Скачать или смотреть Visualizing Real Number Sets: Understanding the Number Line

We use the number line to represent sets of real numbers. A set is a collection of elements that share common properties.

Let's start with the set of natural numbers, {1, 2, 3, ...}. It extends to infinity towards the right, but it has no negative values.

Next, we have the set of integers, {...,-3, -2, -1, 0, 1, 2, 3, ...}. It includes both positive and negative values, and it also extends to infinity in both directions.

Moving on to the set of rational numbers, it includes all numbers that can be expressed as a fraction, such as 1/2, 2/3, 7/4, etc. These numbers can be located on the number line between the integers, and they can be either positive or negative. It may look like that there are no gaps and the number line is complete, but this is not the case.

We need irrational numbers to fill those gaps. Irrational numbers are numbers that cannot be expressed as a fraction, such as PI and square root of 2. They are located between the rational numbers on the number line.

Finally, we have the set of real numbers, which includes all the numbers we have discussed so far. It is the most complete set of numbers.

Notice that the sets of natural, integer, rational, and irrational numbers are all subsets of the set of real numbers, which includes all possible values between negative infinity and positive infinity.

Visualizing sets of real numbers on the number line can help us better understand their properties and relationships.

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