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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

how does the planets on our solar system orbit
cheten Reply
how many Messier objects are there in space
satish Reply
did you g8ve certificate
Richard Reply
what are astronomy
Issan Reply
Astronomy (from Ancient Greek ἀστρονομία (astronomía) 'science that studies the laws of the stars') is a natural science that studies celestial objects and phenomena. It uses mathematics, physics, and chemistry in order to explain their origin and evolution.
Rafael
vjuvu
Elgoog
what is big bang theory?
Rosemary
what type of activity astronomer do?
Rosemary
No
Richard
the big bang theory is a theory which states that all matter was compressed together in one place the matter got so unstable it exploded releasing All its contents in the form of hydrogen
Roaul
I want to be an astronomer. That's my dream
Astrit
Who named the the whole galaxy?
Shola Reply
solar Univers
GPOWER
what is space
Richard
what is the dark matter
Richard
what are the factors upon which the atmosphere is stratified
Nicholas Reply
is the big bang the sun
Folakemi Reply
no
Sokak
bigbang is the beginning of the universe
Sokak
but thats just a theory
Sokak
nothing will happen, don't worry brother.
Vansh
what does comet means
GANGAIN Reply
these are Rocky substances between mars and jupiter
GANGAIN
Comets are cosmic snowballs of frozen gases , rock and dust that orbit the sun. They are mostly found between the orbits of Venus and Mercury.
Aarya
hllo
John
hi
John
qt rrt
John
r u there
John
hey can anyone guide me abt international astronomy olympiad
sahil
how can we learn right and true ?
Govinda Reply
why the moon is always appear in an elliptical shape
Gatjuol Reply
Because when astroid hit the Earth then a piece of elliptical shape of the earth was separated which is now called moon.
Hemen
what's see level?
lidiya Reply
Did you mean eye sight or sea level
Minal
oh sorry it's sea level
lidiya
according to the theory of astronomers why the moon is always appear in an elliptical orbit?
Gatjuol
hi !!! I am new in astronomy.... I have so many questions in mind .... all of scientists of the word they just give opinion only. but they never think true or false ... i respect all of them... I believes whole universe depending on true ...থিউরি
Govinda
hello
Jackson
hi
Elyana
we're all stars and galaxies a part of sun. how can science prove thx with respect old ancient times picture or books..or anything with respect to present time .but we r a part of that universe
Reply
w astronomy and cosmology!
Michele
another theory of universe except big ban
Albash Reply
how was universe born
Asmit Reply
there many theory to born universe but what is the reality of big bang theory to born universe
Asmit
what is the exact value of π?
Nagalakshmi
by big bang
universal
there are many theories regarding this it's on you believe any theory that you think is true ex. eternal inflation theory, oscillation model theory, multiple universe theory the big bang theory etc.
Aarya
I think after Big Bang!
Michele
from where on earth could u observe all the stars during the during the course of an year
Karuna Reply
I think it couldn't possible on earth
Nagalakshmi
in this time i don't Know
Michele
is that so. the question was in the end of this chapter
Karuna
in theory, you could see them all from the equator (though over the course of a year, not at pne time). stars are measured in "declination", which is how far N or S of the equator (90* to -90*). Polaris is the North star, and is ALMOST 90* (+89*). So it would just barely creep over the horizon.
Christopher
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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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