Question

How can a speed of \(100 \mathrm{~m} / \mathrm{s}\) be written so that it has three significant figures?

Short answer

100 m/s is written as \(1.00 \times 10^2\) m/s for three significant figures.

Step-by-step solution

Understanding the Requirements

To express 100 m/s with three significant figures, we need to write it in such a way that clearly indicates three digits are significant. This requires consideration of numerical representation methods.

Using Scientific Notation

Scientific notation is a convenient way to express numbers with a desired number of significant figures. The original number, 100 m/s, can be written as \(1.00 \times 10^2\) m/s to clearly show three significant figures.

Clarifying Significant Figures

In \(1.00 \times 10^2\), the three significant figures are 1, 0, and 0. By using scientific notation, trailing zeros are considered significant because they occur after a decimal point.

Key concepts

Scientific Notation

Scientific notation is a powerful method to express very large or very small numbers straightforwardly. It helps in representing numbers to a specific number of significant figures by using powers of ten.

When writing numbers in scientific notation, a number is rewritten as a product of a decimal number between 1 and 10, and a power of ten. For example, the speed of 100 m/s can be written as \(1.00 \times 10^2\) m/s. Here, 1.00 is the significant portion, while \(10^2\) scales it up to its actual magnitude.

• Scientific notation makes it easy to identify significant figures.
• This format is helpful in maintaining precision, especially in scientific calculations.
• It also allows one to easily interpret and communicate very large or extremely small values.

By clearly showing significant digits with scientific notation, we achieve a precise representation that often reflects the level of accuracy of measurement.

Numerical Representation

Numerical representation is about how we choose to express numbers in different formats. The choice of representation can change how easily others interpret a number and its significance.

Different representation methods include

• standard form,
• scientific notation, and
• fractional form.

From these, scientific notation stands out for precision and clarity. For instance, going back to 100 m/s, while it's seemingly easy, representing it as 100 could mean different things in terms of significant figures. In contrast, \(1.00 \times 10^2\) clarifies that all three digits - 1, 0, and 0 - count as significant figures.
By choosing this representation, we provide clear mathematical precision and avoid ambiguities related to how many figures truly carry significance.

Trailing Zeros Significance

Trailing zeros often lead to confusion about their role in determining the significance of numbers. These zeros can significantly change what a number indicates about precision and accuracy.

Trailing zeros are zeros at the end of a number. Their significance depends on whether there's a decimal point present:

• If a decimal is present, such as in 1.00, trailing zeros are significant because they indicate precision in measurement.
• Without a decimal, such as in 100, trailing zeros may not necessarily be significant.

In the scientific notation \(1.00 \times 10^2\), the two zeros after 1 are significant because they follow a decimal point, which is crucial information in scientific and technical contexts. This distinction ensures that when passing on measurements, the stated precision matches the expected levels important for calculations and interpretations.