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Understanding place value is fundamental to grasping how numbers are structured and operate. In our decimal number system, each digit within a number has a different value depending on its position, or 'place', within the number. Starting from the right, the first place is the 'ones', followed by 'tens', 'hundreds', 'thousands', and so on, each place being ten times the value of the place to its right.

For example, in the numeral 9007, the '7' holds the ones place value, meaning it is worth 7 units. The '0' in the tens place and the '0' in the hundreds place don't contribute anything since zero multiplied by any value remains zero. Finally, the '9' is in the thousands place and represents a value of 9000. Breaking down numbers in this way is crucial for various arithmetic operations and for a better understanding of numbers in general.

The Hindu-Arabic numeral system is a positional decimal numeral system, and it is the most common system for the symbolic representation of numbers in the world today. It employs ten symbols: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9, known as numerals. These numerals were developed in ancient India and later transmitted to the Islamic world, and eventually Europe. What makes this system powerful is its positional notation—where the position of the numeral affects its actual value in the number. A single numeral can stand for different values depending on its place. For instance, '3' can mean thirty in 30 or three hundred in 300, entirely based on its position.

Expanded notation is a way to break down a number to show the value of each of its digits. It involves writing the number as the sum of the values of its digits. When numbers are written in expanded form, each term represents the digit times its place value.

Using our example number 9007, this process involves several steps:

• Recognizing that '9' is in the thousands place and stands for 9 times 1000 or 9000.
• Observing that the '0' in the hundreds place and '0' in tens place, both have a value of 0 (as any number times 0 equals 0).
• Finally, noting that '7' is in the ones place and is simply 7 times 1, or 7.

The entire number is then written in expanded form as 9000 + 0 + 0 + 7, which simplifies to 9000 + 7, since adding zero does not change the value. This form of notation is particularly useful in understanding the importance of each digit's place value within a number.