Problem 41 - Contemporary Precalculus

Question

Draw a picture on the number line of the given interval. $$(-\infty, 0]$$

Step-by-Step Solution

Answer

Answer: To represent the interval \((-\infty, 0]\) on a number line, draw a horizontal line with 0 at the center. Mark the point for 0 and shade the interval by drawing a solid dot at 0, and then shading the line to the left of the solid dot. Add a small arrowhead pointing to the left at the end of the shaded area to indicate that the interval continues indefinitely in the negative direction.

Step 1: Draw a number line

Begin by drawing a horizontal line on your paper or canvas, which will serve as the number line, with 0 at the center.

Step 2: Mark the point for 0

Place a clearly visible mark on the number line where 0 is located, and also write the number '0' below that mark, indicating its position.

Step 3: Shade the interval

The interval is given as \((-\infty, 0]\). Since 0 is included in this interval, you should draw a solid dot at 0. This can be done by shading the circle located above the number 0 on the number line. Begin shading the line to the left of the solid dot, indicating that all numbers from 0 to negative infinity are included in the interval.

Step 4: Arrow to indicate infinity

Since the interval stretches from 0 to negative infinity, it will be helpful to use an arrow. Draw a small arrowhead pointing to the left at the end of the shaded area, which indicates that the interval continues indefinitely in the negative direction.

Step 5: Review the drawing

Take a moment to review the completed number line drawing, ensuring that it accurately represents the given interval \((-\infty, 0]\). The drawing should include a shaded interval stretching from the solid dot at 0 to the left, with an arrow at the end to indicate that it continues indefinitely.

Key Concepts

Number Line Representation

When we think about visually representing numbers, a number line is an essential concept. A number line is essentially a straight, horizontal line with numbers placed at specific intervals. The center of the number line often starts at zero, with positive numbers going to the right and negative numbers heading to the left.
A number line helps us understand the position and relationship of numbers. In this exercise, the focus is on the interval \((-\infty, 0]\), which means it includes all values from negative infinity to zero. On a number line, this interval would start from the far left, pointing towards zero, and stopping there.
To accurately draw this representation, we need a solid dot at zero to show that zero is included in the interval. The area from the dot extends to the left, indicating all negative numbers. To express that it goes indefinitely negative, we continue with an arrow pointing left.

Interval Notation

Interval notation is a simple yet powerful way to describe a set of numbers. In the expression \(( - \infty, 0 ]\), each part gives us vital information about the numbers included.

• The parentheses \(()\) indicate that the boundary, \(-\infty\), is not part of the interval.
• The square bracket \([])\) denotes that zero is indeed part of the interval, meaning the interval includes zero.

Interval notation is favored in mathematics for its clarity and conciseness. It allows us to express a range of numbers quickly. This is useful, especially when dealing with continuous data, because it minimizes confusion about inclusion.
When using interval notation with number lines, it’s crucial to match the notation with the drawing accurately. Open boundaries like \(-\infty\) will not be "closed" on the line with a dot, while closed boundaries like zero will have a solid point to show inclusion.

Infinity in Mathematics

Infinity is a unique concept in mathematics that represents something without an end. It can be difficult to fully grasp because unlike numbers we encounter every day, infinity continues forever.
In mathematical terms, infinity is not a number in the traditional sense, but rather an idea or a concept. There are two types: positive infinity \((+\infty\)) and negative infinity \((-\infty\)). Positive infinity means going forever to the right, while negative infinity implies extending infinitely left on a number line.
In the context of the given interval \((-\infty, 0]\), the presence of \(-\infty\) means the interval starts from way beyond the observable number line, continuing from the negative side, endlessly approaching zero. It reminds us that no actual endpoint exists until the zero itself.

• Infinite intervals need some visual cue, like an arrow, because finite space cannot entirely capture infinite ideas.
• The infinity symbol in our interval doesn't specify an endpoint but suggests how far left our interval travels on the line.

Understanding infinity within mathematical and visual contexts helps us model real-world phenomena that stretch beyond concrete boundaries, such as time or space.