Problem 23 - Thinking Mathematically

Question

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

Step-by-Step Solution

Answer

The Hindu-Arabic numeral equivalent of the given expanded form is 73.

Step 1: Process the first term

Firstly, \(7 \times 10^{1}\) is equivalent to 70. In Hindu-Arabic numerals, this stands for 'Seventy'.

Step 2: Process the second term

Then, \(3 \times 1\) equals to 3. In Hindu-Arabic numerals, this means 'Three'.

Step 3: Combine the results

Combine the processed terms, 70 + 3 = 73. So, the resultant Hindu-Arabic numeral is 73.

Key Concepts

Expanded Form

In mathematics, the expanded form of a number expresses each digit's contribution to the overall number based on its position. It breaks down the number into an addition of its parts. For instance, the number 73 in expanded form is such that each digit is represented by its actual value and the contribution of its position. Here’s what it looks like:

• The tens place is represented as \(7 \times 10^1\), which equals 70, since the digit 7 is in the tens place.
• The ones place is represented as \(3 \times 1\), which equals 3, since the digit 3 is in the ones place.

When we add these together, it gives us the original number: 73. This helps in understanding how the value of each digit contributes to the entire number when expressed in the Hindu-Arabic numeral system.

Base-10 System

The base-10 system, also known as the decimal system, is the foundation of our number system. It is called base-10 because it uses ten digits, from 0 to 9. Each position in a number has a place value that is a power of ten. For example, in the number 73:

• The digit 7 is in the tens place, which is \(10^1\) or 10.
• The digit 3 is in the ones place, which is \(10^0\) or 1.

The base-10 system is intuitive for most people, as it aligns with our ten fingers, making calculations easier to understand and perform. This is why the base-10 system is the most widely used numeral system across the world.

Numeral Conversion

Numeral conversion involves changing numbers expressed in one numeral system to another. For the majority of daily applications, understanding how to express numbers in the Hindu-Arabic numeral system from expanded form is crucial. This process helps in:

• Recognizing what each part of an expanded expression represents.
• Combining these values accurately to find the original number.
• Understanding how numbers work in a more abstract sense, paving the way for learning more complex mathematical concepts.

For example, converting \((7 \times 10^1) + (3 \times 1)\) into the Hindu-Arabic numeral 73 reinforces the connectivity between numerical expression and numerical value.

Mathematics Education

Mathematics education plays a crucial role in helping students understand numerical concepts that are foundational to daily life. Learning about expanded forms, the base-10 system, and numeral conversion is essential for developing number sense. Education encourages:

• Critical thinking skills to solve mathematical problems and puzzles.
• Developing proficiency in arithmetic operations and understanding numerical relationships.
• Enhancing logical reasoning capabilities and fostering a love for mathematical sciences.

Using the Hindu-Arabic numeral system as a teaching tool enables students to easily transition from simple arithmetic to more complex mathematical operations, integrating these concepts into real-world applications effectively.