Problem 1 - Elementary and Intermediate Algebra

Question

Fill in the blanks. \(4.84 \times 10^{5}\) is written in __________ notation. \(484,000\) is written in ___________ notation.

Step-by-Step Solution

Answer

Scientific notation; Decimal notation.

Step 1: Understand Scientific Notation

Scientific notation is a way of expressing numbers that are too big or too small to be conveniently written in decimal form. It is written as the product of a number (usually between 1 and 10) and a power of ten. For example, a number like \(4.84 \times 10^{5}\) is written in scientific notation.

Step 2: Convert from Scientific to Decimal Notation

Decimal notation is the standard way of writing numbers using the base 10 system. To convert \(4.84 \times 10^{5}\) from scientific notation to decimal notation, multiply \(4.84\) by \(10^{5}\). This results in moving the decimal point 5 places to the right, giving the number \(484,000\).

Step 3: Identify Decimal Notation

Decimal notation is a way of writing numbers without exponents, simply using base 10 digits. The number \(484,000\) is in decimal notation because it is expressed as a regular whole number without any exponents.

Key Concepts

Decimal Notation

Decimal notation is the most common way of expressing numbers in our everyday life. It uses a base 10 system, meaning each position in a number represents a power of ten. This allows for straightforward counting and understanding of numbers.
For example, the number 484,000 is written in decimal notation. It does not use exponents and is composed of plain digits arranged in a particular order. Each position corresponds to increasing powers of ten:

• '4' in the hundred-thousand place (4 x 100,000)
• '8' in the ten-thousand place (8 x 10,000)
• '4' in the thousand place (4 x 1,000)
• And so on.

Each digit's position helps clearly define the size of the number, making decimal notation intuitive and accessible for all numerical representations.

Exponents

Exponents are a powerful mathematical shorthand used to express repeated multiplication of the same number. This becomes especially helpful when dealing with very large or very small numbers.
In basic terms, an exponent tells you how many times you need to multiply the base number by itself. For example, in the expression \(10^5\), the number 10 is the base, and 5 is the exponent, meaning you should multiply 10 five times:

• 10 x 10 x 10 x 10 x 10 = 100,000

Exponents reduce complexity and make it easier to handle large calculations, particularly when using scientific notation. In scientific notation, numbers are expressed as a product of a number (between 1 and 10) and an exponential expression, such as \(4.84 \times 10^5\). This method simplifies the representation and manipulation of numbers that would otherwise be cumbersome to write out fully.

Base 10 System

The base 10 system, also known as the decimal system, is the foundation of most number systems used today. It consists of ten digits: 0 through 9. Every number is built through combinations of these digits, positioned differently based on powers of ten.
This positional notation system means that the value of a digit is determined by both the digit itself and its position within the number. For example, the number 563 can be broken down as:

• 5 hundreds (5 x 100)
• 6 tens (6 x 10)
• 3 units (3 x 1)

The base 10 system is intuitive because it corresponds to how we naturally count using our ten fingers. It simplifies arithmetic operations, such as addition, subtraction, multiplication, and division, the main reason why it's universally adopted in daily life and most educational systems.