Problem 80 - Algebra and Trigonometry

Question

\(73-80\) . Write each number in scientific notation. $$ 0.0007029 $$

Step-by-Step Solution

Answer

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

Step 1: Identify the First Non-Zero Digit

Look at the number 0.0007029. Identify the first non-zero digit, which is 7.

Step 2: Move the Decimal Point

Move the decimal point to the right of the first non-zero digit, 7. This turns 0.0007029 into 7.029.

Step 3: Count the Decimal Places Moved

Count how many places you moved the decimal point. Starting from 0.0007029 to 7.029, you moved the decimal point 4 places to the right.

Step 4: Write in Scientific Notation

In scientific notation, the expression is written as the significant figure times 10 raised to the negative power of the number of decimal places moved. So, 0.0007029 is written as \(7.029 \times 10^{-4}\).

Key Concepts

Understanding the Decimal Point

A decimal point is a critical part of expressing numbers, especially in scientific notation. It separates the integer part of a number from its fractional part. In the number 0.0007029, the decimal point is located right after the initial zeros. To convert such a small number into scientific notation, it's essential to move this decimal point. This process helps in expressing the number in a shorter form, which is useful for calculations.
Moving the decimal point changes the value of the number. For example, moving it from 0.0007029 to 7.029 equals changing its scale by a factor of ten. This movement is the first step in expressing the number in scientific notation.
Keep in mind that the number of decimal places moved becomes part of the exponent term when expressing numbers in scientific notation.

Importance of Significant Figures

Significant figures are the digits that carry meaning contributing to a number's precision. When you're working with scientific notation, these are the numbers you keep when writing the original number in its new form. In our exercise, 0.0007029, the significant figures are 7, 0, 2, and 9.
This means, after identifying the first non-zero digit (7), we move the decimal point to directly after it, resulting in 7.029. Each of these digits, including the 0 in between 7 and 2, are crucial as they define the precision of the number.
When the number is rewritten in scientific notation, these significant figures ensure that no meaningful detail or precision is lost. That's how 0.0007029 turns into 7.029 when expressed in scientific notation.

Decoding the Negative Exponent

In scientific notation, exponents play a vital role, especially when dealing with very small or large numbers. A negative exponent means the original number is a fraction, smaller than one. When the decimal is moved to the right in numbers less than one, the exponent becomes negative.
In our example, 0.0007029, the decimal is moved 4 places to the right to make it 7.029. This movement results in a negative exponent in the scientific notation, creating an expression of the form \(7.029 \times 10^{-4}\).
The \(-4\) in the exponent tells us that the decimal point was moved four places, shrinking the original fractional number to its concise form. This not only makes the number easier to handle but also maintains its precision.