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Since algebra uses the same symbols as arithmetic for adding, subtracting, multiplying and dividing, you're already familiar with the basic vocabulary.
In this lesson, you'll learn some important new vocabulary words, and you'll see how to translate from plain English to the "language" of algebra.
The first step in learning to "speak algebra" is learning the definitions of the most commonly used words.
Algebraic Expressions  Variables  Coefficients  Constants  Real Numbers  Rational Numbers  Irrational Numbers  Translating Words into Expressions
Algebraic
Expressions
An algebraic expression is one or more algebraic terms in a phrase.
It can include variables,
constants,
and operating symbols, such as plus and minus signs. It's only a phrase, not
the whole sentence, so it doesn't include an equal sign.
Algebraic
expression:
3x^{2} + 2y + 7xy + 5
In an algebraic expression, terms are the elements separated by the plus or minus signs. This example has four terms, 3x^{2}, 2y, 7xy, and 5. Terms may consist of variables and coefficients, or constants.
Variables
In algebraic expressions, letters represent variables. These letters are actually
numbers in disguise. In this expression, the variables are x and y. We call
these letters "variables" because
the numbers they represent can vary—that
is, we can substitute one or more numbers for the letters in the expression.
Coefficients
Coefficients are the number part of the terms with variables. In 3x^{2}
+ 2y + 7xy + 5, the coefficient of the first term is 3. The coefficient
of the second term is 2, and the coefficient of the third term is 7.
If a term consists of only variables, its coefficient is 1.
Constants
Constants are the terms in the algebraic expression that contain only numbers.
That is, they're the terms without variables. We call them constants because
their value never changes, since there are no variables in the term that can
change its value. In the expression 7x^{2}
+ 3xy + 8 the constant term is "8."
Real
Numbers
In algebra, we work with the set of real numbers, which we can model using
a number line.
Real numbers describe realworld quantities such as amounts, distances, age, temperature, and so on. A real number can be an integer, a fraction, or a decimal. They can also be either rational or irrational. Numbers that are not "real" are called imaginary. Imaginary numbers are used by mathematicians to describe numbers that cannot be found on the number line. They are a more complex subject than we will work with here.
Rational
Numbers
We call the set of real integers and fractions "rational numbers."
Rational comes from the word "ratio"
because a rational number can always be written as the ratio,
or quotient, of two integers.
Examples
of rational numbers
The fraction ½ is the ratio of 1 to 2.
Since three can be expressed as three over one, or the ratio of 3 to one, it is also a rational number.
The number "0.57" is also a rational number, as it can be written as a fraction.
Irrational
Numbers
Some real numbers can't be expressed as a quotient of two integers. We call
these numbers "irrational numbers". The decimal form of an irrational
number is a nonrepeating and nonterminating decimal number. For example,
you are probably familiar with the number called "pi". This irrational
number is so important that we give it a name and a special symbol!
Pi cannot be written as a quotient of two integers, and its decimal form goes on forever and never repeats.
Translating
Words into Algebra Language
Here
are some statements in English. Just below each statement is its translation
in algebra.
the
sum of three times a number and eight
3x + 8
The words "the sum of" tell us we need a plus sign because we're going to add three times a number to eight. The words "three times" tell us the first term is a number multiplied by three.
In this expression, we don't need a multiplication sign or parenthesis. Phrases like "a number" or "the number" tell us our expression has an unknown quantity, called a variable. In algebra, we use letters to represent variables.
the
product of a number and the same number less 3
x(x – 3)
The words "the product of" tell us we're going to multiply a number times the number less 3. In this case, we'll use parentheses to represent the multiplication. The words "less 3" tell us to subtract three from the unknown number.
a number divided by the same number less five
The words "divided by" tell us we're going to divide a number by the difference of the number and 5. In this case, we'll use a fraction to represent the division. The words "less 5" tell us we need a minus sign because we're going to subtract five.
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