Problem 118 - Algebra A Combined Function

Question

Write each number in standard form. Then write the number in scientific notation. The estimated number of Wikipedia users in a day is 0.13 billion.

Step-by-Step Solution

Answer

130,000,000; 1.3 × 10^8

Step 1: Convert to Standard Form

Standard form refers to writing numbers without any additional notations like 'billion'. We know that 1 billion is equal to 1,000,000,000. Therefore, 0.13 billion can be written as 0.13 × 1,000,000,000. Calculate this to find the number in standard form. So, 0.13 × 1,000,000,000 = 130,000,000.

Step 2: Convert to Scientific Notation

Scientific notation is a way of expressing numbers as a product of a number between 1 and 10, and a power of 10. To convert 130,000,000 to scientific notation, you need to express it in the form of a decimal Move the decimal point in 130,000,000 seven places to the left to get 1.3. Therefore, in scientific notation, the number is 1.3 × 10^8.

Key Concepts

Standard Form

When we talk about converting numbers to **Standard Form**, we are referring to the process of expressing a number in its simple numerical form without any extra terms like "million" or "billion." It's the most straightforward version of a number you might encounter in daily mathematics.

• Start by removing any words that indicate a large quantity, such as "billion."
• In our example, 0.13 billion needs conversion into a true numerical value.
• You do this by multiplying 0.13 by the numerical equivalent of a billion, which is 1,000,000,000.

In simpler terms, you calculate:\[0.13 \times 1,000,000,000 = 130,000,000\]Now, the number is in standard form! It is simply expressed as 130,000,000 without any modifiers like "billion." Keeping numbers in standard form allows them to be easily manipulated in arithmetic operations and visually simplifies the concept of large quantities.

Powers of Ten

**Powers of Ten** are crucial in mathematics for understanding large numbers more compactly and conveniently. In scientific notation, every massive or minuscule number can be expressed as a product of a coefficient and a power of ten. The power of ten is determined by how many places the decimal point is moved to convert the number to a simpler form between 1 and 10.

• For the number 130,000,000, we need to convert it to a simpler number like 1.3.
• Start by moving the decimal point seven places to the left from 130,000,000 to get 1.3.

The number 1.3 is now multiplied by 10 raised to the power corresponding to how many places the decimal was moved. Here it is seven places, thus:\[1.3 \times 10^8\]The exponent "8" means that the decimal point in the starting number was moved 8 places back to the right to confirm to 130,000,000. Powers of ten allow large or small numbers to be represented with ease, making operations like multiplication and division much faster and error-free.

Decimal Conversion

In this context, **Decimal Conversion** plays a pivotal role in transforming the meaning of words like "billion" into understandable numerical expressions. It transforms symbolic or word-based values into concrete mathematical value.

• Take 0.13 billion as an example.
• The term "billion" represents the numerical value 1,000,000,000.

By multiplying 0.13 by 1,000,000,000, we translate this word-based figure into a precise decimal form:\[0.13 \times 1,000,000,000 = 130,000,000\]Understanding decimal conversion aids in bridging different forms of numeric representation, simplifying - for instance - economic figures or statistical data into a digestible numerical form. It is the key to demystifying complex terms and converting them into tangible values allowing better analysis and comparison in various fields, from finance to science.