Problem 37 - Chemistry Matter and Change

Question

Challenge Write the numbers 10,100, and 1000 in scientific notation with two, three, and four significant figures, respectively.

Step-by-Step Solution

Answer

10: \(1.0 \times 10^{1}\), 100: \(1.00 \times 10^{2}\), 1000: \(1.000 \times 10^{3}\).

Step 1: Express 10 in Scientific Notation

To write the number 10 in scientific notation, we express it as a product of a number between 1 and 10 and a power of 10. Here, 10 can be expressed as \(1.0 \times 10^{1}\). With two significant figures, it remains \(1.0 \times 10^{1}\).

Step 2: Express 100 in Scientific Notation

Next, to write 100 in scientific notation with three significant figures, we identify that 100 is \(1.00 \times 10^{2}\), which keeps three significant figures as required.

Step 3: Express 1000 in Scientific Notation

Finally, for the number 1000, we write it in scientific notation as \(1.000 \times 10^{3}\) to ensure it has four significant figures.

Key Concepts

Significant Figures

Significant figures are the digits in a number that contribute to its precision. They include all non-zero digits, zeroes between non-zero digits, and trailing zeroes in a decimal number.
In scientific notation, the number of significant figures is crucial because it reflects the accuracy of the value. For example, when converting numbers like 10, 100, and 1000, specifying significant figures ensures a precise expression:

• 10 has two significant figures as 1.0 in scientific notation.
• 100 is written with three significant figures as 1.00.
• 1000 expands to four significant figures as 1.000.

In this context, significant figures help clarify the precision and accuracy of measurements and calculations.

Powers of Ten

The power of ten in scientific notation aligns with moving the decimal point in a number. It simplifies how we handle very large or very small numbers by scaling them conveniently.
Each power of ten indicates how many places the decimal point has shifted:

• A positive power of ten moves the decimal right (making the number larger).
• A negative power moves it left (making the number smaller).

For instance, in scientific notation:

• Number 10 is expressed as \(10^1\).
• Number 100 is expressed as \(10^2\).
• Number 1000 becomes \(10^3\).

Understanding these powers can make handling complex calculations easier and more intuitive.

Mathematical Representation

Mathematical representation in scientific notation involves expressing numbers as a product of a coefficient and a power of ten.
This form is written as \(a \times 10^n\), where:

• a is the coefficient, a number between 1 and 10, defining significant figures.
• n is the exponent, indicating the power of ten and how the decimal point has shifted.

This method simplifies both large and small numbers into a uniform format that is easy to read and compute.
It also aids in accurately conveying the exactness of numbers through significant figures, thereby providing a relatively simple yet powerful way to work mathematically across different scales.