Problem 80 - Algebra and Trigonometry

Question

Write each number in scientific notation. $$ 579,000,000,000,000,000 $$

Step-by-Step Solution

Answer

The scientific notation for the given number 579000000000000000 is \(5.79 \times 10^{17}\).

Step 1: Identifying the Number of Digits

Count the total number of digits in the number which are 579000000000000000 till significant digit after decimal point. In the provided number, there are 18 digits until we reach the significant digit, 5.

Step 2: Moving the Decimal Point

Since the original number is greater than 1, the decimal point should be moved to the left. Position the decimal point after the first significant digit, which is 5. This makes the number 5.79.

Step 3: Writing the Exponential Form

Multiply the shortened form 5.79 by ten raised to the power of (total digit - 1). Here, it would be \( 10^{18-1} = 10^{17} \). Finally, this gives us the scientific notation of the number.

Step 4: Final Scientific Notation

Put it all together. So, the respective scientific notation for the given number is \(5.79 \times 10^{17}\)

Key Concepts

Exponential Form

Understanding exponential form is crucial when working with scientific notation. Exponential form allows us to write very large or very small numbers in a compact way. It involves multiplying a number by a power of ten.
The power, indicated as an exponent, tells you how many times to multiply the number 10 by itself. In scientific notation, an exponent shows where the decimal point has been moved from its original position.
For example, when you write \(5.79 \times 10^{17}\) in exponential form, the number \(5.79\) is being multiplied by \(10\), raised to the power of 17. This simplifies the representation of numbers and makes them easier to work with.

Decimal Point Placement

Decimal point placement is a fundamental aspect of converting numbers into scientific notation. The decimal point in a number indicates where the whole number part ends and the fractional part begins. When converting to scientific notation, the decimal point is strategically placed after the first significant digit of a number.
This step ensures that the representation remains consistent and manageable. For our number, 579,000,000,000,000,000, we move the decimal point so it's after the first significant digit, 5.
This transforms the number to 5.79. The number of positions the decimal point is moved becomes the exponent on the 10 in scientific notation.

Significant Digits

Significant digits, or significant figures, define the precision of a number. They include all non-zero numbers, any zeros between them, and all final zeros in a decimal number.
In the number 579,000,000,000,000,000, the significant digits are 579.
These digits give the number its precision.

• Non-zero numbers are always significant.
• Zeros located between significant digits are significant.
• Final zeros to the right of the decimal point are also significant.

Understanding significant digits is essential because they ensure the scientific notation is both concise and accurate. Knowing which digits are significant helps place the decimal point correctly during conversion.