has the highest value among all the digits in the decimal number system. 

In the same way, if you were looking for the smallest three-digit  number, you’d simply take three digits. The 1’s digit, you could simply fill it with a zero and do the same thing for the 10’s place value. The last place value, 100’s, could not be filled with a 0 otherwise you’d end up with a 000 which is practically a 0 (zero).

So the greatest two-digit number for instance becomes 99, the greatest three-digit number, 999, and so on. The smallest two-digit number is 10, the smallest three-digit number, 100, and so on.

The interesting thing here is that if you,

  • 9 (largest one-digit number) +1 = 10 (smallest two-digit number) = 10 * 1 = 10 * 10^0
  • 99 (largest two-digit number)    + 1 = 100  (smallest three-digit number) = 10 * 10 = 10 * 10^1
  • 999 (largest three-digit number) + 1 = 1000 (smallest four-digit number) = 10 * 100 = 10 * 10^2
  • 9999 (largest four-digit number) + 1 = 10000 (smallest five-digit number) = 10 * 1000 = 10 * 10^3
  • and so on …
  • largest n-digit number + 1 = smallest (n+1)-digit number = 10 * 10^(n-1)

What that means is that, if you add 1 to the largest n-digit number, you’ll always get the smallest (n+1)-digit number. It also means that if you add 1 to the largest n-digit number, that can be expressed as [10 * 10^(n-1)].

These articles are used by the author in a series of mathematics courses that will teach you mathematics from the sixth standard all the way up to 12th standard. The purpose of these courses are to help us understand mathematics so that you can use them professionally well. There is a road map that we have put together about the all the courses included, where to starts, etc. To know more about that and have access to those courses, please visit mathematics page on Great IT Courses. Thank you.

Author: John Raschedian

Web Developer

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