Problem 27 - Thinking Mathematically

Question

Express each expanded form as a Hindu-Arabic numeral. \(\begin{aligned}\left(5 \times 10^{5}\right)+\left(2 \times 10^{4}\right)+(8&\left.\times 10^{3}\right)+\left(7 \times 10^{2}\right) \\\ &+\left(4 \times 10^{1}\right)+(3 \times 1) \end{aligned}\)

Step-by-Step Solution

Answer

The expanded form corresponds to the Hindu-Arabic numeral 528743.

Step 1: Expand each term

Begin by calculating the value of each term. This is done by multiplying the number by ten raised to the exponent. Here are the results: \(5 \times 10^{5} = 500000\),\(2 \times 10^{4} = 20000\),\(8 \times 10^{3} = 8000\),\(7 \times 10^{2} = 700\),\(4 \times 10^{1} = 40\),\(3 \times 1 = 3\).

Step 2: Add the values

Next, add all of the calculated numbers together. This involves adding \(500000 + 20000 + 8000 + 700 + 40 + 3\).

Step 3: Calculate the sum

The sum of all these numbers is calculated to be 528743.

Step 4: Express in Hindu-Arabic numeral

Finally, express this sum as a Hindu-Arabic numeral as 528743.

Key Concepts

Understanding Mathematics

Mathematics is the language of numbers and logic. It helps us solve problems in everyday life. When we talk about mathematics, we can think of it as a toolkit. It's filled with principles and methods.
These include arithmetic, algebra, geometry, and more. In this exercise, we explore one of these principles called the expanded form.

• Mathematics allows us to represent numbers in different ways.
• It helps us understand the value and position of numbers.
• Mathematical expressions can simplify complex ideas.

Recognizing different ways to express numbers, such as in expanded form, is crucial. It offers a deeper grasp of how numbers work together.

Exploring Expanded Form

Expanded form breaks down a number to show the value of each digit. It expresses the number as a sum of each digit multiplied by its matching position value.
For example, the number 528743 in expanded form is represented as:

• \[(5 \times 10^5) + (2 \times 10^4) + (8 \times 10^3) + (7 \times 10^2) + (4 \times 10^1) + (3 \times 1)\]

Expanded form is a useful tool in mathematics.

• It shows how each digit contributes to the whole number.
• It reveals the base-10 system we use in our number system.

Using expanded form allows us to understand the importance of each digit's place.

Importance of Place Value

Place value is essential in understanding numbers. It's what tells us how much each digit in a number is worth.
Each position in a number corresponds to a power of ten. For instance, in 528743, the place value of the digit '5' is linked to \(10^5\) or a hundred thousand.

• '2' in the ten-thousands place reflects \(2 \times 10^4\) or twenty thousand.
• '8' in the thousands place shows \(8 \times 10^3\) or eight thousand, and so forth.

Place value helps us in:

• Determining the size of a number.
• Performing arithmetic calculations correctly.
• Understanding the difference between numbers like 50 and 5000.

Without place value, numbers would lose context and meaning.

Understanding Numerical Expressions

Numerical expressions use numbers and operations to represent a quantity. They can be as simple as a single number or as complex as an equation.
In our exercise, we use a numerical expression to show how the expanded form translates to a Hindu-Arabic numeral. Here,\[(5 \times 10^5) + (2 \times 10^4) + (8 \times 10^3) + (7 \times 10^2) + (4 \times 10^1) + (3 \times 1)\] becomes 528743.

• Numerical expressions can be calculated using operations like addition, subtraction, multiplication, and division.
• They help express large numbers in a simplified form.

Mastering these expressions aids in solving equations and understanding mathematical concepts better. They act as a bridge from basic arithmetic to advanced mathematics.