The language of algebra

INSTRUCTIONAL MODULE FOR
MATHEMATICS
THE LANGUAGE OF ALGEBRA
jrbt 2014

The Language of Algebra
Webster defined language as a systematic means of communicating ideas
and feelings by the use of conventionalized sign, gestures, marks, and
especially articulate vocal sound. Some such signs and symbols have
become internationally known and can be understood by Europeans, Asians,
or Americans. For instance, in the sentence below, can you replace the
symbols with the word or words they suggest?
Man in his pursuit for power should remember to
uphold
and
and fair play in order to preserve
world
which Christ
gave
humanity through his
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I. OBJECTIVES
At the end of this presentation, the students are
expected to
1. Acquire sufficient vocabulary to enable them to
interpret algebraic expressions and to translate
verbal statements to the language of algebra.
2. Appreciate the role of Algebra in expressing
mathematical ideas in a universal language
understood by all people.
3. Use the language of algebra in translating
mathematical phrases and sentences.jrbt 2014

II. MATHEMATICAL SYMBOL
The language of algebra is made up of
a. Digits: 0,1,2,3,4,5,6,7,8,9
b. Letters to represent numbers: x,y,z,etc.
c. Operational symbols as indicated in
the next slide.
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Operational symbols Verbal interpretations
+ plus, the sum of, increased by,
added to, more than, total of
- minus, less, subtracted from,
difference of, taken away from,
decreased by, deducted from,
diminished by
x, ., ( ) multiplied by, the product of,
times, twice, thrice etc.
÷, x/y divided by, the quotient of x
over y
x ▪ x = ( x )2 = ( x )2 The square of a number
x ▪ x ▪ x = ( x )3 = x 3 x to the power, the cube of a
number
√ square root
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> greater than` ≥ greater than or equal to
< less than ≤ less than or equal to
= equal ≠ is not equal to
e. ( ) quantities in the parenthesis to be treated as a
single quantity
Note that most of this symbols are not strange to you.
They are the same symbols used in arithmetic but with
slight deviation. For instance, in algebra the x sign has
ceased to become the usual symbol for multiplication.
Instead the raised dot and the parenthesis have become
the indications for multiplication. Thus:
(1) Three times four is 3 • 4 or 3( 4 )
(2) a times b is denoted as a • b or (a) (b) or just plain abjrbt 2014

Where the letter are involved the sign of multiplication
may be entirely omitted. .
The main difference between the language of algebra and
the language of arithmetic lies in the use of letters to
represent numbers. Such letters are sometimes referred
to as general numbers because they can stand for any
number or any unspecified quantity.
“ Isa at
dalawa”
“ One plus
two”
“ Ono mas
dos”
The people making the above statements could have
understood each other instantly had they use the
language of mathematics.jrbt 2014

