2.2: Symbols and Notations
Variables and Constants
A basic characteristic of algebra is the use of symbols (usually letters) to represent numbers.
A letter or symbol that represents any member of a collection of two or more numbers is called a variable .
A letter or symbol that represents a specific number, known or unknown is called a constant .
In the following examples, the letter \(x\) is a variable since it can be any member of the collection of numbers {35, 25, 10}. The letter \(h\) is a constant since it can assume only the value 5890.
Suppose that the streets on your way from home to school have speed limits of 35 mph, 25 mph, and 10 mph. In algebra we can let the letter \(x\) represent our speed as we travel from home to school. The maximum value of \(x\) depends on what section of street we are on. The letter \(x\) can assume any one of the various values 35,25,10.
Suppose that in writing a term paper for a geography class we need to specify the height of Mount Kilimanjaro. If we do not happen to know the height of the mountain, we can represent it (at least temporarily) on our paper with the letter \(h\). Later, we look up the height in a reference book and find it to be 5890 meters. The letter \(h\) can assume only the one value, 5890, and no others. The value of \(h\) is constant .
Symbols of Operation, Equality, and Inequality
Binary Operation
A binary operation on a collection of numbers is a process that assigns a number to two given numbers in the collection. The binary operations used in algebra are addition, subtraction, multiplication, and division.
If we let \(x\) and \(y\) each represent a number, we have the following notations:
- Addition: \(x + y\)
- Subtraction: \(x - y\)
- Multiplication: \(x \cdot y\)
- Division: \(\dfrac{x}{y}\) \(x/y\) \(x \div y\) \(\sqrt[y]{x}\)
Sample Set A
\(a+b\) represents the sum of \(a\) and \(b\).
\(4+y\) represents the sum of \(4\) and \(y\).
\(8−x\) represents the difference of \(8\) and \(x\).
\(6x\) represents the product of \(6\) and \(x\).
\(ab\) represents the product of \(a\) and \(b\).
\(h3\) represents the product of \(h\) and \(3\).
\((14.2)a\) represents the product of \(14.2\) and \(a\).
\((8) (24)\) represents the product of \(8\) and \(24\).
\(5\cdot6(b)\) represents the product of 5,6, and \(b\).
\(6x\) represents the quotient of \(6\) and \(x\).
Practice Set A
Represent the product of 29 and \(x\) five different ways.
- Answer
-
\(29 \cdot x\), \(29x\), \((29)(x)\), \(29(x)\), \((29)x\)
If we let \(a\) and \(b\) represent two numbers, then \(a\) and \(b\) are related in exactly one of three ways:
Equality and Inequality Symbols:
\(a = b\) \(a\) and \(b\) are equal
\(a > b\) \(a\) is strictly greater than \(b\)
\(a < b\) \(a\) is strictly less than \(b\)
Some variations of these symbols include
\(a \not= b\) \(a\) is not equal to \(b\)
\(a \ge b\) \(a\) is greater than or equal to \(b\)
\(a \le b\) \(a\) is less than or equal to \(b\)
The last five of the above symbols are inequality symbols. We can negate (change to the opposite) any of the above statements by drawing a line through the relation symbol (as in \(a \not= b\)), as shown below:
\(a\) is not greater than \(b\) can be expressed as either
\(a \not > b\) or \(a \le b\)
\(a\) is not less \(b\) than can be expressed as either:
\(a \not < b\) or \(a \ge b\)
\(a < b\) and \(a \not \ge b\) both indicate that \(a\) is less than \(b\)
Grouping Symbols
Grouping symbols are used to indicate that a particular collection of numbers and meaningful operations are to be grouped together and considered as one number. The grouping symbols commonly used in algebra are
Parentheses: ( )
Brackets: [ ]
Braces: { }
Bar: __
In a computation in which more than one operation is involved, grouping symbols help tell us which operations to perform first. If possible, we perform operations inside grouping symbols first.
Sample Set B
(4 + 17) - 6 = 21 - 6 = 15
8(3 + 6) = 8(9) = 72
5[8 + (10 - 4)] = 5[8 + 6] = 5[14] = 70
2{3[4(17 - 11)]} = 2{3[4(6)]} = 2{3[24]} = 2{72} = 144
\(\dfrac{9(5+1)}{24+3}\)
The fraction bar separates the two groups of numbers 9(5 + 1) and 24 + 3. Perform the operations in the numerator and denominator separately.
