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This module is from Elementary Algebra by Denny Burzynski and Wade Ellis, Jr. The basic operations with real numbers are presented in this chapter. The concept of absolute value is discussed both geometrically and symbolically. The geometric presentation offers a visual understanding of the meaning of |x|. The symbolic presentation includes a literal explanation of how to use the definition. Negative exponents are developed, using reciprocals and the rules of exponents the student has already learned. Scientific notation is also included, using unique and real-life examples.Objectives of this module: be able to add numbers with like signs and unlike signs, understand addition with zero.

Overview

  • Addition of Numbers with Like Signs
  • Addition with Zero
  • Addition of Numbers with Unlike Signs

Addition of numbers with like signs

Let us add the two positive numbers 2 and 3. We perform this addition on the number line as follows.

We begin at 0, the origin.
Since 2 is positive, we move 2 units to the right.
Since 3 is positive, we move 3 more units to the right.
We are now located at 5.
Thus, 2 + 3 = 5 .

A number line with arrows on each end, labeled from negative two to eight in increments of one. There is a curved arrow starting from zero, and pointing towards two. There is another curved arrow starting from two, and pointing towards five.

Summarizing, we have

( 2 positive units ) + ( 3 positive units ) = ( 5 positive units )

Now let us add the two negative numbers 2 and 3 . We perform this addition on the number line as follows.

We begin at 0, the origin.
Since 2 is negative, we move 2 units to the left.
Since 3 is negative, we move 3 more units to the left.
We are now located at 5 .

Thus, ( 2 ) + ( 3 ) = 5 .

A number line with arrows on each end, labeled from negative seven to three in increments of one. There is a curved arrow starting from zero, and pointing towards negative two. There is another curved arrow starting from negative two, and pointing towards negative five

Summarizing, we have

( 2 negative units ) + ( 3 negative units ) = ( 5 negative units )

These two examples suggest that

( positive number ) + ( positive number ) = ( positive number ) ( negative number ) + ( negative number ) = ( negative number )

Adding numbers with the same sign

To add two real numbers that have the same sign, add the absolute values of the numbers and associate the common sign with the sum.

Sample set a

Find the sums.

3 + 7

Add these absolute values . | 3 | = 3 | 7 | = 7 } 3 + 7 = 10 The common sign is "+ ."

3 + 7 = + 10 or 3 + 7 = 10

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( 4 ) + ( 9 )

Add these absolute values . | 4 | = 4 | 9 | = 9 } 4 + 9 = 13 The common sign is " ."

( 4 ) + ( 9 ) = 13

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Practice set a

Find the sums.

( 4 ) + ( 8 )

12

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( 36 ) + ( 9 )

45

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14 + ( 20 )

34

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2 3 + ( 5 3 )

7 3

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2.8 + ( 4.6 )

7.4

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Addition with zero

Notice that

Addition with 0

( 0 ) + ( a positive number ) = ( that same positive number ) ( 0 ) + ( a negative number ) = ( that same negative number)

The additive identity is 0

Since adding 0 to a real number leaves that number unchanged, 0 is called the additive identity .

Addition of numbers with unlike signs

Now let us perform the addition 2 + ( 6 ) . These two numbers have unlike signs. This type of addition can also be illustrated using the number line.

We begin at 0, the origin.
Since 2 is positive, we move 2 units to the right.
Since 6 is negative, we move, from the 2, 6 units to the left.
We are now located at 4 .

A number line with arrows on each end, labeled from negative five to five in increments of one. There is a curved arrow starting from zero, and pointing towards two. There is another curved arrow starting from two, and pointing towards negative four.

A rule for adding two numbers that have unlike signs is suggested by noting that if the signs are disregarded, 4 can be obtained from 2 and 6 by subtracting 2 from 6. But 2 and 6 are precisely the absolute values of 2 and 6 . Also, notice that the sign of the number with the larger absolute value is negative and that the sign of the resulting sum is negative.

Adding numbers with unlike signs

To add two real numbers that have unlike signs, subtract the smaller absolute value from the larger absolute value and associate the sign of the number with the larger absolute value with this difference.

