Problem 36 - Elementary and Intermediate Algebra

Question

Write number in scientific notation. 0.04

Step-by-Step Solution

Answer

0.04 in scientific notation is \(4 \times 10^{-2}\).

Step 1: Identify the Decimal

Observe the given number, which is 0.04. Notice where the decimal point is initially placed.

Step 2: Move the Decimal Point

Shift the decimal point to the right until there's only one non-zero digit to its left. For 0.04, the decimal moves two places to the right to become 4.

Step 3: Count the Moves

Count the number of places you moved the decimal point. In this case, the decimal point was moved 2 places to the right.

Step 4: Construct the Scientific Notation

Combine the coefficient from Step 2 (which is 4) with a power of ten. Since we moved the decimal 2 places to the right, the exponent will be -2. Therefore, the scientific notation form is written as \(4 \times 10^{-2}\).

Key Concepts

Understanding Decimal Points

A decimal point is a symbol used to separate the whole number part of a number from the fractional part.
In mathematics, the decimal point is essential to indicate how values are divided.
When working with scientific notation, it's important to correctly identify and manipulate the decimal point.

• The decimal point in 0.04 separates the zero (just before it) from the value 04 after it.
• To convert into scientific notation, typically, you want one non-zero digit to the left of the decimal.
• In 0.04, the decimal point needs to be moved to the right for scientific notation.

Every movement of the decimal point is recorded in the power of ten, which we'll explore more in the following section.

Exploring Powers of Ten

The power of ten represents how many times a number should be multiplied by ten, or divided by ten if the power is negative.
This concept is key to the process of expressing numbers in scientific notation.

• In scientific notation, this is expressed as "\(10^n\)," where \(n\) is the power.
• If we move the decimal to the right, the power will be negative, indicating division by ten for each move.
• In the given example, moving the decimal two places to 4 indicates a multiplier of \(10^{-2}\).

Understanding powers of ten helps create a simplified and standardized format for representing numbers, especially very large or small ones.

Learning Exponents

Exponents in mathematical expressions indicate how many times a number is used in a multiplication.
In the context of scientific notation, it provides a simple way to handle large or small numbers.

• The exponent indicates the magnitude of the power of ten.
• For example, in the scientific notation \(4 \times 10^{-2}\), the exponent is -2.
• This means "move the decimal place 2 steps to the left," which may be seen inversely when starting from a decimal.

Exponents help to succinctly convey the scale of numbers, especially as they rapidly increase or decrease.

Decoding Coefficients

A coefficient in scientific notation is the initial number that is not multiplied by ten.
This number is simplified to a single digit followed by any necessary decimals between one and ten.

• In our exercise, the number 4 is the coefficient of \(4 \times 10^{-2}\).
• The coefficient is obtained by adjusting the decimal so that there is just one non-zero digit to the left.
• It's essentially the main factor in the scientific notation that retains its original value, with changes highlighted through the power of ten.

The coefficient's role is core to maintaining the number's accuracy while using scientific notation, enabling it to present any number within a standardized framework.