Problem 74 - Algebra and Trigonometry

Question

Write each number in decimal notation without the use of exponents. $$ -3.14 \times 10^{-3} $$

Step-by-Step Solution

Answer

-0.00314

Step 1: Understand the exponent

Note that the power of 10 is negative, which means that the number is less than 1. A negative exponent on 10 means that the decimal point needs to be moved to the left by the number of places equal to the exponent's absolute value, in this case, 3 places to the left.

Step 2: Move the decimal point

Starting from -3.14, move the decimal point 3 places to the left, filling in with zeros as needed. The result will be -0.00314.

Step 3: Result in decimal notation

So, -3.14 x 10^-3 in decimal notation is -0.00314.

Key Concepts

Scientific Notation

Scientific notation is a method of expressing very large or very small numbers in a compact form. This system is particularly useful in scientific and engineering contexts where such values are common. The general form of scientific notation is \( a \times 10^n \), where \( a \) is a number greater than or equal to 1 and less than 10, and \( n \) is an integer. In this notation, \( a \) is referred to as the coefficient and \( n \) as the exponent.

When converting a number from scientific to decimal notation, one should adjust the decimal point based on the exponent \( n \) of 10. If \( n \) is positive, the decimal point moves to the right, indicating a value greater than 1. Conversely, if \( n \) is negative, the decimal point moves to the left, indicating a value less than 1.

Negative Exponents

Negative exponents indicate the reciprocal of a base raised to a positive exponent. For example, \( 10^{-n} \) is the same as \( 1/(10^n) \) where \( n \) is a positive integer. This principle is crucial when converting numbers from scientific notation to decimal notation. To understand this, think of moving the decimal point in the opposite direction compared to a positive exponent. A negative exponent means that we count spaces to the left instead of to the right.

For instance, in our exercise \( -3.14 \times 10^{-3} \) implies shifting the decimal three places to the left. Each shift results in another zero being added as a placeholder. This process is vital to obtain the correct decimal equivalent of a scientifically notated number with a negative exponent.

Decimal Notation

Decimal notation is the regular form of a number with digits and decimal points that we use in everyday life. It represents the whole numbers, as well as the fractional parts, without the use of exponents. The main goal when converting scientific notation to decimal notation is to expand the number so it's readable and understandable without any knowledge of scientific notation.

When the conversion task involves negative exponents, such as our example \( -3.14 \times 10^{-3} \) it's important to ensure that the decimal point is accurately placed. After moving the decimal point three spaces to the left, zeros are added as placeholders to fill in the gaps. This process yields \( -0.00314 \) in decimal notation, which is easily understood at a glance without requiring any scientific background or calculator.