Problem 21 - Elementary Algebra

Question

For the following problems, draw a number line that extends from -3 to 3\. Locate each real number on the number line by placing a point (closed circle) at its approximate location. $$1 \frac{1}{2}$$

Step-by-Step Solution

Answer

Answer: The real number $$1 \frac{1}{2}$$ is located between 1 and 2 on the number line.

Step 1: Draw a Number Line

Firstly, we have to draw a number line that ranges from -3 to 3. Make sure to include the 0 in the middle and mark each whole number with a tick.

Step 2: Convert Mixed Fraction to Proper Fraction

The given real number is a mixed fraction: $$1 \frac{1}{2}$$. We can convert this mixed fraction into a proper fraction by multiplying the whole number (1) by the denominator (2) and adding the numerator (1). In this case, it would be: $$1 * 2 + 1 = 3$$ This makes the proper fraction as $$\frac{3}{2}$$.

Step 3: Identify the Location for the Real Number

Now, we need to identify the correct position for the given real number, which is $$\frac{3}{2}$$. Since 1.5 (or $$\frac{3}{2}$$) is between 1 and 2 on the number line, locate the spot between those two whole numbers to place the point.

Step 4: Place a Point (Closed Circle) at the Approximate Location

Finally, place a point or closed circle at the identified location for the real number $$1 \frac{1}{2}$$ or $$\frac{3}{2}$$ on the number line, which is between 1 and 2.

Key Concepts

Mixed Fraction

A mixed fraction combines a whole number with a proper fraction. It's written with the whole number followed by the fraction, such as \( 1 \frac{1}{2} \). Understanding mixed fractions is crucial when it comes to placing them on a number line, as they represent a value greater than '1'. To visualize a mixed fraction on a number line, one needs to first recognize that it exists beyond the first whole number and is part way to the next whole number. For example, \( 1 \frac{1}{2} \) indicates one whole plus half of another whole.

Proper Fraction

Proper fractions are a type of fraction where the numerator (the top number) is less than the denominator (the bottom number), such as \( \frac{1}{2} \). Proper fractions represent numbers between 0 and 1 on a number line. This means that any proper fraction can be placed between the 0 and the 1 on the number line, illustrating that the value is less than a whole but more than nothing. Identifying and converting mixed fractions into proper fractions, if needed, helps in precision when representing these numbers on a number line.

Locating Numbers on Number Line

A number line is a visual representation that helps in understanding the relative positions of numbers. Locating a real number on a number line involves a few key steps. First, determine the integers the number falls between. Then, if the number is not a whole number, it's essential to estimate its exact position based on its fractional value. For instance, to place \( 1 \frac{1}{2} \) on a number line, acknowledge that it falls between 1 and 2, and then mark it halfway between these integers, because a half represents a midway point. This process of estimating and marking positions aids students in concretely grasping the concept of number magnitude and ordering.

Representing Real Numbers

Real numbers include all rational numbers, such as integers, decimals, and fractions. When representing real numbers on a number line, it's important to understand the full scope of number types and their positions relative to one another. For example, rational numbers like \( \frac{3}{2} \) can be easily represented after they are converted into a decimal or a proper fraction. Although some real numbers like irrational numbers cannot be perfectly placed on the number line due to their non-repeating and non-terminating nature, estimations are used. By representing various types of real numbers on a number line, one can visualize and comprehend the vast continuum that is the real number system.