Multiplication and Division of Real Numbers

πŸ†Practice multiplication and division of directed numbers

The method to solve an exercise with real numbers, when it involves multiplication and division, is very similar to the one we use when we have to add or subtract real numbers, with the difference that, in this case, we must make use of the multiplication and division table that we learned in elementary school.

When we have two real numbers with the same sign (plus or minus) we distinguish two cases:

When we have two real numbers with the same sign (plus or minus) we distinguish two cases
  • The product (result of the multiplication) of two positive numbers will be positive. The quotient (result of the division) of two positive numbers will be positive.
    (+2)Γ—(+1)=+2(+2) \times (+1)= +2
    (+2):(+1)=+2(+2) :(+1)= +2
  • The product of two negative numbers will be positive. The quotient of two negative numbers will be positive.
    (βˆ’2)Γ—(βˆ’1)=+2(-2) \times (-1)= +2
    (βˆ’2):(βˆ’1)=+2(-2) :(-1)= +2
  • When we have two numbers with different signs, that is, one with the plus sign and the other with the minus sign, the result of the multiplication or division will always be negative.
    (+2)Γ—(βˆ’1)=βˆ’2(+2) \times (-1)= -2
    (βˆ’2):(+1)=βˆ’2(-2) :(+1)= -2
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Test yourself on multiplication and division of directed numbers!

einstein

What will be the sign of the result of the next exercise?

\( (-2)\cdot(-4)= \)

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Multiplication of Real Numbers

Let's see, before multiplying we prefer to simplify the exercise, we will do it in the way we have learned in the addition and subtraction of real numbers, grouping the signs:

  • When we have two equal signs they are grouped into one positive.

For example:
(+5)Γ—(+5)(+5)\times (+5)Β or Β (βˆ’5)Γ—(βˆ’5)(-5)\times (-5)
will become +25+25


  • When we have two different signs they are grouped into one negative.

For example:
(+5)Γ—(βˆ’5)(+5)\times (-5)Β Β or Β (βˆ’5)Γ—(+5)(-5)\times (+5)
will become βˆ’25-25


For example:

(+10)Γ—(βˆ’5)Γ—(βˆ’3)Γ—(βˆ’6)Γ—(+8)=(+10)\times (-5)\times (-3)\times (-6)\times (+8)=

(βˆ’50)Γ—(βˆ’3)Γ—(βˆ’6)Γ—(+8)=(-50)\times (-3)\times (-6)\times (+8)=

(+150)Γ—(βˆ’6)Γ—(+8)=(+150)\times (-6)\times (+8)=

(βˆ’900)Γ—(+8)=βˆ’7200(-900)\times (+8)=-7200


Likewise, we will carry out the division of real numbers in the same way.

  • When we have two equal signs, they are combined into one positive sign.

For example:
Β (βˆ’5):(βˆ’5)=+1(-5):(-5)=+1Β /Β (βˆ’5)(βˆ’5)=+1{(-5)\over(-5)}=+1
Β (+5):(+5)=+1(+5):(+5)=+1Β /Β (+5)(+5)=+1{(+5)\over(+5)}=+1


  • When we have two different signs, they are combined into one negative sign.

For example:

(βˆ’5):(+5)=βˆ’1(-5):(+5)=-1Β /Β (βˆ’5)(+5)=βˆ’1{(-5)\over(+5)}=-1
Β (+5):(βˆ’5)=βˆ’1(+5):(-5)=-1Β /Β (+5)(βˆ’5)=βˆ’1{(+5)\over(-5)}=-1


For example:

(+100):(βˆ’2):(βˆ’2):(βˆ’5):(βˆ’2.5)=(+100):(-2):(-2):(-5):(-2.5)=

(βˆ’50):(βˆ’2):(βˆ’5):(βˆ’2.5)=(-50):(-2):(-5):(-2.5)=

(+25):(βˆ’5):(βˆ’2.5)=(+25):(-5):(-2.5)=

(βˆ’5):(βˆ’2.5)=+2(-5):(-2.5)=+2


If you are interested in this article, you may also be interested in the following articles:

Positive numbers, negative numbers, and zero

The real line

Opposite numbers

Absolute value

Elimination of parentheses in real numbers

Addition and subtraction of real numbers

On the website of Tutorela you will find a variety of articles on mathematics.


Examples and exercises with solutions for multiplication and division of real numbers

examples.example_title

What will be the sign of the result of the next exercise?

(βˆ’2)β‹…(βˆ’4)= (-2)\cdot(-4)=

examples.explanation_title

It's important to remember: when we multiply a negative by a negative, the result is positive!

You can use this guide:

examples.solution_title

Positive

examples.example_title

What will be the sign of the result of the next exercise?

2β‹…(βˆ’2)= 2\cdot(-2)=

examples.explanation_title

To solve the exercise you need to remember an important rule: Multiplying a positive number by a negative number results in a negative number.

(βˆ’)Γ—(+)=(βˆ’) (βˆ’)Γ—(+)=(βˆ’)
Therefore, if we multiply negative 2 by 2 the result will be negative 4.

