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3.8 Summary of key concepts

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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.This module contains a summary of the key concepts in the chapter "Basic Operations with Real Numbers".

Summary of key concepts

Positive and negative numbers ( [link] )

A number is denoted as positive if it is directly preceded by a " + " sign or no sign at all. A number is denoted as negative if it is directly preceded by a " − " sign.

Opposites ( [link] )

Opposites are numbers that are the same distance from zero on the number line but have opposite signs.

Double-negative property ( [link] )

− ( − a ) = a

Absolute value (geometric) ( [link] )

The absolute value of a number a , denoted | a | , is the distance from a to 0 on the number line.

Absolute value (algebraic) ( [link] )

| a | = { a if   a ≥ 0 − a if   a < 0

Addition of signed numbers ( [link] )

To add two numbers with
like signs , add the absolute values of the numbers and associate the common sign with the sum.
unlike signs , subtract the smaller absolute value from the larger absolute value and associate the sign of the larger absolute value with the difference.

Addition with 0 ( [link] )

0 + any   number = that   particular   number , that is, 0 + a = a for any real number a .

Additive identity ( [link] )

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

Definition of subtraction ( [link] )

a − b = a + ( − b )

Subtraction of signed numbers ( [link] )

To perform the subtraction a − b , add the opposite of b to a , that is, change the sign of b and add.

Multiplication and division of signed numbers ( [link] )

( + )   ( + ) = + ( + ) ( + ) = + ( + ) ( − ) = − ( − )   ( − ) = + ( + )   ( − ) = − ( − )   ( + ) = − ( − ) ( − ) = + ( − ) ( + ) = −

Reciprocals ( [link] )

Two numbers are reciprocals of each other if their product is 1. The numbers 4 and 1 4 are reciprocals since ( 4 )   ( 1 4 ) = 1 .

Negative exponents ( [link] )

If n is any natural number and x is any nonzero real number, then x − n = 1 x n .

Writing a number in scientific notation ( [link] )

To write a number in scientific notation:

  1. Move the decimal point so that there is one nonzero digit to its left.
  2. Multiply the result by a power of 10 using an exponent whose absolute value is the number of places the decimal point was moved. Make the exponent positive if the decimal point was moved to the left and negative if the decimal point was moved to the right.

Converting from scientific notation:
Positive exponent ( [link] )

To convert a number written in scientific notation to a number in standard form when there is a positive exponent as the power of 10, move the decimal point to the right the number of places prescribed by the exponent on the 10.

Negative exponent ( [link] )

To convert a number written in scientific notation to a number in standard form when there is a negative exponent as the power of 10, move the decimal point to the left the number of places prescribed by the exponent on the 10.

Questions & Answers

A golfer on a fairway is 70 m away from the green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an angle of 40° with an initial speed of 20 m/s, how close to the green does she come?
Aislinn Reply
cm
tijani
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John Reply
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Siyaka Reply
A mouse of mass 200 g falls 100 m down a vertical mine shaft and lands at the bottom with a speed of 8.0 m/s. During its fall, how much work is done on the mouse by air resistance
Jude Reply
Can you compute that for me. Ty
Jude
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David Reply
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David
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emma Reply
what is chemistry
Youesf Reply
what is inorganic
emma
Chemistry is a branch of science that deals with the study of matter,it composition,it structure and the changes it undergoes
Adjei
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Adjanou
chemistry could also be understood like the sexual attraction/repulsion of the male and female elements. the reaction varies depending on the energy differences of each given gender. + masculine -female.
Pedro
A ball is thrown straight up.it passes a 2.0m high window 7.50 m off the ground on it path up and takes 1.30 s to go past the window.what was the ball initial velocity
Krampah Reply
2. A sled plus passenger with total mass 50 kg is pulled 20 m across the snow (0.20) at constant velocity by a force directed 25° above the horizontal. Calculate (a) the work of the applied force, (b) the work of friction, and (c) the total work.
Sahid Reply
you have been hired as an espert witness in a court case involving an automobile accident. the accident involved car A of mass 1500kg which crashed into stationary car B of mass 1100kg. the driver of car A applied his brakes 15 m before he skidded and crashed into car B. after the collision, car A s
Samuel Reply
can someone explain to me, an ignorant high school student, why the trend of the graph doesn't follow the fact that the higher frequency a sound wave is, the more power it is, hence, making me think the phons output would follow this general trend?
Joseph Reply
Nevermind i just realied that the graph is the phons output for a person with normal hearing and not just the phons output of the sound waves power, I should read the entire thing next time
Joseph
Follow up question, does anyone know where I can find a graph that accuretly depicts the actual relative "power" output of sound over its frequency instead of just humans hearing
Joseph
"Generation of electrical energy from sound energy | IEEE Conference Publication | IEEE Xplore" ***ieeexplore.ieee.org/document/7150687?reload=true
Ryan
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Maurice Reply
what are the types of wave
Maurice
answer
Magreth
progressive wave
Magreth
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Mohammed
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Mujahid
A string is 3.00 m long with a mass of 5.00 g. The string is held taut with a tension of 500.00 N applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the 3.00 m of the string?
yasuo Reply
Who can show me the full solution in this problem?
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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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