Problem 29 - Thinking Mathematically

Question

Express each expanded form as a Hindu-Arabic numeral. \(\left(7 \times 10^{3}\right)+\left(0 \times 10^{2}\right)+\left(0 \times 10^{1}\right)+(2 \times 1)\)

Step-by-Step Solution

Answer

The Hindu-Arabic numeral of the expanded form equation is 7002.

Step 1: Understand the Power of 10

In base 10, the multiplication by \(10^{3}\), \(10^{2}\), and \(10^{1}\) represent the thousands, hundreds, and tens places, respectively. Thus, \(7 \times 10^{3}\) represents the 'thousands' place, \(0 \times 10^{2}\) is the 'hundreds' place, \(0 \times 10^{1}\) is the 'tens' place, and \(2 \times 1\) is the 'ones' place.

Step 2: Perform the Multiplications

Now, multiply the number by the power of 10 it is associated with. So we get: \(7 \times 10^{3} = 7000\), \(0 \times 10^{2} = 0\), \(0 \times 10^{1} = 0\), and \(2 \times 1 = 2\).

Step 3: Perform the Addition

Add these values together. \(7000 + 0 + 0 + 2 = 7002\).

Step 4: Write the Final Answer

Therefore, the expanded form is equal to 7002.

Key Concepts

Expanded Form

The concept of expanded form is a method used to express numbers as the sum of their digits, each multiplied by its corresponding place value. It's like breaking down a number to show the value of each of its digits. This method helps students understand the value of each digit based on its position. For example, the number 7002 can be expanded to show the role each digit plays in creating the entire number as follows:

• 7 is in the thousands place, so it is expressed as \(7 \times 10^3\).
• 0 is in the hundreds place, contributing \(0 \times 10^2\).
• 0 is in the tens place, contributing \(0 \times 10^1\).
• 2 is in the ones place, so it appears as \(2 \times 1\).

Each part of the expanded form highlights how these positions collectively sum up to arrive at the original number, 7002.
By practicing the expanded form, students can easily understand the distribution of value within a given number, which is fundamental in arithmetic and number theory.

Base 10 System

The base 10 system, also known as the decimal system, is the most commonly used number system in the world. It utilizes ten digits, from 0 to 9, to represent all possible numbers. This system is organized around the idea of powers of 10. Each place value in a number represents a power of 10, which serves as a building block for all larger numbers.
For example, in the number 7002:

• The first place from the right is the ones or \(10^0\), contributing 2 to the total.
• The second place is the tens or \(10^1\), contributing 0.
• The third place is the hundreds or \(10^2\), also contributing 0.
• The fourth place is the thousands or \(10^3\), contributing 7000.

Understanding the base 10 system is crucial because it is foundational to most arithmetical operations and aligns with our everyday counting methods. It simplifies the reading, writing, and understanding of numbers.

Place Value System

The place value system is a numeral system in which the position of a digit in a number determines its value. This system efficiently manages the magnitude of numbers and can represent very large or very small numbers using limited basic elements.
Take the number 7002 as an example of the place value system:

• The digit 7 is placed in the thousands position, indicating its value is \(7 \times 1000\) or 7000.
• The digit 0 in the hundreds place indicates no value at all from that position, contributing 0 to the total.
• Another 0 in the tens place also similarly contributes nothing.
• Finally, the digit 2 in the ones place represents \(2 \times 1\), contributing 2.

The place value system allows numbers to be easily expanded or condensed in expression without changing their intrinsic value. It's a fundamental concept that underpins not only arithmetic but also the practical usage of numbers in various fields such as commerce and science.