Question

How is the number 0.009834 expressed in scientific notation? a) \(9.834 \cdot 10^{4}\) c) \(9.834 \cdot 10^{3}\) b) \(9.834 \cdot 10^{-4}\) d) \(9.834 \cdot 10^{-3}\)

Short answer

Question: Convert the decimal number 0.009834 into scientific notation and choose the correct option: a) \(9.834 \cdot 10^{3}\) b) \(98.34 \cdot 10^{-2}\) c) \(0.09834 \cdot 10^{-2}\) d) \(9.834 \cdot 10^{-3}\) Answer: d) \(9.834 \cdot 10^{-3}\)

Step-by-step solution

Identify the decimal number between 1 and 10

The number 0.009834 is a decimal less than 1. To express it in scientific notation, we want to obtain a decimal number between 1 and 10 multiplied by a power of 10. In other words, we need to write 0.009834 as \(A \cdot 10^B\) where 1 ≤ A < 10 and B is an integer.

Determine the decimal number in scientific notation

To create the decimal between 1 and 10, we must move the decimal point in the number 0.009834 to obtain a number between 1 and 10. In this case, we must move the decimal point 3 places to the right to get 9.834.

Determine the power of 10 in scientific notation

Since we moved the decimal point 3 places to the right in step 2, to compensate for this, we need to multiply the number 9.834 by a power of 10 that will move the decimal point back 3 places. To do this, we need to multiply by \(10^{-3}\): \(9.834 \cdot 10^{-3} = 0.009834\) Now we have the number in scientific notation: \(9.834 \cdot 10^{-3}\).

Match the scientific notation with the given options

We have determined that the scientific notation for 0.009834 is \(9.834 \cdot 10^{-3}\). Looking at the given options, we can see that the correct option is: d) \(9.834 \cdot 10^{-3}\)

Key concepts

Expressing Decimals in Scientific Notation

Imagine needing to handle very small or large numbers, such as the thickness of a cell membrane or the distance between galaxies. Writing out all the zeros is impractical, not to mention error-prone. This is where scientific notation becomes exceedingly useful.
Scientific notation is a way to express numbers that are too big or too small to be conveniently written in decimal form. It is used by scientists and engineers to make computations simpler and clearer.

How to Express a Decimal in Scientific Notation

Take a typical decimal number that is less than one, such as 0.009834. To express this number in scientific notation, follow these steps:

• Find a number between 1 and 10. For 0.009834, this would be 9.834.
• Count how many places you move the decimal to get that number. Here, we move the decimal three places to the right.
• Attach a power of ten that indicates the number of places you've moved the decimal point. Since we've moved it to the right, we use a negative exponent.
• Combine the two to express 0.009834 as 9.834 times a power of ten. Ultimately, you write 0.009834 as \(9.834 \cdot 10^{-3}\).

Powers of Ten

The concept of 'powers of ten' is essential in mathematics, particularly when dealing with large-scale or microscale quantities.
The power of ten denotes how many times to multiply the number ten by itself. The exponent in a power of ten indicates whether you're scaling the number up or down.

Understanding Positive and Negative Powers of Ten

A positive exponent, like \(10^{3}\), means we multiply ten by itself three times: \(10 \cdot 10 \cdot 10 = 1,000\). In contrast, a negative exponent, such as \(10^{-3}\), tells us we're dealing with a fraction — it's like dividing 1 by ten three times: \(\frac{1}{10 \cdot 10 \cdot 10} = \frac{1}{1000}\).
Using powers of ten, scientists can neatly express measurements, whether they are exceptionally large or incredibly minute, by shifting the decimal point to create a new value that is easier to comprehend and work with.

Place Value Adjustment

Place value in mathematics is a fundamental concept that dictates the value of each digit in a number based on its position. When dealing with decimal numbers, adjusting the place value is how we keep the original number's value intact while representing it in a different form, such as scientific notation.

Adjusting Decimal Place Value for Scientific Notation

Let's apply place value adjustment to our previous example, 0.009834. By moving the decimal point to the right three places, the place value of each digit increases. The original thousandths place (0.009) becomes the units place (9).
Every shift of the decimal point equates to a corresponding power of ten. Shifting three places to the right is like dividing by \(10^{3}\), therefore, we use the inverse operation, multiplying by \(10^{-3}\), to balance the equation.
Understanding these adjustments is crucial when transitioning between standard decimal notation and scientific notation, and ensures that the value of the number remains accurate after the conversion.