Key takeaway: Integer rules are the sign patterns that tell whether the result of adding, subtracting, multiplying or dividing positive and negative whole numbers is positive or negative.
Grade 77.NS.A.1Numbers and Operations
Definition of Integer Rules
The integer rules describe what happens to the sign and size of an answer when whole numbers that may be positive, negative or zero are combined. Learning these integer rules once saves you from sketching a number line for every problem.
Think of gains and losses in a card game. Two gains make a bigger gain; a gain of 9 and a loss of 4 leave you 5 ahead. Subtraction turns into adding the opposite. With products and quotients, only the signs matter: a pair of matching signs yields a positive, a mismatched pair yields a negative. Learn the integer rules together with the order of operations, and long expressions become routine.
Formula and notation
Sign patterns for multiplying or dividing:
\[(+)(+)=+,\quad (-)(-)=+,\quad (+)(-)=-,\quad (-)(+)=-\]
Subtraction rewrites as addition: \(a – b = a + (-b)\). So \(5 – (-3) = 5 + 3 = 8\).
Key facts
- Taking away a debt of 6 is the same as gaining 6, so \(x – (-6) = x + 6\).
- A product with an even count of negative factors is positive; an odd count is negative.
- Dividing follows the multiplication sign pattern, and zero divided by a nonzero integer is zero.
- Adding opposites always gives zero, whatever their size.
Integer Rules example
Evaluate \((-8)(-5) + 36 : (-9) – (-12)\).
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Multiply: \((-8)(-5) = 40\), since two negatives give a positive.
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Divide: \(36 : (-9) = -4\), since the signs differ.
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Rewrite: \(40 + (-4) – (-12) = 40 – 4 + 12\).
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Add from left to right: \(36 + 12 = 48\).
The value of the expression is 48.
Common mistake
Watch out
Omar writes \(-3 – (-8) = -11\), because “two minus signs make it more negative.”
Why it is wrong
Subtracting a negative removes a debt, so the value goes up. Rewrite the problem as adding the opposite: \(-3 + 8 = 5\). On a number line, start at \(-3\) and walk 8 steps to the right, landing on 5, not on \(-11\). Two minus signs side by side act like a double negative in a sentence: they cancel each other.
Related terms and lessons
- Grade 7 · Adding and Subtracting Integers and Rational NumbersLessonWorksheetQuiz
- Grade 7 · Multiplying and Dividing Integers and Rational NumbersLessonWorksheetQuiz
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