Question

35. the set, s consists of 900,000,000 whole numbers, each being the same number of digits long. howmany digits long is a number from s? (hint: use the fact that a whole number cannot start with the digit 0.)

Answer 1

• 9 digit long

Let the number of digits be n and the number of elements in set be s.

When n = 1

• The set contains 1-digit numbers, 1 through 9,
• The set consists of 10 - 1 = 9 numbers.

When n = 2

• The set contains 2-digit numbers, 10 through 99,
• The set contains 100 - 10 = 90 numbers.

When n = 3

• The set contains 3-digit numbers, 100 through 999,
• The set contains 1000 - 100 = 900 numbers.

The pattern we see helps us determine the relationship between s and n as follows.

When set contains n-digit numbers, the set contains:

• s = 10ⁿ - 10ⁿ⁻¹ = 10ⁿ⁻¹(10 - 1) = 9*10ⁿ⁻¹ elements

We have s known, substitute it into equation above and solve for n:

• 900000000 = 9*10ⁿ¹
• 100000000 = 10ⁿ⁻¹
• 10⁸ = 10ⁿ⁻¹
• n - 1 = 8
• n = 9

The numbers in the set s are 9-digit long.