Problem 56 - Prealgebra and Introductory Algebra

Question

Write the expression in words. $$-(-d)$$

Step-by-Step Solution

Answer

The word expression for '-(-d)' is 'the opposite of the negative of d', 'negative negative d' or 'it negates the negative of d'.

Step 1: Identify the mathematical operation

The mathematical expression involves a negative sign before parentheses containing another negative sign and a variable 'd'. This is a double negation of the variable 'd'.

Step 2: Translate to English

Now we'll translate our mathematical expression into English. The negative sign in mathematics is often expressed as 'negative' when spoken. Hence, 'negative of something' implies the negation of that something. When we have 'negative of negative something', it is interpreted as 'the opposite of the negative of something'. Therefore, '-(-d)' can be translated into words as 'the opposite of the negative of d' or 'negative negative d'. It's also correct to say 'it negates the negative of d'.

Key Concepts

Double Negation in Algebra

When you encounter mathematical expressions like \(-(-d)\), you're looking at an example of double negation. In algebra, negation refers to the operation of multiplying by \(-1\), and it essentially changes the sign of the number, variable, or expression it is applied to. Double negation, therefore, is when this operation is applied twice in succession.

Imagine you're holding a balloon. If someone tells you to 'let go,' and then immediately says 'don't let go,' you're back to holding the balloon as if nothing changed. Similarly, in algebra, when you negate a negation, it's like doing nothing at all – you end up with the original value. Mathematically, this is because multiplying by \(-1\) twice is the same as multiplying by \(+1\), or simply said, \(-1 \times -1 = 1\). So, \(-(-d)\) simplifies to \(d\), revealing that double negation restores the original term.

Mathematical Notation

Mathematical notation is a language of its own, used to communicate complex ideas succinctly and unambiguously. The notation \(-(-d)\) is made up of symbols: the minus sign \(-\), parentheses \(( )\), and the variable \(d\). Each symbol has a specific meaning.

• The minus sign indicates negation or subtraction.
• Parentheses group terms and operations, showing what should be computed first.
• Variables like \(d\) represent numbers or quantities that can change.

Understanding mathematical notation is crucial for translating expressions into verbal language and for overall mathematical comprehension.

Expressing Algebraic Terms Verbally

Communicating algebraic expressions like \(-(-d)\) verbally can be as much of an art as it is a science. To express \(-(-d)\), you could say 'negative negative d' or 'the opposite of the negative of d.' But, the art of communication also demands clarity – so it could be even better to explain that it means 'the negation of the negation of d' or simply 'positive d' since the two negatives cancel out.

When expressing algebraic terms verbally, try to keep the language simple and the concepts clear. This not only helps in understanding but also in identifying possible simplifications that could lead to solving algebra problems more efficiently.