Problem 12 - Thinking Mathematically

Question

Write each Hindu-Arabic numeral in expanded form. 698

Step-by-Step Solution

Answer

The expanded form of 698 is 600 + 90 + 8.

Step 1: Identifying the Value of Each Digit

Break down the number 698 into its constituent digits and determine their respective place values. In 698, there are three digits - 6, 9, and 8. The 6 is in the hundreds place, 9 is in the tens place and 8 is in the ones place.

Step 2: Expanded Representation

Take each digit in the number and multiply it by the place value it represents. Therefore, 6 is 6*100 = 600, 9 is 9*10 = 90 and 8 is 8*1 = 8.

Step 3: Writing the Final Expanded form

The expanded form of a number is represented by adding the multiplies from Step 2. So, 698 = 600 + 90 + 8.

Key Concepts

Place Value

Understanding place value is essential when converting numbers into expanded form. Place value tells us the position of a digit in a number. It determines the value of that digit based on its position. For example, in the number 698, the digit 6 is in the hundreds place. This means it represents 6 hundreds, or 600. The digit 9, being in the tens place, signifies 9 tens, or 90. Finally, the digit 8 is in the ones place, making it simply 8.

• The hundreds place multiplies the digit by 100.
• The tens place multiplies the digit by 10.
• The ones place multiplies the digit by 1.

Thus, place value helps us understand how each digit contributes to the total value of the number.

Hindu-Arabic Numerals

The Hindu-Arabic numeral system is the number system we use daily. It is a positional numeral system, which means the position of each digit affects its value. This system includes 0-9 digits, allowing us to express any number. Because of its straightforward and efficient structure, it replaced many older systems such as Roman numerals.

• This positional system makes it easy to perform arithmetic operations.
• The numeral system is also the foundation for decimal-based systems, including our currency.
• The structure facilitates easy conversion into expanded forms, as seen with the example of 698.

The ease of using Hindu-Arabic numerals, with its simple representation of numbers, is why it is so widespread and forms the basis of modern mathematics.

Mathematical Notation

Mathematical notation is a system used to convey numerical and logical concepts. It uses symbols to express these numbers or operations concisely and understandably. When expressing numbers like 698 in expanded form, mathematical notation helps us break down complex ideas into simpler parts.
For example, the expanded form of 698 uses basic mathematical operations:

• Each digit is multiplied by its place value, shown as: \(6 \times 100 = 600\), \(9 \times 10 = 90\), \(8 \times 1 = 8\).
• The results are then summed up: \(600 + 90 + 8\).

This usage of notation makes it easy to understand how each digit plays a role through multiplication and addition. With the help of mathematical notation, even bigger or more complex numbers can be easily represented and understood in expanded form.