Question

SCIENTIFIC NOTATION Rewrite the number in scientific notation. $$0.000432$$

Short answer

The scientific notation for 0.000432 is \(4.32 \times 10^{-4}\).

Step-by-step solution

Identify a and n

In the decimal \(0.000432\), the number that lies between 1 and 10 is 4.32, and the decimal place moved 4 places to the right to achieve this.

Apply Scientific Notation

The number 4.32 is represented as \(a\). Next, we record how many places the decimal point moved to set \(a\) between 1 and 10. Since the original number is a small decimal number (less than 1), the exponent \(n\) expressing the movement of the decimal will be a negative integer because we actually moved the decimal point to the right. So we have \(n = -4\). Then the number is rewritten as \(a \times 10^n\).

Final Result

The number 0.000432 into scientific notation is \(4.32 \times 10^{-4}\).

Key concepts

Exponents

Understanding exponents is crucial when working with scientific notation. At its most basic, an exponent tells us how many times to multiply a base number by itself. In scientific notation, this principle is used to simplify how we write very large or very small numbers.

Let's look at an example: If you have the number 10 raised to the power of 3, written as \(10^3\), it means that you multiply 10 by itself three times: \(10 \times 10 \times 10 = 1000\). Similarly, negative exponents, like \(10^{-3}\), indicate division by the base number repeated as many times as the value of the exponent. So, in this case, it would mean \(1 \div (10 \times 10 \times 10) = 0.001\), or in decimal notation, \(1/1000\).

When converting to scientific notation, you will move the decimal point to create a new number between 1 and 10, then use an exponent to show how many places the decimal moved and in which direction. A positive exponent shows a move to the left (for large numbers), while a negative exponent shows a move to the right (for small numbers).

Decimal Notation

Decimal notation is the standard form of writing numbers that includes a decimal point. Most numbers we deal with day-to-day are written in decimal notation. It's simple and intuitive – every place to the right of the decimal point represents a fractional part of ten.

Take for example the number \(0.5\), which is the decimal notation for the fraction \(\frac{1}{2}\). Each move to the right in decimal notation represents a 'tenths', 'hundredths', 'thousandths', etc. Moving the decimal point to the right in the number decreases its value, as each position is ten times smaller than the one preceding it. In the context of scientific notation, this decimal point movement is key to providing the exponent value necessary to accurately convey the magnitude of the number.

Algebraic Expression

An algebraic expression is a mathematical phrase that can contain ordinary numbers, exponents, and variables. It does not have an equals sign like an equation, but it can be simplified or used to represent a value in terms of variables. In the realm of scientific notation, the algebraic expression describes how we rewrite numbers.

The expression \(a \times 10^n\) is an algebraic expression where \(a\) is a number between 1 and 10 and \(n\) is the exponent indicating the number of places the decimal moved. Here, \(a\) represents the significant figures of the original number, while the exponent \(n\) tells us the scale or magnitude. Thus, scientific notation is a concise algebraic way to express very large or small numbers in a standardized format that is easy to read and understand.