Problem 86 - Algebra and Trigonometry

Question

\(81-88\) Write each number in decimal notation. $$ 6 \times 10^{12} $$

Step-by-Step Solution

Answer

6000000000000

Step 1: Understand Scientific Notation

To convert a number from scientific notation to decimal notation, we need to adjust the base number by moving the decimal point according to the power of 10.

Step 2: Identify the Base Number

The base number in the scientific notation \(6 \times 10^{12}\) is \(6\).

Step 3: Move the Decimal Point

Because the exponent is \(12\), we move the decimal point of \(6.0\) twelve places to the right to convert it into a standard form.

Step 4: Write in Decimal Notation

Moving the decimal point to the right 12 times gives us \(6000000000000\). This is the number in decimal notation.

Key Concepts

Decimal Notation

Decimal notation refers to representing numbers using the base-10 numeral system, which is the standard system for denoting integer and non-integer numbers. In this system, each number is composed of digits ranging from 0 to 9.
This is how we most commonly see numbers in everyday usage, such as when counting objects or expressing measurements. For instance, the number 3,567 represents three thousand five hundred sixty-seven in decimal notation.
To convert from scientific notation to decimal notation, we often manipulate the position of the decimal point depending on the power of ten associated with the number. This means moving the decimal point to the right for positive exponents and to the left for negative exponents.

• The starting point is the base number before the multiplication operation.
• The direction and the number of positions to move the decimal point are determined by the exponent of ten.
• A positive exponent indicates moving the decimal point to the right, thereby increasing the value in decimal notation.
• A negative exponent indicates moving the decimal point to the left, reducing the number's size.

Understanding how to switch back and forth between these notations is crucial for various mathematical and scientific calculations.

Exponentiation

Exponentiation is a mathematical operation where a number, referred to as the base, is multiplied by itself a certain number of times, indicated by an exponent. For example, when faced with the expression \(6 \times 10^{12}\),everything revolves around the exponent of 12.
Here, the base (10) raised to the power of 12 means we multiply 10 by itself 12 times.

• The term 'exponent' signifies the number of times the base is used as a multiplier in the operation.
• An expression like \(10^{12}\) symbolizes a long multiplication, 10 multiplied by itself repeatedly, resulting in a very large number.
• Exponentiation is a fundamental concept in scientific notation, simplifying the representation of particularly large or small numbers.

This is especially handy in fields like science and engineering, where one faces extremely large or minuscule figures regularly. When you convert from scientific to decimal notation, the exponent can seem daunting, but it ultimately guides only how many places you'll move the decimal point.

Place Value System

The place value system is how the value of digits in a number is determined by their position. Each digit in a base-10 numeral has a value based on its place, delineating hundreds, tens, units, etc.
The number 6000000000000 is an outcome of applying the place value system to \(6 \times 10^{12}\).Here, 6 takes the place value related to trillions because the exponent moves the decimal point to the right 12 times, thus making it 6000000000000.

• This system relies on the base-10 understanding, where each column corresponds to a power of 10.
• The rightmost digit holds a place value of \(10^0\) or 1, the next left is \(10^1\), and so on.
• For very large numbers, such as those derived from scientific notation, this system is critical to understanding and ensures correct magnitude representation.

Grasping place value is fundamental when translating numbers from scientific notation back to decimal notation, accurately reflecting their scope and ensuring calculations remain correct in practical applications.