\(\dfrac{9(5+1)}{24+3}=\dfrac{9(6)}{24+3}=\dfrac{54}{24+3}=\dfrac{54}{27}=2\)
Practice Set B
Use the grouping symbols to help perform the following operations.
\(3(1 + 8)\)
- Answer
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27
\(4[2(11 - 5)]\)
- Answer
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48
\(6{2[2(10 - 9)]}\)
- Answer
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24
\(\dfrac{1+19}{2+3}\)
- Answer
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4
The following examples show how to use algebraic notation to write each expression.
9 minus \(y\) becomes \(9 - y\)
46 times \(x\) becomes \(46x\)
7 times \((x + y)\) becomes \(7(x + y)\)
4 divided by 3, times \(z\) becomes \((\dfrac{4}{3})z\)
\((a - b)\) times \((b-a)\) divided by (2 times \(a\)) becomes \(\dfrac{(a-b)(b-a)}{2a}\)
Introduce a variable ( any letter will do but here we'll let \(x\) represent the number) and use appropriate algebraic symbolds to write the statement: A number plus 4 is strictly greater than 6. The answer is \(x + 4 > 6\).
The Order of Operations
Suppose we wish to find the value of \(16 + 4 \cdot 9\). We could
add 16 and 4, then multiply this sum by 9.
\(16 + 4 \cdot 9 = 20 \cdot = 180\)
multiply 4 and 9, then add 16 to this product.
\(16 + 4 \cdot 9 = 16 + 36 = 52\)
We now have two values for one number. To determine the correct value we must use the standard order of operations.
- Perform all operations inside grouping symbols, beginning with the innermost set.
- Perform all multiplications and divisions, as you come to them, moving left-to-right.
- Perform all additions and subtractions, as you come to them, moving left-to-right.
As we proceed in our study of algebra, we will come upon another operation, exponentiation, that will need to be inserted before multiplication and division.
Sample Set C
Use the order of operations to find the value of each number.
\(16 + 4 \cdot 9\). Multiply first.
= \(16 + 36\) Now add.
= \(52\)
\((27 - 8) + 7(6 + 12)\). Combine within parentheses.
= \(19 + 7(18)\) Multiply.
= \(19 + 126\) Now add.
= 145
\(8 + 2[4 + 3(6-1)]\). Begin with the innermost set of grouping symbols, ( ).
= \(8 + 2[4 + 3(5)]\) Now work within the next set of grouping symbols, [ ].
= \(8 + 2[4 + 15]\)
= \(8 + 2[19]\)
= \(8 + 38\)
= \(46\)
\(
\begin{aligned}
\dfrac{6+4[2+3(19-17)]}{18-2[2(3)+2]} &=\dfrac{6+4[2+3(2)]}{18-2[6+2]} \\
&=\dfrac{6+4[2+6]}{18-2[8]} \\
&=\dfrac{6+4[8]}{18-16} \\
&=\dfrac{6+32}{2} \\
&=\dfrac{38}{2} \\
&=19
\end{aligned}
\)
Practice Set C
Use the order of operations to find each value.
\(25 + 8(3)\)
- Answer
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49
\(2 + 3(18 - 5 \cdot 2)\)
- Answer
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26
\(4 + 3[2 + 3(1 + 8 \div 4\)
- Answer
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37
\(\dfrac{19+2\{5+2[18+6(4+1)]\}}{5 \cdot 6-3(5)-2}\)
- Answer
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17
Exercises
For the following problems, use the order of operations to find each value.
