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Chapter 1: Problem 9

Graph the numbers on a number line. Label each. $$6,-4, \frac{3}{4}, 0,-1 \frac{1}{2}$$

Short Answer

Expert verified
To graph the numbers 6, -4, \(\frac{3}{4}\), 0, and \(-1 \frac{1}{2}\) on a number line, first draw a long straight line and label the midpoint with 0. Place the positive numbers (6 and \(\frac{3}{4}\)) to the right of 0 and the negative numbers (-4 and \(-1 \frac{1}{2}\)) to the left of 0. Mark the points for each number, ensuring equal spacing between whole numbers and placing fractions and mixed numbers between whole numbers accordingly.

Step by step solution

01

Draw a number line

To begin, let's draw a long straight line using a ruler or a straightedge. This line will represent our number line. Place an arrowhead on the right end of the line to indicate that it extends indefinitely to the right for all positive numbers.
02

Mark and label 0

The next step is to mark the 0 on the number line. To do this, choose a point near the middle of the line, and label it with the number 0. When placing numbers on the number line, it's crucial to keep space for all five numbers provided in the exercise.
03

Place positive numbers on the right side

Now let's place the positive numbers on the number line. Starting from 0, move right and mark the point where the whole number 6 should be placed, trying to keep equal distances between the whole numbers 1, 2, 3, 4, 5, and 6. Label the point with the number 6. The positive fraction 3/4 should be between 0 and 1. Estimate the location from 0 that is approximately 75% of the way to 1 (as 3/4 = 0.75) and mark the point for 3/4. Label the point as \(\frac{3}{4}\).
04

Place negative numbers on the left side

Now, let's place the negative numbers on the left side. From 0, move to the left and mark the point where the whole number -4 should be placed, also keeping equal distances between the numbers -1, -2, -3, and -4. Label this point with the number -4. The negative mixed number -1 1/2 is equivalent to -1.5. This number should be between -1 and -2. Estimate the location between -1 and -2 that is halfway (as 1/2 = 0.5), mark the point, and label it as \(-1 \frac{1}{2}\).
05

Review the number line

Now that we have placed all five numbers on the number line, review it to ensure that the positive numbers are on the right of 0, while the negative numbers are on the left. Also, ensure that the distances between the whole numbers are somewhat equal and that the fractions and mixed numbers are placed correctly between the whole numbers.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Positive and Negative Numbers
Positive and negative numbers are fundamental in mathematics. They express quantities that can either add to a value or represent something less than zero. On a number line, these numbers help us visualize their position relative to 0.
  • Positive numbers are greater than zero and are usually found to the right of zero on a number line.
  • Negative numbers are less than zero and appear to the left of zero.
  • Zero itself is neutral; it's neither positive nor negative.
To graph these numbers on a number line, balance is key. Every number should maintain an equal distance from its neighbors to clearly depict its value. For instance, -4 and 6 placed on a number line show -4 on the left of 0 and 6 on the right. Understanding this visualization helps in operations like addition and subtraction where direction matters.
Graphing Numbers
Graphing numbers on a line helps us see their order and relative sizes. Here's how it works:
  • Start with drawing a straight line. This will be your number line.
  • Place zero near the center. This serves as your starting point.
  • Mark positive numbers to the right and negative numbers to the left. Each point should be equidistant from the next.
For example, placing numbers like 6, -4, \( \frac{3}{4} \), and \(-1 \frac{1}{2}\), requires estimating their exact positions. Positive fractions such as \( \frac{3}{4} \) should appear between 0 and 1 proportionally. As for \(-1 \frac{1}{2}\), place it midway between -1 and -2. Graphing helps us easily identify which numbers are greater and how far apart they are from one another.
Equivalent Fractions
Equivalent fractions are different fractions that represent the same value. They may look different but occupy the same point on a number line.
  • For example, \( \frac{1}{2} \) is the same as \( \frac{2}{4} \) or \( \frac{3}{6} \) because dividing 1 by 2 equals the same value as dividing 3 by 6.
  • On a number line, these fractions would all land on the same spot.
  • The key is making the numerator and denominator proportional so that the overall value remains constant.
When graphing fractions, it's essential to understand their equivalents, ensuring the fractions are correctly placed. This aids in comparing and calculating fractions easily, and develops a clearer concept of common denominators.
Mixed Numbers
Mixed numbers combine whole numbers with fractions. They are written in the form of a number followed by a fraction, such as \(-1 \frac{1}{2}\).
  • Mixed numbers represent quantities between whole numbers.
  • For example, \(-1 \frac{1}{2}\) reflects a point halfway between -1 and -2.
  • When converting a mixed number, subtract the whole number and add the fractional part.
On the number line, these need careful placement: find the whole number, then add the fraction's equivalent distance. Mixed numbers are useful for practical measurements, and understanding them assists in various arithmetic operations such as addition and subtraction.

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