Problem 30 - Fundamentals of Mathematics

Question

Write each expression in words. $$ -(-(-3)) $$

Step-by-Step Solution

Answer

Negative three.

Step 1: Understanding the Expression

The given expression is \(-(-(-3))\). It contains three negative signs and a number. Our task is to simplify this expression step by step.

Step 2: Simplifying the Expression

We start with the innermost part of the expression: - Start with the number inside: \((-3)\) which is 'negative three'.- Apply the first negative sign: The expression \(-(-3)\) becomes 'the opposite of negative three', which simplifies to 'three'.

Step 3: Applying the Outer Negative Sign

Now take the result from Step 2 (three) and apply the remaining negative signs: - Apply one more negative sign: \(-3\) becomes 'the opposite of three', which is 'negative three'.- The final expression \(-(-3)\) simplifies to three. We already have one more negative left, so the expression \(-(-(-3))\) becomes 'the opposite of three', simplifying back to 'negative three'.

Step 4: Writing the Expression in Words

The simplification showed the expression \(-(-(-3))\) can be verbalized as 'the opposite of the opposite of negative three'. This ultimately simplifies back to 'negative three'.

Key Concepts

Expressions in Words

When dealing with mathematical expressions, it's often helpful to translate them into words. This process allows us to understand the core meaning behind the mathematical symbols we see. Let's consider the expression \(-(-(-3))\). At first glance, this can seem confusing with its multiple negative signs. However, when we write it in words, it becomes much clearer.

- The innermost part \((-3)\) is simply 'negative three.'- Moving outward, \(-(-3)\) translates to 'the opposite of negative three,' which simplifies to 'three.'- Finally, one more negative sign, \(-(-(-3))\), means 'the opposite of three,' reverting back to 'negative three.'

By using words, we transform the complex appearance of the expression into a more understandable narrative, making it easier to process each step logically.

Simplifying Expressions

Simplifying expressions, especially those involving negative numbers, requires a systematic approach. Let’s break down the expression \(-(-(-3))\). Each negative sign represents taking the 'opposite' of a number.

• Begin with the innermost part: \((-3)\), or 'negative three.'
• Apply the first negative sign: \(-(-3)\). The 'opposite of negative three' becomes positive 'three.'
• Apply the second negative: \(-3\). Taking the 'opposite of three' results in 'negative three.'

An important rule to remember is that applying an odd number of negatives results in a negative number, while an even number results in a positive number. As in our example, the odd count of negative signs (three) leaves us with the negative number 'negative three.'

Mathematical Operations

When faced with complex-looking expressions, understanding the basic mathematical operations is essential. In our example \(-(-(-3))\), the primary operation is applying negative signs, sometimes called 'negation'. This reverses the sign of the number each time.

Here’s the breakdown:

• Negation flips the sign of a number: negative becomes positive and vice versa.
• Multiple negations follow the rule of opposites becoming double opposites, which reverts the sign again.
• Apply negation sequentially to simplify step by step: first negate \(-3\) to get 'three', negate 'three' to get '-3'.

Understanding these principles allows you to approach any similar problems with confidence. The operation of negating is simple but powerful, as it can swiftly change the outcome of expression calculations.