Sign rules are a set of principles that govern the signs (positive or negative) of the results when multiplying or dividing integers. These rules help determine the overall sign of the product or quotient based on the signs of the individual factors or divisors.
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When multiplying two integers with the same sign (both positive or both negative), the result is positive.
When multiplying two integers with different signs (one positive and one negative), the result is negative.
When dividing an integer by a positive integer, the sign of the quotient is the same as the sign of the dividend.
When dividing an integer by a negative integer, the sign of the quotient is the opposite of the sign of the dividend.
The sign rules for multiplication and division of integers are important for simplifying algebraic expressions and solving equations.
Review Questions
Explain the sign rule for multiplying two integers with the same sign.
When multiplying two integers with the same sign, either both positive or both negative, the sign of the result is positive. For example, when multiplying 3 × 5 or -2 × -4, the sign of the product is positive. This is because the same signs, either both positive or both negative, indicate that the magnitudes of the numbers are moving in the same direction, resulting in a positive product.
Describe the sign rule for dividing an integer by a negative integer.
When dividing an integer by a negative integer, the sign of the quotient is the opposite of the sign of the dividend. For instance, if you divide -12 by -3, the quotient is positive 4. This is because the negative sign of the divisor (-3) cancels out the negative sign of the dividend (-12), resulting in a positive quotient. The sign rule for division with negative integers is important for simplifying algebraic expressions and solving equations involving negative numbers.
Analyze the relationship between the signs of the factors and the sign of the product when multiplying integers.
The sign of the product when multiplying integers is determined by the combination of the signs of the individual factors. If the factors have the same sign, either both positive or both negative, the product will be positive. However, if the factors have different signs, one positive and one negative, the product will be negative. This pattern is crucial to understand when working with integer operations, as it allows you to predict the sign of the result without needing to perform the actual multiplication. Mastering the sign rules for multiplication is a key skill in elementary algebra.