Problem 26 - Thinking Mathematically

Question

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

Step-by-Step Solution

Answer

The Hindu-Arabic numeral equivalent of the given expanded form is 753.

Step 1: Carry out Multiplications

Start by performing all the individual multiplications within the brackets. This means: \(7 \times 10^{2} = 700\), \(5 \times 10^{1} = 50\), and \(3 \times 1 = 3\).

Step 2: Summing the Products

The next step is to sum up the products from step 1 to convert the expanded form into Hindu-Arabic numeral. It means: \(700 + 50 + 3 = 753\)

Key Concepts

Place Value

The Hindu-Arabic numeral system is based on place value. Each digit in a number has a specific value depending on its position in the number. For example, consider the number 753. Here, the digit 7 is in the hundreds place, the digit 5 is in the tens place, and the digit 3 is in the ones place. This means:

• The 7 indicates 700 (or 7 hundreds)
• The 5 indicates 50 (or 5 tens)
• The 3 indicates 3 (or 3 ones)

This system allows you to break down and understand numbers by evaluating the contribution of each digit based on its position.

Expanded Form

Understanding the expanded form of a number helps to emphasize its place values. In expanded form, you express a number as the sum of each digit multiplied by its respective place value. For example, the number 753 can be written in expanded form like this:

• \(7 \times 10^{2}\) - Here, 7 is multiplied by 10 squared, which equals 700.
• \(5 \times 10^{1}\) - Next, 5 is multiplied by 10 to the power of 1, equaling 50.
• \(3 \times 1\) - Finally, 3 is multiplied by 1, remaining as 3.

By summing these values, you return to the original number: \[(7 \times 10^{2}) + (5 \times 10^{1}) + (3 \times 1) = 753\] This breakdown makes it easier to understand how total values are constructed by individual digits.

Numerical Conversion

Numerical conversion involves changing a number from one form to another. For instance, translating an expanded form like \[(7 \times 10^{2}) + (5 \times 10^{1}) + (3 \times 1)\] back to its standard Hindu-Arabic numeral is an example of conversion. To perform this:

• First, complete the calculations in the expanded form. That means solving each multiplication:
  • \(7 \times 10^{2} = 700\)
  • \(5 \times 10^{1} = 50\)
  • \(3 \times 1 = 3\)
• Then, add the results of these calculations: \[700 + 50 + 3 = 753\]

By following these steps, expanded forms are successfully and accurately translated back to standard numerical representations. Understanding this process is essential for interpreting various mathematical notations effectively.