Question

Find the reciprocal of the number. $$-8$$

Short answer

The reciprocal of -8 is -0.125.

Step-by-step solution

Identify the number

The number given in the problem is -8.

Find Reciprocal

The reciprocal of any number \(x\) is calculated as \(1/x\). Apply this formula here to find the reciprocal of -8. Therefore, the reciprocal of -8 is \(1/(-8) = -0.125\).

Key concepts

Reciprocal of a Number

The concept of a reciprocal is integral to various branches of mathematics including algebra and calculus. In simple terms, the reciprocal of a number refers to a value which, when multiplied by the original number, results in 1. The formula to find the reciprocal of a number 'n' is given by \(1/n\).

For instance, the reciprocal of 5 is \(1/5\) or 0.2 because \(5 \times 0.2 = 1\). It's a straightforward concept, but requires careful attention when the number is negative or a fraction. The reciprocal of a negative number will also be negative, as it retains the sign while inverting the value. This leads us to an interesting property: any number multiplied by its reciprocal gives the multiplicative identity, which is 1.

Negative Numbers

Negative numbers are an essential part of the number system, representing values less than zero. They come into play in various real-life contexts like debt, temperature below zero, or levels below sea level. In mathematics, dealing with negative numbers requires understanding their properties, especially concerning how they interact with positive numbers and other negatives.

One key aspect is that the product of two negative numbers is positive, whilst multiplying a positive number with a negative one results in a negative number. This rule also applies when finding the reciprocal of a negative number. Since the reciprocal of a positive number is also positive, when you find the reciprocal of a negative number, you maintain the negative sign, and only the absolute value of the number is reciprocated.

Arithmetic Operations

Arithmetic operations form the foundation of mathematics, involving addition, subtraction, multiplication, and division. These operations are rules we apply to numbers to perform calculations. For example, multiplication involves repeated addition, and division is essentially repeated subtraction or the method of finding out how many times one number is contained within another.

In our case of finding a reciprocal, we are concerned with multiplicative operations. The reciprocal is directly related to division since finding a reciprocal is dividing 1 by the number in question. It's crucial to recognize that division by zero is undefined in arithmetic, which means every number except zero has a reciprocal. It’s also worth noting that reciprocal is the flip side of multiplication, as the reciprocal of a reciprocal brings you back to the original number.