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When working with negative numbers, it's essential to remember that they behave opposite to how we typically think of positive numbers. To compare negative numbers, you need to consider them in terms of their distance from zero on a number line. The larger the negative number (in absolute value), the farther it is from zero, and thus the smaller it is when comparing it with other negative numbers.

For instance, when comparing \( -1.5 \) to \( -1.4 \) , \( -1.5 \) is smaller because it is further from zero. This concept can be counterintuitive at first, as in the positive number range, a larger number signifies being greater. Understanding the rules for comparing negative numbers is crucial when arranging them in order. Always move from the furthest away from zero to the closest when ordering from least to greatest.

Fundamentals of Square Roots

A square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 9 is 3 because \( 3 \times 3 = 9 \). Knowing how to work with square roots is essential when they appear in various math problems, including when ordering rational numbers.

Usually, square roots that are not perfect squares (where the square root isn't a whole number) will require an approximation when comparing them with other numbers. \( \sqrt{2} \) is an example of a non-perfect square. It is approximately 1.4142. When we are dealing with negative square roots, such as \( -\sqrt{2}\), we apply the same approximation method, resulting in a negative decimal, in this case, approximately -1.4142.

The decimal representation of numbers plays a crucial role when ordering rational numbers, which include fractions, integers, and irrational numbers that can all be represented by decimals. Decimals extend to the right of the decimal point and can be used to accurately compare and order numbers because they follow a uniform positional format.

To order decimals, look at the digits from left to right, starting after the decimal point. The first place is the tenths, followed by the hundredths, and so on. With negative decimals, consider the absolute values and order them as you would positive decimals, but remember: the higher the absolute value, the lower the actual negative number. For instance, \( -1.5 \) has a larger absolute value than \( -1.4 \) but is less when ordering because it is further to the left on the number line.