Problem 35 - College Algebra

Question

The set, \(S\) consists of 900,000,000 whole numbers, each being the same number of digits long. How many digits long is a number from S? (Hint: use the fact that a whole number cannot start with the digit 0.)

Step-by-Step Solution

Answer

The numbers in set \( S \) are 9 digits long.

Step 1: Determine the Range of Each Number Length

To find the number of digits, we must find the range of numbers for specific lengths. Numbers with 1 digit range from 1 to 9, numbers with 2 digits range from 10 to 99, numbers with 3 digits range from 100 to 999, and so on. Generally, numbers with 'n' digits start from 10^{n-1} to (10^n)-1.

Step 2: Use Set Cardinality to Find the Digit Count

The set \( S \) has 900,000,000 numbers. We need to find an \( n \) such that the entire range of numbers with \( n \) digits fits this quantity. For \( n \) digits, the count of numbers is \( 9 \times 10^{n-1} \). For it to be 900,000,000, solve the equation \( 9 \times 10^{n-1} = 900,000,000 \).

Step 3: Solve for 'n'

Start by dividing both sides of the equation by 9: \( 10^{n-1} = 100,000,000 \). This indicates that the power of 10 on the right side is 8 (since 100,000,000 is \( 10^8 \)). Therefore, \( n-1 = 8 \), leading to \( n = 9 \).

Step 4: Confirm the Number of Digits

Re-examine. With 9 digits, numbers start from \( 10^8 \) which is 100,000,000, and end at \( 10^9 - 1 \) which is 999,999,999. Total count: \( 999,999,999 - 100,000,000 + 1 = 900,000,000 \). This confirms that all numbers in \( S \) have 9 digits.

Key Concepts

Whole Numbers

Whole numbers are a fundamental concept in number theory. They include all the non-negative numbers, beginning from 0 and going upwards without any fractional or decimal part.
The simplest way to think about whole numbers is to consider the sequence: 0, 1, 2, 3, and so on. Each number in this sequence is a whole number.

• Whole numbers do not include any negative numbers.
• They are used to count, label, and understand quantities in the most basic way.
• If you think about the set of whole numbers, it is infinite since you can always add 1 to any whole number to get the next one.

Whole numbers become especially important when working with digit ranges and determining number lengths, such as the 9-digit numbers discussed in the exercise.

Digit Range

The digit range of a number specifies the set of all numbers that can be formed with a specific number of digits. For instance, with one digit, you have numbers ranging from 1 to 9.
These are all the possibilities with one place-value position.

• For two-digit numbers, the range expands from 10 to 99.
• As you add more digits, you start with a higher base number, like 100 for three digits, and go up to just below the next power of ten.

Understanding digit ranges helps in tackling problems that involve specific quantities of numbers, such as finding how many numbers of a certain digit length fit into a set like in the problem we are exploring.

Set Cardinality

Set cardinality is simply a term used to describe the number of elements in a set. When dealing with numbers, it is about counting the numbers in a particular set.
In the context of the problem, the set cardinality is 900,000,000, indicating there are this many numbers in set S.

• Understanding the cardinality of a set helps solve problems related to digit ranges, as it provides an idea of how many numbers should be accounted for.
• It's crucial for finding solutions where you need to determine how numbers fit within a defined set, as seen when confirming counts of 9-digit numbers.

By determining that the set consists of 9-digit numbers, we align the cardinality with the correct range of digits.

Exponents

Exponents are a mathematical notation used to represent multiplications of the same number. They show how many times a number, known as the base, is used as a factor.
In our exercise, exponents help express the range boundaries for numbers with a certain digit range. For instance, a number with 'n' digits begins at \(10^{n-1}\).

• Exponents simplify expressions of large numbers; for example, \(10^8\) represents 100,000,000.
• They aid in comprehending the scale and size of numbers, which is crucial when determining the set cardinality or digit range.

In the solution, understanding that \(10^{n-1} = 100,000,000\) leads to the conclusion that these numbers are 9 digits long through basic power calculations.