Problem 36 - Astronomy: At Play in the Cosmos

Question

Humanity has existed in its culturally and anatomically modern form for approximately 50,000 years. Write this number in scientific notation

Step-by-Step Solution

Answer

50,000 is written as \( 5 \times 10^4 \) in scientific notation.

Step 1: Understand the problem

We need to express the number 50,000 in scientific notation. Scientific notation is writing a number as a product of a number (usually between 1 and 10) and a power of ten.

Step 2: Identify the significant figures

Identify the significant figures in 50,000. The significant figure here is 5.

Step 3: Determine the power of 10

Consider how many places the decimal point needs to move to get the significant figure (5) to lie between 1 and 10. In the number 50,000, the decimal point is after the last zero. Moving it 4 places to the left results in 5.

Step 4: Write in scientific notation

Combine the significant figure with the appropriate power of 10. Since we moved the decimal 4 places, the exponent is 4. Thus, write 50,000 as \( 5 \times 10^4 \).

Key Concepts

Powers of Ten

The concept of "Powers of Ten" is crucial when dealing with large or small numbers, as it helps simplify expressions and makes them easier to understand. In scientific notation, numbers are expressed as a product of a base number and a power of ten. The base number is usually between 1 and 10, and the power of ten essentially tells you how many places the decimal point has moved.

For example, in the number 50,000, we need to determine the power of ten involved. Since 50,000 equals 5 followed by four zeros, we express this in scientific notation as:

• The base number being 5, a single significant digit.
• The decimal point is moved four places to get from 50,000 to 5.

Thus, 50,000 is expressed as \( 5 \times 10^4 \). The exponent 4 represents the four places the decimal point has moved from its original position, making it simple to describe the magnitude of the number with minimal digits.

Significant Figures

Significant figures are the digits in a number that contribute to its accuracy. In the context of scientific notation, they are essential for preserving the precision of the number. When you write a number like 50,000, determining which figures are significant is key to correctly representing the number in scientific notation.

In 50,000, the significant figure to consider is 5. This is the only non-zero number and is prominent in expressing the value's magnitude. When converting into scientific notation, only one significant figure is needed if it represents the entire measurement accurately. However, the choice of significant figures can vary depending on the precision required in different fields or problems.

The benefit of focusing on significant figures is that it allows one to ignore zeros that simply serve as placeholders, which do not add to the number's precision. This is why leading and trailing zeros are often not considered significant unless otherwise indicated or measured during calculations.

Decimal Point Placement

Understanding where to place the decimal point is crucial for writing numbers in scientific notation. The position of the decimal can drastically change a number's value. In the number 50,000, for example, the default position of the decimal is at the end of the number, even though it is not explicitly shown.

To convert to scientific notation, the decimal point is moved such that only the significant figure (here, 5) remains before the decimal. This gives a clearer, more concise form:

• Originally, the decimal position is after the last zero in 50,000.
• We move it 4 places to the left to position it between 5 and 0.

The number thus becomes 5, simplifying the representation and ensuring clarity when expressing large quantities or measurements. The process of moving the decimal and counting how many places it travels help determine the exponent in the power of ten, effectively showcasing the number's scale.