Question

Express these numbers in scientific notation. a) -890,000 b) 602,000,000,000 c) 0.0000004099 d) 0.000000000000011

Short answer

a) \(-8.9 \times 10^5\); b) \(6.02 \times 10^{11}\); c) \(4.099 \times 10^{-7}\); d) \(1.1 \times 10^{-14}\).

Step-by-step solution

Understanding Scientific Notation

Scientific notation is a way to express very large or very small numbers. A number is expressed in the form \( a \times 10^n \), where \( 1 \leq a < 10 \) and \( n \) is an integer. We need to adjust the number so the leading number is between 1 and 10, then count how many places we moved the decimal point to determine \( n \).

Solve for Number (a)

For \(-890,000\), we find the leading number to be \(-8.9\). We move the decimal point 5 places to the left, so the number in scientific notation is \(-8.9 \times 10^5\).

Solve for Number (b)

For \(602,000,000,000\), we find the leading number \(6.02\). We move the decimal point 11 places to the left, so the number in scientific notation is \(6.02 \times 10^{11}\).

Solve for Number (c)

For \(0.0000004099\), we find \(4.099\), after moving the decimal point 7 places to the right. Since the original number is less than 1, \( n \) is negative: \(4.099 \times 10^{-7}\).

Solve for Number (d)

For \(0.000000000000011\), we find \(1.1\), after moving the decimal point 14 places to the right. Again, as the number is less than 1, \( n \) is negative: \(1.1 \times 10^{-14}\).

Key concepts

Large Numbers

Working with large numbers can sometimes feel overwhelming, especially if they run into the billions or even more. Scientific notation provides a simpler way to express these large numbers. By using scientific notation, we can transform a lengthy sequence of zeroes into a concise mathematical expression.
In scientific notation, we change a large number into the form of \(a \times 10^n\). Here, \(a\) is a number that's at least 1, but less than 10. The exponent \(n\) indicates how many places the decimal point has been shifted to the left.
Consider the example of 602,000,000,000. To convert it into scientific notation:

• First, find a number between 1 and 10, which is 6.02, by moving the decimal point leftward until a value in this range is attained.
• The decimal point was moved 11 places to the left. Hence, \(n\) equals 11.

This gives us the scientific notation: \(6.02 \times 10^{11}\). Such simplifications make calculations and interpretations much more manageable.

Small Numbers

When dealing with small numbers that are close to zero, scientific notation is equally helpful. It allows us to express these numbers more succinctly by eliminating excessive decimal places.
With small numbers, scientific notation takes the form \(a \times 10^n\), where \(a\) is a number between 1 and 10, and \(n\) is a negative integer, indicating how many places the decimal point has been shifted right.
Take, for instance, the number 0.0000004099. Here's how to convert it:

• Move the decimal point rightward until it results in a number between 1 and 10. In this case, it's 4.099.
• The decimal was shifted 7 places to the right, resulting in a negative exponent \(n = -7\).

Thus, the scientific notation becomes \(4.099 \times 10^{-7}\). This expression simplifies handling such small values, making them easier to comprehend and use in calculations.

Scientific Notation Process

The process of converting numbers to scientific notation involves a systematic series of steps. This allows us to handle both large and small numbers efficiently.
Let's break down the process:

• Identify the leading number: Begin by identifying a number between 1 and 10, traditionally the first non-zero digit in the original number.
• Determine the direction and count: Decide how far the decimal needs to be moved. If dealing with a large number, move the decimal to the left. Conversely, for a small number, shift it to the right.
• Ascertain the exponent: Count the total shifts made with the decimal and set your exponent \(n\) accordingly. Positive exponents signify large numbers, while negative ones indicate small numbers.
• Express in scientific format: Finally, represent the number as \(a \times 10^n\) for simplicity and precision.

This process demands attention to detail but ultimately results in an accurate and useful representation of numbers, regardless of their size.By practicing the scientific notation process, you'll find it easier to manage and manipulate numbers for various mathematical tasks.