Sample set b

Find the following sums.

7 + ( 2 )

| 7 | = 7 Larger absolute value . Sign is " + " . | 2 | = 2 Smaller absolute value .

Subtract absolute values: 7 2 = 5. Attach the proper sign: " + " .

7 + ( 2 ) = + 5 or 7 + ( 2 ) = 5

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3 + ( 11 )

| 3 | = 3 Smaller absolute value . | 11 | = 11 Larger absolute value . Sign is " " .

Subtract absolute values: 11 3 = 8. Attach the proper sign: " " .

3 + ( 11 ) = 8

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The morning temperature on a winter's day in Lake Tahoe was 12 degrees. The afternoon temperature was 25 degrees warmer. What was the afternoon temperature?

We need to find 12 + 25 .

| 12 | = 12 Smaller absolute value . | 25 | = 25 Larger absolute value . Sign is "+" .

Subtract absolute values: 25 12 = 13. Attach the proper sign: " + " .

12 + 25 = 13

Thus, the afternoon temperature is 13 degrees.

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Use a calculator. Add 147 + 84 .                                                      Display Reads

Type 147 147 Press + / 147 Press + 147 Type 84 84 Press = 63

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Practice set b

Find the sums.

1345.6 + ( 6648.1 )

7993.7

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Exercises

Find the sums for the the following problems.

( 3 ) + ( 12 )

15

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( 4 ) + ( 8 )

12

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( 16 ) + ( 8 )

24

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( 3 ) + ( 12 )

15

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9 + ( 6 )

15

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16 + ( 9 )

25

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5 + ( 12 ) + ( 4 )

21

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1221 + ( 44 )

1265

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47.03 + ( 22.71 )

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1.998 + ( 4.086 )

6.084

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[ ( 3 ) + ( 4 ) ] + [ ( 6 ) + ( 1 ) ]

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[ ( 2 ) + ( 8 ) ] + [ ( 3 ) + ( 7 ) ]

20

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[ ( 3 ) + ( 8 ) ] + [ ( 6 ) + ( 12 ) ]

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[ ( 8 ) + ( 6 ) ] + [ ( 2 ) + ( 1 ) ]

17

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[ 4 + ( 12 ) ] + [ 12 + ( 3 ) ]

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[ 5 + ( 16 ) ] + [ 4 + ( 11 ) ]

18

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[ 2 + ( 4 ) ] + [ 17 + ( 19 ) ]

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[ 10 + ( 6 ) ] + [ 12 + ( 2 ) ]

14

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14 + [ ( 3 ) + 5 ]

16

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[ 2 + ( 7 ) ] + ( 11 )

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[ 14 + ( 8 ) ] + ( 2 )

4

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In order for a small business to break even on a project, it must have sales of $ 21 , 000 . If the amount of sales was $ 15 , 000 , how much money did this company fall short?

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Suppose a person has $ 56.00 in his checking account. He deposits $ 100.00 into his checking account by using the automatic teller machine. He then writes a check for $ 84.50 . If an error causes the deposit not to be listed into this person's account, what is this person's checking balance?

$ 28.50

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A person borrows $ 7.00 on Monday and then $ 12.00 on Tuesday. How much has this person borrowed?

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A person borrows $ 11.00 on Monday and then pays back $ 8.00 on Tuesday. How much does this person owe?

$ 3.00

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Exercises for review

( [link] ) Simplify 4 ( 7 2 6 2 3 ) 2 2 .

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( [link] ) Simplify 35 a 6 b 2 c 5 7 b 2 c 4 .

5 a 6 c

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( [link] ) Simplify ( 12 a 8 b 5 4 a 5 b 2 ) 3 .

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( [link] ) Determine the value of | 8 | .

8

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( [link] ) Determine the value of ( | 2 | + | 4 | 2 ) + | 5 | 2 .