Verbal Expression Symbols
1. a. Eight increased by nine gives
seventeen
b. The sum of eight and nine is
seventeen
c. Nine added to eight gives
seventeen
8 ÷ 9 = 17
2. Five times the sum of six and twelve 5 ( 6 + 12 )
3. a. Certain number increased by ten.
b. Ten more than a number
c. Juan’s age ten years hence.
d. Lynss’s money if she has ten
pesos more the Remy
x + 10
( Any letter can take the place of that
certain number; thus n + 10 is another
translation. )
4. a. A number decreased by five.
b. Five less than a certain number.
c. The difference between a certain
number and five.
d. Carmen’s age five years ago
x - 5
5. The quotient of a divided by b added
to the product of a and b.
a/b + ab
6. Twice a certain number increased by 2x + 5 = 15
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Note that in every expression there are at least two
quantities which are related to each other by some
operation. When all the quantities are specified , a
numerical sentence or phrase is obtained as in example
1 and 2
When one of the quantities is not specified, a letter is
used to represent it and an open phrase , as in example
3, or an open sentence as in example 6, may be
obtained. One symbol can have different equivalents.
Thus the plus sign (+) could stand for “increased by,
added to, “etc. Likewise, an open phrase may have
varied interpretations. So aside from what has been
stated in example 3, n + 10 may mean 10 more green
apples than the number of red apples.
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PRE - TEST
A. Translate each of the following word phrases or
sentences into mathematical symbols.
1. Two subtracted from ten is eight.
2. Two times the sum of six and twelve is the same as
the product of nine times four.
3. Nine more than six times a number.
4. Twice a number decreased by seven.
5. The number of inches in f feet.
6. The cost of a dozen books at x pesos a book.
7. The number of months in n year.
8. The square of a certain number a.
9. The cost of one book if a dozen books cost p pesos.
10. A number added to one-half of itself.
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B. Interpret in the best way you can the meaning of the
following algebraic expressions.
1. n-2
2. (1/3 ) y
3. 5a
4. (xy)2
5. a/b
6. 3x + 8
7. 9 ( a + d)
8. c2 + d2
9. ( c+ d )
10. 8x + 3
5
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III. THE VARIABLE AND ITS
DOMAIN
Mathematical Expressions
In Mathematics as in English we have
expressions called phrases and sentences. A
comparison of English phrases and sentences
with mathematical phrases and sentences is
shown in the next slide.
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In English In Mathematics
English expressions can be classified
into
Mathematical expressions can be
classified into
I. A phrase – a group of words which
does not express a complete
thought.
Example: Most beautiful Woman in
the world
1. Arithmetical expressions composed
of:
A. Numerical Phrases
Example:
a. 6 – 1
b. 3 x 2 + 6 - 8
II. A sentence – a word or group of
words that conveys a compulete
thought.
Example:
a. Go.
b. She is one of the most
beautiful woman in the world.
This sentence is neither true of false
until we give the antecedent or
replacement of she. It is an open
sentence.
c. Mt. Mayon is the most beautiful
volcano in the world.
B. Numerical or arithmetical
sentences.
Example:
1. 3 x 2 = 4 + 2
2. 12 – 19 > 7
II. Algebraic expressions composed of :
A. Open phrase
Examples:
1. ? + 9
2. n – 8
B. Open Sentences
Examples:
1. 2x – 8 = 4
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only 5, and 3x2+6-8 is a symbol for 4 and only 4.
An arithmetical or a numerical phrase sentences is a
statement of relationship between two phrases. The
relationship might be one of equality of inequality. The
sentence may be true or false.
An open phrase, sometimes called algebraic phrase, is a
numeral for an indefinite number. N + 9 stands for an
indefinite number. Its value depends, upon the replacement
of the unspecified quantity n. It is 9 when n = 0; 24 when n =
15. An open phrase is a mathematical expression which
contains one or more letters to represent the unspecified
number.
An open sentence, is a statement of relationship between
two mathematical phrases, at least one of which is an open
phrase. It is a sentence which is neither true of false. 2x – 8=
4 states the equality between the open phrase 2x-8 and the
arithmetical phrase 4. its truth or falsity will be determined by
the given value of x.
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Unknowingly, you have been making open sentences.
Actually the sentences, “a certain number is increased
by five equals three” is an open sentence. The phrase a
“certain number” can be indicated by other symbols
that will denote the missing link needed to make the
sentence true. Some symbols are given below.
_____ + 5 = 3 * + 5 = 3
? + 5 = 3 ∆ + 5 = 3
Instead of using (____), ( ? ), (* ), or (∆), mathematics
often use a letter in place of the unknown number. If the
letter is used, then the sentence becomes
x + 5
or n + 5
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makes the sentence true is called the truth set or solutions set.
In x + 5 = 3, the truth set or solution set is -2. the truth set or
solution set is selected from a given set is called the universal
set or the domain of the variable. Since the elements of the
domain are possible replacement for the variable, it is also
known as the replacement set. To illustrate the use of the
domain in x + 5 = 3,
Example 1:
Domain: {counting numbers}
Truth set:{ } because no counting number will satisfy the
equation
Example 2:
Domain: {rational numbers}
Truth set: {-2}
When the domain of a variable is not specified; let us agree to
use the rational numbers.
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IV. COEFFICIENTS,
EXPONENTS, POWER, AND
SCIENTIFIC NOTATION
3x ( a) x 6.99 x 108 3 x 4
= 12
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Coefficients
When one number can be divided by another with a remainder of zero,
the second number is called a factor of the first. Thus 4 is a factor of 24
because 24 ÷ 4 + 6 with a remainder of zero and because 6 x 4 = 24.
But 4 is not a factor of 10 because 10 + 4 has a remainder and also 4 x 2
does not give 10. In symbols, we say a and b are factors of c if a • b = c
and c/a = b; c/b = a,
In algebra when tow or more number symbols are multiplied to form a
product, each symbol is called the coefficient of the other. Thus;
a. in 2x 2 is the coefficient of x;
x is the coefficient of 2
b. in 3ay 3 is the coefficient of ay;
a is the coefficient of 3y
y is the coefficient of 3x
The number factor is called the numerical coefficient. The letter factor is called
the literal coefficient . When no number is written before a letter as in x or xy, the
numerical coefficient is understoo to by 1. unless otherwise specified, the term
“coefficient” generally refers to numerical coefficient. Thus, the coefficient of 5cd
is 5.
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in many situations. Take the addition to a number to itself;
n+n+n+n+n can be solved very easily and written even more
quickly if we take n five times and write it 5n. Similarly the
multiplication of three or more numbers can simply be
expressed by its indicated products as abcd instead of
a.b.c.d, etc. This shortening process comes in more
effectively when we wish to multiply a number several times
by itself. Hence, 3 x 3 can be expressed a 32; and 2x2x2 as
23 read as “three squares” and “two cubes” respectively,
Using variables, we have x.x = x y.y.y = y read as “x square”
and “y cube” .
This process of multiplying a number by itself two or more
times is called raising to a power. Hence, the process of
multiplying 3 by itself 2 times is indicated as (3)2 , meaning
3x3= 9; that of taking 2 five times by itself is (2)5 = 25 or 32,
obtained by performing 2x2x2x2x2. Note that (3)2 indicates
that 3 is to be squared while 32 is the answer and is only
another name of 9.
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This process of raising a number to a power
involves 3 terms; the base, the exponents and
the power. Discover the meaning of each term
from the given examples.
Multiplication Raising to a
power
Base Exponent Power
4 • 4 = 16 (4)2 = 42 or
16
4 2 42 or 16
2 • 2 • 2 = 8 (2)3 = 23 or 8 2 3 23 Or 8
x • x •x • x (x)4 = x4 x 4 x4
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Scientific Notation
Writing a number by scientific notations is one of the
useful applications of powers and exponents. By this
method one can express a very large or very small
number in a compact way. Since only positive exponents
have been considered, this section will be limited to
expressing only very large numbers by scientific notation.
Do you know that
a. Light travel at the rate of 669,600,000 miles per hour?
b. The diameter of the Earth is 41,800,000 fet?
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 Big numbers can be easily expressed and read
in scientific notation or standard notation. By this
method a number is renamed as the products of
two factors. One factor is a power of 10 and the
other factor is a number ≥ 1 < 10
Key Concept
Given
Number
1st Factor (the
large number
espressed as a
number > but <
10
2nd Fator
Power of
10
Scientific
Notation
a. 669,600,0
0
b. 41,800,00
0
6.696
4.180
108
107
6.696 x 108
4.180 x 107
jrbt 2014

