Problem 37 - Prealgebra and Introductory Algebra

Question

What is twice \(-20 ?\)

Step-by-Step Solution

Answer

-40

Step 1: Identify the multiplication

Understand that 'twice' means 'two times'. Therefore, twice -20 can be written as '2 * -20'.

Step 2: Perform the multiplication

Multiply 2 by -20. Remember the rule that multiplying two numbers with different signs (one positive and one negative) results in a negative number.

Key Concepts

Multiplying Integers

When it comes to multiplying integers, the process is the same as with positive numbers, but with an additional rule to keep in mind concerning the signs. If you're multiplying two integers with the same sign, whether they are both positive or both negative, the result is always positive. Conversely, if the two integers have different signs, one being positive and the other negative, the result will be negative.

For example, when multiplying \(2 \times -20\), you have a positive integer 2 and a negative integer -20. According to our rule, since the signs are different, the result will be negative. Thus, \(2 \times -20 = -40\). This concept is vital in understanding mathematical operations and is frequently used across various topics in mathematics, making it a foundational element in learning algebra.

Basic Algebra

Diving into the world of basic algebra, one encounters operations that often involve letters, known as variables, alongside numbers. However, before getting to variables, it's crucial to be comfortable with the numerical operations such as addition, subtraction, multiplication, and division.

Understanding how to multiply integers, as described above, helps in solving algebraic equations. For instance, knowing that multiplying a positive number by a negative number yields a negative result is essential. When you come across an algebraic expression such as \(x \times -4\), and you know that \(x = 5\), applying the rule quickly tells you that the result is \(5 \times -4 = -20\). Basic algebra lays the foundation for more complex mathematical concepts, helping students unlock the ability to solve more elaborate equations and understand how numbers interact within equations.

Negative Numbers

The concept of negative numbers might seem challenging at first, but it's quite intuitive once you get the hang of it. Negative numbers are located to the left of zero on the number line and represent values less than zero. These numbers are essential when it comes to expressing values below a certain point, like temperatures below freezing, elevations below sea level, or debts.

Understanding Multiplication with Negative Numbers

When multiplying a negative number by a positive number, the rules of integer multiplication apply. In our original problem, \(2 \times -20\), the positive integer indicates the 'quantity' being taken, and the negative sign indicates that the 'direction' or 'quality' of the number is opposite that of a positive number. Hence, even if you took 'twice' of something, if it's 'twice' a negative, the result is still negative. Recognizing how negative numbers behave in mathematical operations is a fundamental part of grasping more extensive concepts of mathematics and applying them to various settings in both academic and real-world scenarios.