Q. A positive minus a negative equals.

Q: What is the rule for "a positive minus a negative"?

The rule is a-(-b)=a+b. Subtracting a negative is the same as adding its positive because -(-b)=b.

Q: Give a simple numeric example.

For example 5-(-3)=5+3=8. You turn the minus negative into plus positive.

Q: Why does subtracting a negative become addition?

Because subtraction means adding the opposite: a-b=a+(-b). If b is negative, -b is positive, so you add.

Q: How can I visualize this on a number line?

Start at a. Subtracting -b means move right b units (since -b is to the left, its opposite sends you right), so you end at a+b.

Q: How does this work with algebraic expressions?

Replace subtraction with addition of the opposite: x-(-y)=x+(-(-y))=x+y. Simplify -(-y)=y.

Q: What about negative minus negative, e.g., -2 - (-5)?

What about negative minus negative, e.g., -2 - (-5)?

Q: Is this related to multiplying by -1?

Yes. -(-b)=(-1)\cdot(-1)\cdot b=1\cdot b=b. Two negatives multiply to a positive, so subtracting a negative adds.

Answer

Subtracting a negative is the same as adding its positive, because \(-(-b)=b\). Hence \(
a-(-b)=a+b
\). Example: \(
5-(-3)=8
\).

Detailed Explanation

Problem. What does a positive minus a negative equal?

Introduce symbols and clarify signs.

Let the positive number be \(p\) with \(p>0\). Let the negative number be written as \(-q\) where \(q>0\). Writing the negative number as \(-q\) makes explicit that its absolute value is \(q\).
Write the subtraction to be evaluated.

The expression “a positive minus a negative” becomes

\(p – (-q).\)
Use the rule for subtraction: subtracting a number is adding its opposite.

By the definition of subtraction and additive inverses, subtracting a number is the same as adding its opposite. The opposite (additive inverse) of \(-q\) is \(q\). Therefore

\(p – (-q) = p + q.\)

Here we used the identity \( -(-q) = q\), which holds for every real number \(q\).
Interpretation and sign result.

Since \(p>0\) and \(q>0\), their sum \(p+q\) is positive. So the result of a positive minus a negative is a positive number equal to the sum of the two absolute values.
Numerical example (illustration).

If \(p=5\) and the negative number is \(-3\) (so \(q=3\)), then

\(5 – (-3) = 5 + 3 = 8.\)

Final answer. A positive minus a negative equals the sum of the positive and the absolute value of the negative; algebraically, \(p – (-q) = p + q\), which is positive when \(p>0\) and \(q>0\).

See full solution

Simplify every task with Edubrain help

AI homework helper

FAQs

What is the rule for "a positive minus a negative"?

The rule is \(a-(-b)=a+b\). Subtracting a negative is the same as adding its positive because \(-(-b)=b\).

Give a simple numeric example.

For example \(5-(-3)=5+3=8\). You turn the minus negative into plus positive.

Why does subtracting a negative become addition?

Because subtraction means adding the opposite: \(a-b=a+(-b)\). If \(b\) is negative, \(-b\) is positive, so you add.

How can I visualize this on a number line?

Start at \(a\). Subtracting \(-b\) means move right \(b\) units (since \(-b\) is to the left, its opposite sends you right), so you end at \(a+b\).

How does this work with algebraic expressions?

Replace subtraction with addition of the opposite: \(x-(-y)=x+(-(-y))=x+y\). Simplify \(-(-y)=y\).

What about negative minus negative, e.g., \(-2 - (-5)\)?

Is this related to multiplying by \(-1\)?

Yes. \(-(-b)=(-1)\cdot(-1)\cdot b=1\cdot b=b\). Two negatives multiply to a positive, so subtracting a negative adds.

Any quick mnemonic to remember this?

Think "minus a negative = plus": two negatives cancel. Or convert subtraction to addition of the opposite and simplify signs.

A positive minus a negative adds up.
Try our tools below.

185,791+ happy customers
Math, Calculus, Geometry, etc.

Quadratic Formula Calculator Math AI Calculus AI