Multiplication and division of negative numbers. Multiplication : It is said to be the abbreviation for the addition of equal addends. They are related to the words: product, double, triple or Cartesian product of sets : 9 + 9 + 9 + 9 + 9 is the same as 5 times 9; that is, 5 X 9, its elements are: factors and product.
Division : Some think that this operation is selfish, because it is related to separate. On the contrary, it is a clear expression of justice. She is in charge of distributing and always does it in equal parts. Its elements are: dividend, divisor and quotient.
Summary
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- 1 Historical Review
- 2 Multiplication
- 1 Properties of multiplication
- 2 Examples
- 3 Rule of signs
- 4 Division
- 5 References
- 6 Source
Historical review
The integers positive and negative, are the natural result of the operations adds and subtracts . Its use, although with various notations, dates back to antiquity.
The name of integers is justified because these numbers , whether positive or negative, always represented a number of non-divisible units (for example, people).
It was not until the 17th century that they were accepted in European scientific works, although Italian Renaissance mathematicians such as Tartaglia and Cardano had already warned them in their works about solving third-degree equations. However, the rule of signs was already previously known to Indian mathematicians .
Multiplication was considered a very difficult operation in Europe before the 16th century , since Roman numerals were still used , and operations with large numbers are more difficult in this numbering system than with the positional decimal system .
Multiplication
Properties of multiplication
The multiplication of whole numbers also has properties similar to that of natural numbers :
- Associative property. Given three integers a , b, and c , the products ( a × b ) × c and a × ( b × c ) are equal.
- Commutative property. Given two integers a and b , the products a × b and b × a are equal.
- Distributive property. It states that multiplying a sum by a number gives the same result as multiplying each sum by the number and then adding all the products.
- Neutral element. All integers a remain unchanged when multiplied by 1: a × 1 = a .
Examples
Associative property :
[(−7) × (+4)] × (+5) = (−28) × (+5) = −140
(−7) × [(+4) × (+5)] = (−7) × (+20) = −140
Commutative property:
(−6) × (+9) = −54
(+9) × (−6) = −54
The addition and multiplication of integers are related, like natural numbers, by the distributive property:
Distributive property:
Given three integers a , b, and c , the product a × ( b + c ) and the sum of products ( a × b ) + ( a × c ) are identical.
Neutral element
a · 1 = a
Multiplication of integers, like addition, requires separately determining the sign and absolute value of the result.
In the multiplication of two integers, the absolute value and sign of the result are determined as follows:
The absolute value is the product of the absolute values of the factors.
The sign is “+” if the signs of the factors are the same, and “-“ if they are different.
To remember the sign of the result, the sign rule is also used :
Rule of signs
(+) × (+) = (+) More for more equal to more.
(+) × (-) = (-) More by minus equals minus.
(-) × (+) = (-) Minus plus equals minus.
(-) × (-) = (+) Minus by minus equals more.
Examples
(+4) × (−6). The sign of the factors is different, so the sign of the result is “-“. The product of the absolute values is 4 × 6 = 24. That is: (+4) × (−6) = −24.
(+5) × (+3). The sign of the factors is identical, so the sign of the result is “+”. The product of the absolute values is 5 × 3 = 15. That is: (+5) × (+3) = +15.
(−7) × (+8). The sign of the factors is different, then the sign of the result is “-“. The product of the absolute values is 7 × 8 = 56. That is: (−7) × (+8) = −56.
(−9) × (−2). The sign of the factors is the same, so the sign of the result is “+”. The product of the absolute values is 9 × 2 = 18. That is: (- 9) × (−2) = +18.
Division
The rules that were obtained for the multiplication work perfectly in the case of the division of the rational numbers.
Division of equal signs gives a positive sign and division of different signs gives a negative sign.
In fraction division , changing the divisor and dividend (numerator and denominator) changes the result of the division, but not the sign.
