Problem 114 - Algebra A Combined Function

Question

Write each number in standard form. Then write the number in scientific notation. Facebook has more than 24 million photos uploaded daily.

Step-by-Step Solution

Answer

24 million in scientific notation is \( 2.4 \times 10^7 \).

Step 1: Understand Standard Form

A number is in standard form if it is written as a regular numeral without any additional formatting like commas or powers of ten. Here, '24 million' means 24,000,000.

Step 2: Write in Standard Form

To convert 24 million into standard form, you write it as 24,000,000. This simply means the digit 24 followed by six zeros.

Step 3: Understand Scientific Notation

Scientific notation is a method of writing very large or very small numbers using powers of ten. A number is written in the form \( a \times 10^n \), where \( 1 \leq a < 10 \) and \( n \) is an integer.

Step 4: Convert to Scientific Notation

To write 24,000,000 in scientific notation, you need to express it as a number between 1 and 10 multiplied by a power of 10. Move the decimal point 7 places to the left: \( 2.4 \times 10^7 \).

Key Concepts

Standard Form

Standard form is how we usually write numbers in everyday life. It's the familiar way of writing a number with digits, sometimes with commas separating groups of three digits to make larger numbers easier to read. For example, the number twenty-four million in standard form is 24,000,000.

This means the digit 24 is followed by six zeros. Even for very large numbers, standard form makes it easy to understand the exact value.

Key points to remember about standard form:

• It's also called "numerical form" or "expanded form."
• Helps in writing numbers without using any special mathematical symbols.
• Useful for counting and showing quantities in real-world situations.

Powers of Ten

Powers of ten are essential in understanding scientific notation. They are simply numbers written with a base of 10 raised to an exponent. In the term \(10^n\), the "n" represents how many times 10 is multiplied by itself.

For example:

• \(10^1 = 10\)
• \(10^2 = 100\)
• \(10^3 = 1000\)

When dealing with large numbers, like 24 million, expressing them using powers of ten simplifies calculations and representations.

In scientific notation, 24,000,000 becomes \(2.4 \times 10^7\). Here's why: the decimal point moves 7 places to the left, leaving \(2.4\), and you multiply it by \(10^7\) to maintain the original number's value.

Large Numbers

Large numbers can be tricky to read and understand, especially when they become very large. The concept of large numbers in mathematics involves structuring a way to make them manageable.

Several techniques help us deal with large numbers:

• Standard Form: Example is 24,000,000 which is easy to visualize.
• Scientific Notation: Useful for compact representation. 24 million is \(2.4 \times 10^7\).
• Estimation and Rounding: Simplifies calculations without needing every precise detail.

By representing large numbers in these ways, we make it easier to perform operations, communicate concepts, and understand vast quantities in various fields, from astronomy to data analytics.