What Is $|-1|$? □ \square □

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Introduction

In mathematics, absolute value is a fundamental concept that deals with the magnitude or size of a number, without considering its sign. It is denoted by two vertical lines on either side of the number, like this: $|x|$, where $x$ is the number inside the absolute value bars. In this article, we will explore the concept of absolute value and calculate the value of $|-1|$.

Understanding Absolute Value

Absolute value is a way to represent the distance of a number from zero on the number line. It is always non-negative, meaning it can never be negative. For example, the absolute value of $-3$ is $3$, because $-3$ is $3$ units away from zero on the number line. Similarly, the absolute value of $5$ is also $5$, because $5$ is $5$ units away from zero on the number line.

Calculating $|-1|$

Now that we understand the concept of absolute value, let's calculate the value of $|-1|$. To do this, we need to find the distance of $-1$ from zero on the number line. Since $-1$ is $1$ unit away from zero on the number line, the absolute value of $-1$ is $1$.

Properties of Absolute Value

Absolute value has several important properties that we need to understand. Here are a few:

  • Non-Negativity: The absolute value of any number is always non-negative. In other words, $|x| \geq 0$ for all $x$.
  • Identity of Indiscernibles: If $|x| = |y|$, then $x = y$ or $x = -y$. This means that if the absolute value of two numbers is equal, then the numbers themselves must be equal or negatives of each other.
  • Triangle Inequality: For any numbers $x$ and $y$, $|x + y| \leq |x| + |y|$. This means that the absolute value of the sum of two numbers is less than or equal to the sum of their absolute values.

Applications of Absolute Value

Absolute value has many applications in mathematics and real-world problems. Here are a few examples:

  • Distance: Absolute value is used to calculate distances between points on a number line or in a coordinate plane.
  • Temperature: Absolute value is used to represent temperature differences between two points.
  • Finance: Absolute value is used to calculate the magnitude of financial losses or gains.
  • Science: Absolute value is used to represent the magnitude of physical quantities such as velocity, acceleration, and force.

Conclusion

In conclusion, absolute value is a fundamental concept in mathematics that deals with the magnitude or size of a number, without considering its sign. We have calculated the value of $|-1|$ and explored the properties and applications of absolute value. Absolute value has many real-world applications and is an essential tool for problem-solving in mathematics and other fields.

Frequently Asked Questions

Q: What is the absolute value of $-5$?

A: The absolute value of $-5$ is $5$, because $-5$ is $5$ units away from zero on the number line.

Q: What is the absolute value of $0$?

A: The absolute value of $0$ is $0$, because $0$ is $0$ units away from zero on the number line.

Q: What is the absolute value of $x + y$?

A: The absolute value of $x + y$ is less than or equal to the sum of their absolute values, i.e., $|x + y| \leq |x| + |y|$.

References

  • [1] "Absolute Value" by Math Open Reference. Retrieved 2023-12-01.
  • [2] "Absolute Value" by Khan Academy. Retrieved 2023-12-01.
  • [3] "Properties of Absolute Value" by Wolfram MathWorld. Retrieved 2023-12-01.

Further Reading

  • "Introduction to Absolute Value" by IXL Math. Retrieved 2023-12-01.
  • "Absolute Value: Definition and Examples" by Mathway. Retrieved 2023-12-01.
  • "Absolute Value: Properties and Applications" by Brilliant. Retrieved 2023-12-01.

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Frequently Asked Questions

Q: What is the absolute value of a negative number?

A: The absolute value of a negative number is always positive. For example, the absolute value of $-3$ is $3$, because $-3$ is $3$ units away from zero on the number line.

Q: What is the absolute value of a positive number?

A: The absolute value of a positive number is the number itself. For example, the absolute value of $5$ is $5$, because $5$ is $5$ units away from zero on the number line.

Q: What is the absolute value of zero?

A: The absolute value of zero is zero. This is because zero is not considered to be a distance from zero on the number line.

Q: How do I calculate the absolute value of a number?

A: To calculate the absolute value of a number, you can simply remove the negative sign if it exists. For example, the absolute value of $-5$ is $5$, because you remove the negative sign to get $5$.

Q: What is the absolute value of a fraction?

A: The absolute value of a fraction is the absolute value of the numerator divided by the absolute value of the denominator. For example, the absolute value of $\frac{-3}{4}$ is $\frac{3}{4}$, because you take the absolute value of the numerator and the denominator separately.

Q: What is the absolute value of a decimal?

A: The absolute value of a decimal is the decimal itself. For example, the absolute value of $-3.5$ is $3.5$, because you simply remove the negative sign to get $3.5$.

Q: Can the absolute value of a number be negative?

A: No, the absolute value of a number can never be negative. This is because absolute value represents the distance of a number from zero on the number line, and distance is always non-negative.

Q: What is the absolute value of a complex number?

A: The absolute value of a complex number is the distance of the complex number from the origin in the complex plane. It is calculated using the formula $|a + bi| = \sqrt{a^2 + b^2}$, where $a$ and $b$ are the real and imaginary parts of the complex number.

Q: What is the absolute value of a vector?

A: The absolute value of a vector is the magnitude of the vector, which is calculated using the formula $|v| = \sqrt{x^2 + y^2 + z^2}$, where $x$, $y$, and $z$ are the components of the vector.

Absolute Value Formulas

Formula 1: Absolute Value of a Number

|x| = x$ if $x \geq 0

|x| = -x$ if $x < 0

Formula 2: Absolute Value of a Fraction

∣xy∣=∣x∣∣y∣\left|\frac{x}{y}\right| = \frac{|x|}{|y|}

Formula 3: Absolute Value of a Decimal

∣x.y∣=x.y|x.y| = x.y

Formula 4: Absolute Value of a Complex Number

∣a+bi∣=a2+b2|a + bi| = \sqrt{a^2 + b^2}

Formula 5: Absolute Value of a Vector

∣v∣=x2+y2+z2|v| = \sqrt{x^2 + y^2 + z^2}

Absolute Value Examples

Example 1: Absolute Value of a Negative Number

Find the absolute value of $-3$.

∣−3∣=3|-3| = 3

Example 2: Absolute Value of a Positive Number

Find the absolute value of $5$.

∣5∣=5|5| = 5

Example 3: Absolute Value of a Fraction

Find the absolute value of $\frac{-3}{4}$.

∣−34∣=34\left|\frac{-3}{4}\right| = \frac{3}{4}

Example 4: Absolute Value of a Decimal

Find the absolute value of $-3.5$.

∣−3.5∣=3.5|-3.5| = 3.5

Example 5: Absolute Value of a Complex Number

Find the absolute value of $3 + 4i$.

∣3+4i∣=32+42=5|3 + 4i| = \sqrt{3^2 + 4^2} = 5

Example 6: Absolute Value of a Vector

Find the absolute value of the vector $\begin{pmatrix} 3 \ 4 \ 0 \end{pmatrix}$.

∣(340)∣=32+42+02=5\left|\begin{pmatrix} 3 \\ 4 \\ 0 \end{pmatrix}\right| = \sqrt{3^2 + 4^2 + 0^2} = 5

Conclusion

In conclusion, absolute value is a fundamental concept in mathematics that deals with the magnitude or size of a number, without considering its sign. We have answered some frequently asked questions about absolute value and provided examples to illustrate the concept. Absolute value has many real-world applications and is an essential tool for problem-solving in mathematics and other fields.