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Understanding place value is fundamental in learning mathematics, especially when dealing with numbers. Essentially, place value refers to the value of a digit based on its position within a number. Each position in a number has a different value, and these values are powers of 10 in the base-ten number system, which is the system we use for most mathematical calculations.

For example, the number 34,569 can be broken down into its place values: the '3' is in the ten thousands place, the '4' is in the thousands place, the '5' is in the hundreds place, the '6' is in the tens place, and the '9' is in the ones place. Each position represents a value that is ten times greater than the position to its right. The knowledge of place value helps in operations such as addition, subtraction, and especially when writing numbers in expanded form, which breaks down a number into each of its place value components.

The system that we use today for writing numbers is the Hindu-Arabic numeral system. It's a decimal (base-ten) numeral system that was developed by Indian mathematicians around the 6th century and later transmitted to the Islamic world, and finally to Europe. This system uses ten symbols: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9, called digits.

These digits are combined to form all other numbers. The beauty of the Hindu-Arabic numeral system lies in its use of place value and the concept of zero, which allows for the expression of very large and very small numbers efficiently and clearly. This numeral system is universally recognized, which makes it an essential tool for global communication in math and science.

Mathematical notation consists of symbols and numbers arranged in a manner that conveys mathematical operations, relationships, and quantities. This precise language is used to perform calculations and to solve problems. When we express numbers in expanded form, we use mathematical notation to break down and simplify numbers based on place values.

For instance, the number 34,569 in expanded form is written as \(30000 + 4000 + 500 + 60 + 9\). In this notation, the plus sign indicates addition, and each term represents a component of the original number according to its place value. Good mathematical notation conveys a lot of information in a compact form and is essential for clear communication of mathematical concepts.