Problem 78 - Algebra and Trigonometry

Question

\(73-80\) . Write each number in scientific notation. $$ 7,259,000,000 $$

Step-by-Step Solution

Answer

7,259,000,000 is \(7.259 \times 10^9\) in scientific notation.

Step 1: Understanding Scientific Notation

Scientific notation is a way of expressing numbers that are too big or too small in a compact form. It involves representing a number as a product of a number between 1 and 10, and a power of 10.

Step 2: Identifying the Significant Figures

To write the number 7,259,000,000 in scientific notation, first identify the significant figures: 7, 2, 5, and 9. This gives us the number 7.259.

Step 3: Determining the Power of 10

Now, count the number of places the decimal needs to be moved to go from the original number to the position between 7 and 2. In this case, we move it 9 places to the left.

Step 4: Writing the Final Scientific Notation

Using these values, the number 7,259,000,000 can be written in scientific notation as \(7.259 \times 10^9\).

Key Concepts

Understanding Significant Figures

In scientific notation, significant figures are critical since they tell us which digits are essential to expressing the precision of the number. Essentially, significant figures are the digits in a number that contribute to its accuracy. When converting a number like 7,259,000,000 to scientific notation, we focus on these key digits. They are 7, 2, 5, and 9 in this case.
- Significant figures do not include any leading or trailing zeros unless they are between two non-zero digits or after a decimal point. - Having fewer significant figures does not mean the number is less accurate; rather, it reflects the precision with which the measurement was made.
Understanding significant figures helps ensure the accuracy and consistency of scientific notation, crucial when dealing with large or small numbers.

The Power of 10 Concept

The power of 10 in scientific notation represents how many times you must multiply or divide the base number to return to its original scale. This concept helps us handle really large or very small numbers efficiently without writing too many zeros.
- When you see a power of 10, such as in the expression \(10^9\), it means multiplying 10 by itself nine times. It's a method to express as much detail as possible without cumbersome long form. - The exponent signifies the number of decimal places you move to the right (for positive powers) or left (for negative powers).
For the number 7,259,000,000, moving the decimal 9 places to the left creates our power of 10, which is \(10^9\), simplifying our expression while maintaining accuracy.

Expressing Large Numbers Easily

Scientific notation is especially helpful for expressing large numbers succinctly, allowing you to convey them with fewer digits and less room for error. For instance, instead of writing 7,259,000,000 with all zeros, scientific notation condenses it to \(7.259 \times 10^9\).
- This concise format not only facilitates easier communication of large figures but also highlights the number's most significant elements immediately. - Reducing the number of zeros can minimize counting errors and better highlight significant digits.
Expressing large numbers in scientific notation can simplify calculations and comparisons significantly, which is an essential tool for mathematicians, scientists, and students alike.