Addition and Subtraction of Real Numbers

πŸ†Practice addition and subtraction of directed numbers

The addition and subtraction of real numbers are based on certain key principles. All principles will be explained using two real numbers, but certainly, the numbers in the exercise do not influence the method of resolution, therefore, these principles can be applied to any number in the exercise.

  • When we have two real numbers with the same sign (plus or minus), this sign will remain in the result, which will, in fact, be the result of the addition. That is, if both numbers have a plus sign the result of the addition will also be positive. If both numbers have a minus sign the result of the subtraction will also be negative.
    +6+4=+10+6+4=+10
    βˆ’6βˆ’4=βˆ’10-6-4=-10

  • When we have two numbers with different signs it is crucial to determine which of the two has the greater absolute value (absolute: the distance from zero). The larger number will determine the sign of the result and, in fact, we will perform a subtraction operation.
    +6βˆ’4=+2+6-4=+2
    βˆ’6+4=βˆ’2-6+4=-2

  • When we have an exercise with a sequence of two signs (usually separated by parentheses) we will differentiate between several cases:

  • When the sequence is of two plus signs the result will also be positive
    6+(+4)=+106+(+4)=+10

  • When the sequence is of two minus signs the result will also be positive
    6βˆ’(βˆ’4)=+106-(-4)=+10

  • When the sequence is of minus and plus or of plus and minus the result will be negative.
    6+(βˆ’4)=+26+(-4)=+2
    6βˆ’(+4)=+26-(+4)=+2

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Test yourself on addition and subtraction of directed numbers!

einstein

\( (-10)-(+13)= \)

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Association of Operations in Exercises with Addition and Subtraction of Real Numbers

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After studying real numbers, it's time to learn how to use them in an equation. Initially, our goal with equations is to simplify them to make it easier to solve problems, and we do this by grouping operations and adding and subtracting real numbers. We just need to remember two rules:

  • When the mathematical operation and the sign of the following real number are of the same type, we group them into a sum.
  • For example: 5+(+5)5+(+5) / 5βˆ’(βˆ’5)5-(-5)
    5+55+5 will become
  • When the mathematical operation and the sign of the following real number are of different types, we group them into an operation that will give the difference between them. For example: 5+(βˆ’5)5+(-5) / 5βˆ’(+5)5-(+5)
    5βˆ’55-5 will become

For example:

10+(+5)βˆ’(+3)βˆ’(βˆ’6)+(βˆ’8)=10+(+5)-(+3)-(-6)+(-8)=
10+5βˆ’3+6βˆ’8=1010+5-3+6-8=10


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The Elevator Method for Adding and Subtracting Real Numbers

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There is a very well-known tactic that helps to understand the topic of real numbers in the best way, it's called the elevator method and it serves to clarify the addition and subtraction of real numbers. With this method, we imagine that the exercise is like a journey in an elevator that goes through the floors. Observe the following exercise:

βˆ’5βˆ’(+1)βˆ’(βˆ’8)+(βˆ’3)=-5-(+1)-(-8)+(-3)=

Before using the elevator method we have to group the signs to simplify the exercise

βˆ’5βˆ’1+8βˆ’3=-5-1+8-3=

Now look at the first number. In fact, you start the exercise on floor -5 and now you are asked to go down one floor. This way you reach floor -6.
Now you are asked to go up 8 floors. So, if we were on floor -6 we will arrive at floor 2. Finally, you are asked to go down 3 floors, therefore, you end up on floor -1, which is the result of the exercise

βˆ’5βˆ’1+8βˆ’3=βˆ’1-5-1+8-3=-1

We will give consistency to the principles outlined through the following examples:

(+3)+(+4)+(+5)=3+4+5=+12(+3) + (+4) + (+5) = 3+4+5= +12

(βˆ’3)+(βˆ’4)+(βˆ’5)=βˆ’3βˆ’4βˆ’5=βˆ’12(-3) + (-4) + (-5) = -3-4-5= -12

βˆ’10+2=βˆ’8-10+2= -8

6βˆ’20=βˆ’146-20= -14

(βˆ’10)βˆ’(βˆ’100)=βˆ’10+100=90(-10)-(-100)= -10+100= 90

8+(βˆ’4)=8βˆ’4=48+(-4)= 8-4= 4


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

Positive, negative numbers and zero

Multiplicative inverse

The real number line

Opposite numbers

Absolute value

Elimination of parentheses in real numbers

Multiplication and division of real numbers

In the Tutorela blog you will find a variety of articles about mathematics.


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Exercises on Addition and Subtraction of Real Numbers

Exercise 1

Assignment

βˆ’27βˆ’(βˆ’7)+(βˆ’6)+2βˆ’11= -27-\left(-7\right)+\left(-6\right)+2-11=

Solution

First, we resolve the multiplication points, that is, the points that have a plus or minus sign before another sign.

βˆ’27+7βˆ’6+2βˆ’11= -27+7-6+2-11=

Now we solve it as a common exercise:

βˆ’27+7βˆ’6+2βˆ’11=βˆ’35 -27+7-6+2-11=-35

Answer

βˆ’35-35


Exercise 2

Assignment

?βˆ’(βˆ’12)=βˆ’40 \text{?}-(-12)=-40

Solution

First, let's note that the two minuses turn into a plus.

?+12=βˆ’40 \text{?+}12=-40

We will move the 12 12 to the right side

?=βˆ’40βˆ’12 \text{?}=-40-12

Finally, we solve

?=βˆ’52 \text{?}=-52

Answer:

βˆ’52 -52


Do you know what the answer is?

Exercise 3

Assignment

βˆ’36+6= -36+6=

Solution

We use the laws of addition and subtraction to solve accordingly.

βˆ’36+6=βˆ’30 -36+6=-30

Answer:

30βˆ’ 30-


Exercise 4

12βˆ’(βˆ’2)= 12-\left(-2\right)=

Solution

Pay attention to the fact that the minus and minus signs become plus, and we solve the exercise accordingly.

12+2=14 12+2=14

Answer

14 14


Check your understanding

Exercise 5

Assignment

Given:

a a Negative number

b b Negative number

What is the sum of a+b a+b ?

Solution

When we add two negative numbers, the result we will obtain is a negative number.

Answer

Negative


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