A. identify the terms described or defined in each of the
following;
1. It is the shorthand of mathematics.
2. The letter or symbol used to represent a number in
algebraic expression.
3. The set of numbers from which to choose permissible
values of the variable.
4. A mathematical expression which contains a variable and
which represents an unspecified number.
5. An expression which contains a variable and which states
the relationship between two quantities.
6. A statement of equality or inequality between two phrases.
7. The 2 in 2x4
8. The base in (2x)3.
9. The coefficient of n.
10. The method of writing 6,100,000,000 as 6.1 x 10jrbt 2014

B. Choose the correct meaning of the given symbols.
1. 6x means 4. (x+3) means
a. 6.x.x.x a. x + 3
b. 6x.6x.6x b. (x+3) (x+3)
c. 6 times x+x+x+x c. 2x +6
2. 10 a2 b3 means 5. The symbol for -8xxyyy is
a. 10 .ab.ab.ab.ab.ab a. (-8xy)5
b. (10.a.a) (10.b.b.b) b. –8(xy)3
c. 10.a.a.b.b.b c. -8x2 y3
3. 6 + 3(4x-7) means
a. 9 times (4x-7)
b. 6 plus the product of 3 multiplied by (4x-7)
c. none of these
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C. Express each of the following phrases and
sentences as a mathematical expression.
1. Two subtracted from ten is eight.
2. The sum of x and y.
3. Three times a certain number.
4. Nine less than twice a number.
5. One-half of a certain number of students.
6. The square of twice a certain number.
7. The number of centavo in p pesos.
8. Nene’s age ten years ago.
10 The distance traveled by Mr. Reyes after a certain
number of hours if the he travels at a uniform speed of
50km per hour
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D. Express each of the following in scientific notation.
.
1. 6,000,000
2. 80,150,000
3. 2,700,000,000,000
E. Express each of the following in full.
1. 3.17 x 105
2. 5.6 x 108
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Reference
MATHEMATICS II
BASIC ALGEBRA FOR SECONDARY SCHOOLS
(Revised Edition)
Numidas O. Limjap
Carmen R. dela Peña
Coordinator
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Answer Key
PRE TEST
A.
1.) 10 – x 6. 2x
2. 2 (6+12) = 9 • 4 7. m/y
3.)9x 8. a2
4.)2x -7 9. cp/12
5. in/ft 10. (x/2) + x = 3x/2
B.
1. a certain number decreased by two
2. a certain number y divided by three
3. a certain number a multiplied by five
4. the product of x and y raised to the power of two
5. the quotient of a and b
6. three times a certain number x increased by eiight
7. the sum of a and d multiplied by nine
8. the sum of the square of c and the square of d
9. the sum of c and d
10. the sum of eight times a certain number x added by three and
divided by five.jrbt 2014

Answer Key
POST TEST
A.
1. algebra 6. equation
2. variable 7. base
3. domain 8. 2
4.algebraic expression 9. 1
5. numerical expression 10. scientific notation
B.
1. a 3. a 5. c
2. c 4. a
C.
1. 10 -2 6. (2x)2
2. x + y 7. c/p
3. 3x 8. a – 10
4. 2x – 9 9. d/w
5. x/2 10. d = 50h
D.
1.) 6.0 x 106 2. ) 8.015 x 107 3. ) 2.7 x 1012
E.
1.) 317, 000 2.) 560,000,000
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End of Presentation
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God Bless..
jrbt 2014

The language of algebra

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The language of algebra