Problem 32 - Algebra for College Students

Question

Write each of the following in ordinary decimal notation. For example, \((3.18)(10)^{2}=318\). \((6)(10)^{-9}\)

Step-by-Step Solution

Answer

0.000000006

Step 1: Understand Scientific Notation

The exercise involves converting a number from scientific notation to ordinary decimal notation. In scientific notation, numbers are expressed as the product of a number and a power of ten, like \((6)(10)^{-9}\), which indicates that we need to adjust the decimal place according to the power of ten.

Step 2: Identify Parts of the Notation

In the expression \((6)(10)^{-9}\):- The number 6 is the coefficient.- The exponent \(-9\) shows the power of ten, indicating we move the decimal 9 places to the left because it's negative.

Step 3: Convert to Ordinary Decimal Notation

To convert \((6)(10)^{-9}\) to ordinary decimal notation, we start with 6. Since the exponent is \(-9\), we move the decimal point left 9 places:- Starting as 6 (written as 6.0).- Move the decimal 9 places left: 0.000000006.

Key Concepts

Decimal Notation

Decimal notation is a way of expressing numbers that uses the base-10 system. This is the most familiar way of writing numbers, where digits are placed based on their place value. In decimal notation, each digit in a number represents an integer multiplied by a power of ten:

• Units: \( 10^0 \) - or 1
• Tens: \( 10^1 \)
• Hundreds: \( 10^2 \)

When writing numbers like these, the position or place of each digit tells you the power of 10 that is multiplied by the digit. For instance, the number 345 means \(3 \times 10^{2} + 4 \times 10^{1} + 5 \times 10^{0}\).

Decimal notation is used to write down numbers as they are usually read or used, such as the number 1567.89, which in expanded form is expressed as \(1 \times 10^3 + 5 \times 10^2 + 6 \times 10^1 + 7 \times 10^0 + 8 \times 10^{-1} + 9 \times 10^{-2}\). This helps in reading and writing numbers in everyday contexts.

Powers of Ten

The concept of powers of ten is crucial in understanding both scientific and decimal notation. A power of ten is simply ten raised to an exponent, and it determines the number of zeros after the 1 or how the decimal is shifted:

• A positive exponent, like \(10^3\), means the decimal point moves three places to the right.
• A negative exponent, such as \(10^{-2}\), indicates the decimal point moves two places to the left.
• An exponent of zero, \(10^0\), equals 1 and does not move the decimal point.

For example, if you have the number \(3.5 \times 10^{4}\), it converts to 35,000 by moving the decimal four places to the right, adding zeros as each place is moved.

Hence, the power of ten is directly related to how you shift the decimal point in a number, depending on whether the exponent is positive or negative, and is integral to both scientific conversions and notations.

Negative Exponents

Negative exponents in scientific notation indicate that the number needs to be moved to the left on the decimal scale. Unlike positive exponents, which stretch the number larger by moving the decimal point to the right, negative exponents do the opposite:

• They represent fractions or parts of a whole by showing how many times the decimal point should be moved left.
• The larger the negative exponent, the smaller the number becomes.

For example, \(10^{-1}\) equals 0.1, and \(10^{-5}\) converts to 0.00001. These are fractions that showcase how moving the decimal point left reduces the size of the number.

Understanding negative exponents is essential because they frequently appear in many scientific and mathematical contexts, particularly in expressing very small measurements or values in fields like chemistry and physics. They help to handle and understand extremely small quantities clearly and concisely.