\(2 + 3(6)\)
- Answer
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\(20\)
\(18 - 7(8 - 3)\)
\(8 \cdot \div 16 + 5\)
- Answer
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\(7\)
\((21 + 4) \div 5 \cdot 2\)
\(3(8 + 2) \div 6 + 3\)
- Answer
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\(8\)
\(6(4 + 1) \div (16 \div 8) - 15\)
\(6(4 - 1) + 8(3 + 7) - 20\)
- Answer
-
\(78\)
\((8)(5) + 2(14) + (1)(10)\)
\(61 - 22 + 4[3(10) + 11]\)
- Answer
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\(203\)
\(\dfrac{(1+16-3)}{7} + 5(12)\)
\(\dfrac{8(6+20)}{8} + \dfrac{3(6+16)}{22}\)
- Answer
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\(29\)
\(18 \div 2 + 55\)
\(21 \div 7 \div 3\)
- Answer
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\(1\)
\(85 \div 5 \cdot 5 - 85\)
\((300 - 25) \div (6 - 3)\)
- Answer
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\(91\dfrac{2}{3}\)
\(4 \cdot 3 + 8 \cdot 28 - (3 + 17) + 11(6)\)
\(2{(7 + 7) + 6[4(8 + 2)]}\)
- Answer
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\(508\)
\(0 + 10(0) + 15[4(3) + 1]\)
\(6.1(2.2 + 1.8)\)
- Answer
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\(22.4\)
\(\dfrac{5.9}{2} + 0.6\)
\((4 + 7)(8 - 3)\)
- Answer
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\(55\)
\((10 + 5)(10 + 5) - 4(60 - 4)\)
\((\dfrac{5}{12} - \dfrac{1}{4}) + (\dfrac{1}{6} + \dfrac{2}{3})\)
- Answer
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\(1\)
\(4(\dfrac{3}{5} - \dfrac{8}{15}) + 9(\dfrac{1}{3} + \dfrac{1}{4})\)
\(\dfrac{0}{5} + \dfrac{0}{1} + 0[2 + 4(0)]\)
- Answer
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\(0\)
\(0 \cdot 9 + 4 \cdot 0 \div 7 + 0[2(2-2)]\)
For the following problems, state whether the given statements are the same or different.
\(x \ge y\) and \(x > y\)
- Answer
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Different
\(x < y\) and \(x \not \ge y\)
\(x = y\) and \(y = x\)
- Answer
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Same
Represent the product of 3 and \(x\) five different ways.
Represent the sum of \(a\) and \(b\) two different ways.
- Answer
-
\(a + b\), \(b + a\)
For the following problems, rewrite each phrase using algebraic notation.
Ten minus three
\(x\) plus sixteen
- Answer
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\(x + 16\)
51 divided by \(a\)
81 times \(x\)
- Answer
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\(81x\)
3 times \(x + y\)
\((x + b)\) times \((x + 7)\)
- Answer
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\((x + b)(x+7)\)
3 times \(x\) times \(y\)
\(x\) divided by (7 times \(b\))
- Answer
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\(\dfrac{x}{7b}\)
\((a + b)\) divided by \((a + 4)\)
For the following problems, introduce a variable (any letter will do) and use appropriate algebraic symbols to write the given statement.
A number minus eight equals seventeen
- Answer
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\(x - 8 = 17\)
Five times a number, minus one, equals zero.
A number divided by six is greater than or equal to forty-four.
- Answer
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\(\dfrac{x}{6} \ge 44\)
Sixteen minus twice a number equals five.
Determine whether the statements for the following problems are true or false.
\(6 - 4(4)(1) \le 10\)
- Answer
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true
\(5(4 + 2 \cdot 10) \ge 110\)
\(8 \cdot 6 - 48 \le 0\)
- Answer
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true
\(\dfrac{20+4.3}{16} < 5\)
\(2[6(1 + 4) - 8] > 2(11 + 6)\)
- Answer
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false
\(6[4 + 8 + 3(26 - 15) \not \le 3[7(10 - 4)]\)
The number of different ways 5 people can be arranged in a row is \(5 \cdot 4 \cdot 3 \cdot 2 \cdot 1\). How many ways is this?
- Answer
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120
A box contains 10 computer chips. Three chips are to be chosen at random. The number of ways this can be done is
\(\dfrac{10 \cdot 9 \cdot 8 \cdot 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1}{3 \cdot 2 \cdot 1 \cdot 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1}\)
How many ways is this?
The probability of obtaining four of a kind in a five-card poker hand is
\(\dfrac{13 \cdot 48}{(52 \cdot 51 \cdot 50 \cdot 49 \cdot 48) \div(5 \cdot 4 \cdot 3 \cdot 2 \cdot 1)}\)
What is this probability?
- Answer
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\(0.00024\), or \(\dfrac{1}{4165}\)
Three people are on an elevator in a five story building. If each person randomly selects a floor on which to get off, the probability that at least two people get off on the same floor is
\(1 - \dfrac{1 \cdot 4 \cdot 3}{5 \cdot 5 \cdot 5}\)
What is this probability?