That is, the result is negative.

+2Γ—βˆ’2=βˆ’4 +2\times-2=-4

examples.solution_title

Negative

examples.example_title

βˆ’aβ‹…b= -a\cdot b=

Replace and calculate if a=βˆ’3,Β b=5 a=-3\text{, }b=5

examples.explanation_title

First, we replace the data in the exercise

-(-3)*5 =Β 

To better understand the minus sign multiplied at the beginning, we will write it like this:

-1*-3*5 =Β 

Now we see that we have an exercise that is all multiplication,

We will solve according to the order of arithmetic operations, from left to right:

-1*-3 = 3

3*5 = 15

examples.solution_title

15 15

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Review Questions

What are the rules for multiplying and dividing integers?

When talking about integers, we can observe that we will deal with positive and negative numbers, therefore, to be able to perform certain mathematical operations such as multiplication and division, certain rules must be followed which are known as the law of signs or sign rules, which will be described below:

For multiplication, there are two cases

Same signs, that is when multiplying numbers with the same sign

  • If we multiply two positive numbers the result will be positive.
  • If we multiply two negative numbers the result will be positive.

Different sign

  • If we multiply a positive number and a negative one, the result will be negative.
  • If we multiply a negative number and a positive one, the result will be negative.

We can summarize this in the following way:

(+)Γ—(+)=+ \left(+\right)\times\left(+\right)=+

(βˆ’)Γ—(βˆ’)=+ \left(-\right)\times\left(-\right)=+

(+)Γ—(βˆ’)=βˆ’ \left(+\right)\times\left(-\right)=-

(βˆ’)Γ—(+)=βˆ’ \left(-\right)\times\left(+\right)=-

In the same way for division, the same rule is followed,

Same sign, that is when dividing numbers with the same sign:

  • If we divide two positive numbers the result will be positive.
  • If we divide two negative numbers the result will be positive.

Different sign

  • If we divide a positive number by a negative one, the result will be negative.
  • If we divide a negative number by a positive one, the result will be negative.

Summarizing these four statements, we can visualize it in the following way:

(+):(+)=+ \left(+\right):\left(+\right)=+

(βˆ’):(βˆ’)=+ \left(-\right) : \left(-\right)=+

(+):(βˆ’)=βˆ’ \left(+\right):\left(-\right)=-

(βˆ’):(+)=βˆ’ \left(-\right):\left(+\right)=-


How is multiplication of real numbers performed?

As we well know one of the arithmetic operations with real numbers besides addition and subtraction, is the multiplication of real numbers in this case to be able to perform this type of mathematical operations, we must take into account two things, the first is to remember the multiplication tables studied in elementary school and the second is to use the sign laws mentioned before, since when working with real numbers we will find positive and negative numbers.

Example 1.

Task. Perform the following multiplication (6)Γ—(βˆ’3)= \left(6\right)\times\left(-3\right)=

Solution:

We can observe that the 6 6 is a positive number and the βˆ’3 -3 is negative therefore (+)Γ—(βˆ’)=βˆ’ \left(+\right)\times\left(-\right)=-

Then the result will be negative, and we just use the multiplication tables, therefore

(6)Γ—(βˆ’3)=βˆ’18 \left(6\right)\times\left(-3\right)=-18

Result

βˆ’18 -18

Example 2.

Task. Perform the following multiplication (βˆ’5)Γ—(βˆ’7)= \left(-5\right)\times\left(-7\right)=

Solution:

We can observe that the βˆ’5 -5 is a negative number and the βˆ’7 -7 is also negative therefore (βˆ’)Γ—(βˆ’)=+ \left(-\right)\times\left(-\right)=+

Then the result will be positive, and we just use the multiplication tables, therefore

(βˆ’5)Γ—(βˆ’7)=35 \left(-5\right)\times\left(-7\right)=35

Result

35 35


How is division in real numbers done?

We have seen that for the division of real numbers the same sign law is followed, so to be able to do the division of real numbers it is enough to make the quotient of said numbers and respect the corresponding sign law, let's see some examples:

Example 1.

Task. Perform the following division (14):(βˆ’7)= \left(14\right):\left(-7\right)=

Solution:

We can observe that the 14 14 is a positive number and the βˆ’7 -7 is negative therefore (+):(βˆ’)=βˆ’ \left(+\right):\left(-\right)=-

Then the result will be negative, and we just do the division

(14):(βˆ’7)=βˆ’2 \left(14\right):\left(-7\right)=-2

Result

βˆ’2 -2

Example 2.

Task. Perform the following division (βˆ’100):(25)= \left(-100\right):\left(25\right)=

Solution:

We can observe that the βˆ’100 -100 is a negative number and the 25 25 is positive therefore (βˆ’):(+)=βˆ’ \left(-\right):\left(+\right)=-

Then the result will be negative, and we just make the quotient, therefore

(βˆ’100):(25)=βˆ’4 \left(-100\right):\left(25\right)=-4

Result

βˆ’4 -4


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