Problem 6 - Applied Physics

Question

Write each number in scientific notation. \(24.97\)

Step-by-Step Solution

Answer

24.97 in scientific notation is \( 2.497 \times 10^1 \).

Step 1: Understand Scientific Notation

Scientific notation is a way of expressing numbers that are too large or too small to be conveniently written in decimal form. It is expressed as \( a \times 10^n \), where \( 1 \leq a < 10 \) and \( n \) is an integer.

Step 2: Identify the Coefficient

For the number 24.97, identify a value \( a \) such that \( 1 \leq a < 10 \). In this case, we can rewrite 24.97 as 2.497 by moving the decimal point one place to the left.

Step 3: Determine the Exponent

Since we moved the decimal point one place to the left to convert 24.97 to 2.497, the exponent will be \( n = 1 \), making the number larger as originally it was when expressed in standard form.

Step 4: Write in Scientific Notation

Combine the coefficient and the exponent to express the number in scientific notation. Thus, 24.97 can be written as \( 2.497 \times 10^1 \).

Key Concepts

Expressing Numbers

Expressing numbers in different ways allows us to better understand and work with them, especially when they are either very large or extremely small. Scientific notation is a handy method for this.
Instead of writing long sequences of zeros, we express numbers as a product of a coefficient and a power of ten. This transformation is particularly useful in scientific and engineering fields.

• The coefficient is a number usually between 1 and 10.
• The power of ten simplifies the original number, indicating its scale.

For instance, expressing 24.97 in scientific notation involves transforming it into the format \( a \times 10^n \). Converting numbers to this format can make calculations more manageable, revealing the number's precise magnitude.

Decimal Form

Understanding decimals is crucial when learning about scientific notation. A decimal number has digits on both sides of the decimal point.
For example, the number 24.97 encompasses the whole number '24' on the left and the fractional '0.97' on the right.
This representation signifies numbers less than a whole (1) and more than a complete (24) unit.

• Each position to the right of the decimal point represents tenths, hundredths, and so forth.
• Decimal numbers make up a part of everyday math, used to measure precisely.

In scientific notation, decimals are modified by adjusting the decimal placement, like turning 24.97 into 2.497. By repositioning the decimal point, it prepares the number for scientific expression as a simpler format, such as \( 2.497 \times 10^1 \). Understanding how decimal points affect values helps in mastering number representation.

Exponent

An exponent in scientific notation serves a vital role in defining the scale of a number. It's like a marker showing how many places a number has moved from its original standing.
For scientific notation, the exponent indicates how many times the number 10 must multiply to create the full number.
In the example of 24.97, when the decimal moves one place to the left to become 2.497, the exponent reflects this shift.

• A positive exponent indicates a movement left or a larger original number.
• A negative exponent signifies moving to the right, or a smaller original number.

Thus the number \( 2.497 \times 10^1 \) keeps its size by reflecting in the exponent's value that has been elevated just right, making magnitude interpretations straightforward. Understanding exponents is essential, as it allows quick reading of how vast or minuscule a number is, just by looking at the exponent alone.