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Questions & Answers

Three charges q_{1}=+3\mu C, q_{2}=+6\mu C and q_{3}=+8\mu C are located at (2,0)m (0,0)m and (0,3) coordinates respectively. Find the magnitude and direction acted upon q_{2} by the two other charges.Draw the correct graphical illustration of the problem above showing the direction of all forces.
Kate Reply
To solve this problem, we need to first find the net force acting on charge q_{2}. The magnitude of the force exerted by q_{1} on q_{2} is given by F=\frac{kq_{1}q_{2}}{r^{2}} where k is the Coulomb constant, q_{1} and q_{2} are the charges of the particles, and r is the distance between them.
Muhammed
What is the direction and net electric force on q_{1}= 5µC located at (0,4)r due to charges q_{2}=7mu located at (0,0)m and q_{3}=3\mu C located at (4,0)m?
Kate Reply
what is the change in momentum of a body?
Eunice Reply
what is a capacitor?
Raymond Reply
Capacitor is a separation of opposite charges using an insulator of very small dimension between them. Capacitor is used for allowing an AC (alternating current) to pass while a DC (direct current) is blocked.
Gautam
A motor travelling at 72km/m on sighting a stop sign applying the breaks such that under constant deaccelerate in the meters of 50 metres what is the magnitude of the accelerate
Maria Reply
please solve
Sharon
8m/s²
Aishat
What is Thermodynamics
Muordit
velocity can be 72 km/h in question. 72 km/h=20 m/s, v^2=2.a.x , 20^2=2.a.50, a=4 m/s^2.
Mehmet
A boat travels due east at a speed of 40meter per seconds across a river flowing due south at 30meter per seconds. what is the resultant speed of the boat
Saheed Reply
50 m/s due south east
Someone
which has a higher temperature, 1cup of boiling water or 1teapot of boiling water which can transfer more heat 1cup of boiling water or 1 teapot of boiling water explain your . answer
Ramon Reply
I believe temperature being an intensive property does not change for any amount of boiling water whereas heat being an extensive property changes with amount/size of the system.
Someone
Scratch that
Someone
temperature for any amount of water to boil at ntp is 100⁰C (it is a state function and and intensive property) and it depends both will give same amount of heat because the surface available for heat transfer is greater in case of the kettle as well as the heat stored in it but if you talk.....
Someone
about the amount of heat stored in the system then in that case since the mass of water in the kettle is greater so more energy is required to raise the temperature b/c more molecules of water are present in the kettle
Someone
definitely of physics
Haryormhidey Reply
how many start and codon
Esrael Reply
what is field
Felix Reply
physics, biology and chemistry this is my Field
ALIYU
field is a region of space under the influence of some physical properties
Collete
what is ogarnic chemistry
WISDOM Reply
determine the slope giving that 3y+ 2x-14=0
WISDOM
Another formula for Acceleration
Belty Reply
a=v/t. a=f/m a
IHUMA
innocent
Adah
pratica A on solution of hydro chloric acid,B is a solution containing 0.5000 mole ofsodium chlorid per dm³,put A in the burret and titrate 20.00 or 25.00cm³ portion of B using melting orange as the indicator. record the deside of your burret tabulate the burret reading and calculate the average volume of acid used?
Nassze Reply
how do lnternal energy measures
Esrael
Two bodies attract each other electrically. Do they both have to be charged? Answer the same question if the bodies repel one another.
JALLAH Reply
No. According to Isac Newtons law. this two bodies maybe you and the wall beside you. Attracting depends on the mass och each body and distance between them.
Dlovan
Are you really asking if two bodies have to be charged to be influenced by Coulombs Law?
Robert
like charges repel while unlike charges atttact
Raymond
What is specific heat capacity
Destiny Reply
Specific heat capacity is a measure of the amount of energy required to raise the temperature of a substance by one degree Celsius (or Kelvin). It is measured in Joules per kilogram per degree Celsius (J/kg°C).
AI-Robot
specific heat capacity is the amount of energy needed to raise the temperature of a substance by one degree Celsius or kelvin
ROKEEB
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Source:  OpenStax, Elementary algebra. OpenStax CNX. May 08, 2009 Download for free at http://cnx.org/content/col10614/1.3
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