Question

SCIENTIFIC NOTATION Rewrite in scientific notation. $$ 0.000459 $$

Short answer

The number 0.000459 written in scientific notation is \(4.59 \times 10^{-4}\)

Step-by-step solution

Identify \(a\) and \(n\)

The number to convert into scientific notation is less than 1, which means that the scientific notation format will have a negative power of 10. The task here is to determine the specific values of \(a\) and \(n\) for the given number. The decimal point needs to be moved 4 places to the right to transform the number to the form where \(1 \leq |a| < 10\), resulting in \(a = 4.59\). The fact that the decimal point was moved four places to the right means that \(n\) equals -4.

Convert to Scientific Notation

Now it's time to write the number in scientific notation using the values obtained for \(a\) and \(n\). To do this, simply replace \(a\) with 4.59 and \(n\) with -4 in the expression \(a \times 10^n\).

Key concepts

Rewriting in Scientific Notation

Understanding how to rewrite numbers in scientific notation can be truly empowering when dealing with very large or very small numbers. Scientific notation is a method used to express these numbers in a more manageable form, which can be particularly useful in scientific and engineering contexts.

To write a number in scientific notation, we follow a standard format: a number between 1 and 10 multiplied by a power of 10. This number is always written as: \( a \times 10^n \), where \( a \) is a number between 1 and 10 and \( n \) is an integer. By condensing the number into this format, it's easier to read, compare, and perform calculations.

Let's say we have the number 0.000459. To convert this to scientific notation, we need to make \( a \) fall between 1 and 10. That means we have to move the decimal point to the right until the number is 4.59. Once we've done this, we've found our \( a \) value, which is the first piece of the puzzle.

Negative Powers of 10

In the context of scientific notation, negative powers of 10 are instrumental. They allow us to compactly represent numbers that are less than one. When moving the decimal point to the right in our number (as we do for numbers less than one), the exponent or power of 10 will be negative.

Powers of 10 in mathematics are a way to express repeated multiplication of 10. For instance, \( 10^3 \) equals 1000, because \( 10 \times 10 \times 10 = 1000 \). Conversely, negative exponents signify division. Therefore, \( 10^{-3} \) equals \( 1/10^3 = 1/1000 \), which is 0.001. When we move the decimal point four places to the right in the number 0.000459, becoming 4.59, we are effectively dividing by 10,000 (\( 10^4 \)). To balance this operation in scientific notation, we use \( 10^{-4} \), which is the second critical piece of the puzzle.

Decimal Point Movement

Decimal point movement is a critical step in rewriting numbers into scientific notation. It's the physical action we take to reformat our number such that it fits within the range we need for \( a \) in the scientific notation format \( a \times 10^n \).

For numbers less than one, the decimal point moves to the right; for numbers greater than one, it moves to the left. The number of places the decimal point moves becomes the absolute value of our exponent \( n \). If we move the decimal point to the right (as with numbers less than one), the exponent will be negative, indicating a small number. Conversely, if we move the decimal point to the left, the exponent will be positive, reflecting a larger number.

This is the action step in the conversion process. For our example with the number 0.000459, the decimal needs to move 4 places to the right to stand between the first non-zero digit and the second, turning it into 4.59 and thus giving us \( n = -4 \) as the third and final piece of our notation puzzle.