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Place value is crucial when comparing decimals because it helps us understand the value of each digit in a number, depending on its position. In decimals, the place immediately after the decimal point is the tenths place, followed by the hundredths place, and so on. Each position is ten times smaller than the position before it.

To grasp this concept, imagine a number written in a place value chart. The decimal point acts as a 'home base', with positions to the left indicating whole units and positions to the right indicating parts of a whole. For example, in the number 0.407, the '4' is in the tenths place, which means we have four tenths, the '0' is in the hundredths place indicating no hundredths, and the '7' is in the thousandths place, signifying seven thousandths.

The idea is to compare decimal numbers by starting with the leftmost digit and moving rightwards until a difference is found. If the first digit is the same, as in the numbers 0.047 and 0.407, both have '0' in the ones place, we look at the digit in the tenths place to distinguish which number is greater.

Decimal notation is a way to write numbers that include fractions in a simple form. It uses a decimal point to separate the whole number part from the fractional part of the number. In decimals, each fraction is a part of 10, 100, 1000, and so on. It's a base-10 numbering system which most of us are familiar with.

For example, the decimal number 0.407 represents 0 whole units plus 4/10 (four tenths) plus 0/100 (zero hundredths) plus 7/1000 (seven thousandths). Practicing decimal notation involves understanding not just the place value but also the division of a whole into parts that are powers of 10. This makes comparing the magnitude of two decimals straightforward, by looking at their place values from left to right until a difference appears.

Inequality symbols (< h4>Inequality Symbols: Understanding ) and (>) are used to compare the sizes of numbers, whether whole numbers or decimals. The symbol () points towards the smaller number and the open side faces the larger number. For decimals, after identifying which number has the greater place value digit, where the first difference occurs from left to right, we use these symbols to show their relationship.

In the given exercise, after comparing digit by digit, we found that the first number, 0.047, has zero in the tenths place while the second number, 0.407, has four in the tenths place. This tells us that 0.407 has a larger value in the tenths place, making it the greater number overall. Therefore, we place the () symbol between the two numbers like this: 0.047 < h4>Correct Symbol Placement) 0.407, effectively communicating that 0.047 is less than 0.407.