Problem 49 - Contemporary Precalculus

Question

Express the given numbers (based on 2006 estimates) in scientific notation. Population of the world: 6,506,000,000

Step-by-Step Solution

Answer

Question: Express the world population in 2006 (6,506,000,000) in scientific notation. Answer: 6.506 * 10^9

Step 1: Identify the First Non-Zero Digit

Find the first non-zero digit in the given number (in this case, it is 6).

Step 2: Rewrite the Number as a Decimal Multiplied by an Integer

Move the decimal point between the first and second non-zero digits (in this case, 6 and 5) and rewrite the number with the new decimal point: 6.506

Step 3: Find the Correct Power of 10

Count how many times the decimal point was moved in Step 2 (in this case, 9 times to the left, so it is a positive exponent).

Step 4: Combine the Decimal and Power of 10

Combine the decimal found in Step 2 and the power of 10 found in Step 3 into the final scientific notation: 6.506 * 10^9 So, the population of the world in 2006, given as 6,506,000,000, can be expressed in scientific notation as 6.506 * 10^9.

Key Concepts

Understanding Exponents

Exponents are a shorthand way to represent repeated multiplication. When you see a number like \(10^2\), it means 10 multiplied by itself twice: \(10 \times 10\). In scientific notation, exponents of 10 are used to express large or small numbers in a more manageable way.
For instance, \(10^9\) indicates a billion because the number 10 is multiplied by itself 9 times.

• Exponents make it easier to write large numbers compactly, reducing the chances of errors.
• They offer an easy method to compare sizes of numbers since the exponent indicates the order of magnitude.

Recognizing and working with these exponents helps you visualize and calculate large quantities efficiently, which is essential in fields like science and engineering.

Decimal Place Value

Understanding decimal place value is crucial when converting numbers into scientific notation. Each digit in a number has a specific value dependent on its position, which is what place value is about.
The place value is a foundational concept based on powers of 10. For instance, in the number 6,506,000,000:

• The digit 6 stands for 6 billion, or \(6 \times 10^9\).
• The digit 5 represents 500 million, or \(5 \times 10^8\).

When we've identified the first non-zero digit, the remaining digits are positioned after a decimal to form the significant figures in scientific notation. In our example, 6.506 forms the base of the scientific notation after identifying each digit's contribution based on place value.

Large Numbers Representation

Representing large numbers like the population of the world in scientific notation helps simplify understanding and calculations. Instead of writing out 6,506,000,000, we write \(6.506 \times 10^9\). This format provides clarity and eases communication for such large values.
Using scientific notation in contexts like global data, which often involves vast numbers, offers several benefits:

• Saves space and reduces the potential for error in manual calculations.
• Facilitates comparisons between tremendously different scales by focusing on the significant digits and order of magnitude.

It’s a precise tool for scientists, economists, and anyone dealing with large-scale data to convey findings in a simplified and standardized manner.