Question

On a number line show the numbers \(-\pi, 0\), \(\sqrt{2},-\sqrt{3},|-0.5|,-(3 !)\) and \(\frac{11}{19}\)

Short answer

Short Answer: To represent the given numbers \(-\pi, 0, \sqrt{2}, -\sqrt{3}, |{-0.5}|, -(3!), \frac{11}{19}\) on the number line, we need to understand the values these numbers represent. The approximate values are \(-3.14, 0, 1.41, -1.73, 0.5, -6,\) and \(0.58\), respectively. Draw a horizontal number line with equal intervals representing increments of 1 and place a mark for 0. Place the positive numbers to the right of 0 and the negative numbers to the left. Locate and label the approximate positions of the given numbers based on their values. Once all numbers are properly placed and labeled, the number line representation is complete.

Step-by-step solution

Understand the given numbers

We are given seven numbers: \(-\pi, 0, \sqrt{2}, -\sqrt{3}, |{-0.5}|, -(3!), \frac{11}{19}\). We need to understand what each number represents and where it would be located on the number line. 1. \(-\pi\): This is a negative number, and its value is roughly \(-3.14\). 2. \(0\): This is the neutral value on the number line. 3. \(\sqrt{2}\): This is a positive number, and its value is roughly \(1.41\). 4. \(-\sqrt{3}\): A negative number with an approximate value of \(-1.73\). 5. \(|{-0.5}|\): This represents the absolute value of \(-0.5\), which gives \(0.5\). 6. \(-(3!)\): We first find the factorial of \(3\) (\(3! = 3 \times 2 \times1\)), and then take its negation. The result is \(-6\). 7. \(\frac{11}{19}\): This is a fraction, and its value is roughly \(0.58\).

Draw the number line

Now that we have all the numbers and their values, we can draw a number line. Start by drawing a straight horizontal line and marking the position of \(0\). From there, mark off equal intervals on each side to represent increments of 1. Be sure to label the positive numbers to the right of \(0\) and the negative numbers to the left of \(0\).

Place the numbers on the number line

Once the number line is drawn, it's time to place the given numbers on it. Locate and label the approximate position of each number on the number line: 1. \(-\pi\): Position it between \(-3\) and \(-4\) (closer to \(-3\)). 2. \(0\): Mark the \(0\) on the number line. 3. \(\sqrt{2}\): Place this number between \(1\) and \(2\) (closer to \(1\)). 4. \(-\sqrt{3}\): Position it between \(-1\) and \(-2\) (closer to \(-2\)). 5. \(|{-0.5}|\): Mark \(0.5\) on the number line between \(0\) and \(1\). 6. \(-(3!)\): Place this number at \(-6\) on the number line. 7. \(\frac{11}{19}\): This is a positive number between \(0\) and \(1\), close to \(0.6\). Once all numbers are properly placed and labeled, the number line representation is complete.

Key concepts

Number Line

A number line is a straight, horizontal line that represents the set of all real numbers. It serves as a visual aid for understanding and organizing numbers, making it easier to see their relative position and size. The center point of the number line is usually marked as zero.

• Positive numbers are placed to the right of zero.
• Negative numbers are placed to the left.
• Intervals between numbers are typically equidistant.

On a number line, each point corresponds to a real number, and the distance between points represents the difference in their value. For example, the number distance from any positive or negative whole number to zero is equal to the number itself. When placing numbers such as de \(-\pi\), \(0\), or \(-\sqrt{3}\), onto a number line, it helps to estimate the approximate position by considering their decimal approximations for precise placement.

Absolute Value

The absolute value of a number is its distance from zero on the number line, disregarding whether it is positive or negative. It is usually denoted by two vertical bars. For any real number, \(a\), the absolute value is written as \(|a|\).

• \( |0.5| = 0.5 \) and \( |-0.5| = 0.5 \)
• Negative values become positive, while positive values remain the same.
• Always non-negative because distance cannot be negative.

The concept of absolute value helps in identifying how far numbers are from zero, regardless of direction. For example, the absolute value of \(-0.5\) is 0.5, as it is half a unit away from zero on the number line.

Factorial

Factorial, denoted by an exclamation mark \(!\), is a mathematical operation involving the multiplication of all positive integers up to a given number. The factorial of a number \(n\) is represented as \(n!\). It's like unraveling the values in a fun math party.

• \(3! = 3 \times 2 \times 1 = 6\)
• Used in permutations, combinations, and various fields of math.
• An interesting property is that \(0! = 1\).

For a number like \(-3!\), compute \(3!\) to get 6, then add the negative sign, making it -6. This shows how factorials can be applied even in operations involving negative values.

Square Root

The square root of a number is a value that, when multiplied by itself, gives the original number. It's an intriguing way of 'un-multiplying' numbers.
For a non-negative number \(a\), the square root is denoted as \(\sqrt{a}\).

• \(\sqrt{4} = 2\), since \(2 \times 2 = 4\).
• Approximately, \(\sqrt{2} \approx 1.41\)
• \(\sqrt{3} \approx 1.73\), but \(-\sqrt{3} = -1.73\)

Square roots are crucial in various mathematical calculations, offering ways to explore areas, solve equations, and more. It's like peeking inside a number to see its simplest roots, whether positive or negative. In everyday math, they help harness the power hidden within numbers for various applications.