Problem 71 - College Algebra

Question

Write each number in decimal notation. $$ 3.18 \times 10^{-6} $$

Step-by-Step Solution

Answer

The number \(3.18 \times 10^{-6}\) in decimal notation is 0.00000318

Step 1 - Understand the scientific notation

Scientific notation is a way of writing numbers that are too big or too small to be conveniently written in decimal form. It has the general format \(a \times 10^n\), where \(a\) is a number greater than or equal to 1 and less than 10, and \(n\) is an integer. The number \(3.18 \times 10^{-6}\) indicates that the decimal point in 3.18 should be moved 6 places to the left.

Step 2 - Convert scientific notation to decimal notation

To convert the scientific notation to decimal notation, start by writing down 3.18 and moving the decimal point 6 places to left, adding zeros as necessary. The conversion thus becomes 0.00000318.

Key Concepts

Decimal Notation

Decimal notation is a standard way of representing numbers where values are based on units of ten. For instance, the number 123.45 is in decimal notation with 123 being the whole number part and .45 representing the fractional part. Each position to the left or right of the decimal point signifies a power of ten, either positive or negative.
When converting scientific notation to decimal notation, such as with the number provided in the exercise, \(3.18 \times 10^{-6}\), we're dealing with a number much smaller than one, as indicated by the negative exponent.
To successfully convert it, imagine the decimal point starting at its original place directly after the number 3 in 3.18. As per the exponent \(10^{-6}\), you move the decimal point six places to the left. Since there are not enough digits before the 3 to move the decimal six places, you add zeros, resulting in the appropriately converted number, 0.00000318.

Exponential Notation

Exponential notation, commonly referred to as scientific notation, is a concise way to express very large or very small numbers. It is written as a product of two parts: a coefficient and a power of ten. The general form is \(a \times 10^n\), where \(a\) is any real number, often between 1 and 10, and \(n\) is an integer called the exponent.
In the given example, \(3.18 \times 10^{-6}\), \(3.18\) is the coefficient and \(10^{-6}\) indicates the decimal point will move six places to the left, as the negative sign of the exponent implies.
It's vital to understand this notation because it simplifies the writing and calculation of extremely precise numbers, which is especially useful in fields like science and engineering where such quantities often occur.

Algebraic Expressions

An algebraic expression is a mathematical phrase that includes numbers, variables, and operation symbols. These expressions can represent quantities in a variety of a real-world context or purely within mathematical reasoning.
In the realm of scientific notation, an algebraic expression can embody the relationship between the coefficient and the power of ten. For example, the scientific notation \(3.18 \times 10^{-6}\) can be seen as an algebraic expression. Here \(3.18\) and \(10^{-6}\) are the terms of the expression, multiplied together to represent a number.
Understanding how to manipulate these expressions is fundamental in algebra, enabling you to simplify and solve equations and problems that model real-world scenarios.