Problem 58 - Algebra and Trigonometry

Question

Write each number in scientific notation. $$ 0.0007029 $$

Step-by-Step Solution

Answer

0.0007029 is \(7.029 \times 10^{-4}\) in scientific notation.

Step 1: Identify the Non-zero Digits

Write down all the non-zero digits of the number. For 0.0007029, the non-zero digits are 7029. These will be the significant figures in the scientific notation.

Step 2: Place the Decimal

Place the decimal point after the first non-zero digit. For the number 7029, place the decimal point after the 7 to get 7.029.

Step 3: Count the Decimal Places

Count how many places you moved the decimal point from its original position to its new position. In 0.0007029, the decimal point was moved 4 places to the right (from before the first 7 to between 7 and 0).

Step 4: Write in Scientific Notation

Combine the significant figure and the power of ten. Since the original decimal was moved 4 places to the right, it means multiplying by \(10^{-4}\). Thus, 0.0007029 in scientific notation is \(7.029 \times 10^{-4}\).

Key Concepts

Significant Figures

When working with scientific notation, understanding significant figures is essential. Significant figures are the digits in a number that contribute to its precision. This includes all non-zero digits and any zeros that are between them or are trailing in the decimal portion.
For example, in the number 0.0007029, identifying the significant figures involves finding all non-zero digits. These are 7029, and they form the main component of the scientific notation.

• Always start counting significant figures from the first non-zero digit.
• Include all subsequent digits, whether they're zeros or non-zeros, as part of the significant figures.

In scientific notation, significant figures help maintain the same level of accuracy as the original number, allowing us to express large or small numbers efficiently.

Decimal Point

The decimal point is a crucial part of interpreting numbers in scientific notation. It's used to determine where we "move" the digits to form a new number between 1 and 10.
To write a number in scientific notation, such as 0.0007029, the decimal point is placed immediately after the first non-zero digit. This results in making sure only one non-zero digit appears before the decimal, transforming it into 7.029.

• Shifting the decimal point changes the power of ten, which corrects the scale of the number.
• This step ensures the significant part of the number is clearly evident while keeping the magnitude manageable with powers of ten.

By consistently following this rule, numbers can quickly be converted into a standardized scientific format.

Power of Ten

Scientific notation involves not only significant figures but also expressing numbers as a product of a decimal and a power of ten. The power of ten indicates how many positions, and in which direction, the decimal point has been moved.
In our example, 0.0007029, the decimal was originally positioned to the left of three zeros. To convert it into scientific notation, we moved it four places to the right. This movement is expressed as a power of ten:

• Every shift to the right means multiplying by a negative power of ten.
• In this case, it translates to \(10^{-4}\), indicating the decimal moved four places to the right.

This method maintains the number's value while making it simpler to work with mathematically.

Non-Zero Digits

Non-zero digits are the main players in any given number and hold their value irrespective of their position. They are critical in forming significant figures and heavily influence the placement of the decimal point in scientific notation.
Taking the number 0.0007029, we identify the non-zero digits as 7, 0, 2, and 9. In the context of scientific notation:

• The first non-zero digit dictates where the decimal point is placed.
• All other non-zero digits following it contribute to the accuracy and precision of the number.

Understanding non-zero digits is fundamental, and once identified, they guide us through how we write the scientific notation correctly. This process allows us to streamline large or small numbers without losing any critical detail.