Evaluation of expressions

preliminaries

Modern mathematical notation, which we often call algebra, has evolved over relatively recent centuries and continues to change as the need arises. 

We trace the use of symbols to express mathematics, for example, to the Arab mathematician al-Khwarizmi in the 9th century. The 'equals' sign $=$= was introduced much later, in the 16th century, by Robert Recorde, an English physician and mathematician.

In the absence of mathematical notation, it would be necessary to use ordinary language to express the steps of a calculation or the explanation for a more general statement. Students sometimes find it hard to believe that algebra exists to make mathematics easier but this is indeed its purpose. 

Mathematical expressions begin to be comprehensible when we understand the meanings of the symbols and the rules by which they can be combined.

The symbols include those for the usual arithmetical operations $+$+, $-$−, $\div$÷, $\times$×, as well as those for powers and roots $\sqrt{ }$√ and other special functions, and also the symbols that mean 'equals' $=$=, 'less than' $<$< and 'greater than' $>$>. Brackets, $\left(\right)$() or $\left[\right]$[] or $\left\{\right\}${} are used to help clarify the correct order for performing the various operations.

We use letters from various alphabets to stand for numbers that are either unknown or variable. We can often express a relationship between numbers even when the particular numbers are not fixed.

Expressions and equations

For example, the equation $y=2x$y=2x means 'Whatever $x$x is, the quantity $y$y is twice as much'. And the equation $y=x+2$y=x+2 means we get the value of $y$y by adding $2$2 to the value of $x$x. 

(A number next to a symbol means the two are to be multiplied.)

The symbol combinations $2x$2x and $x+2$x+2 by themselves, without an equals sign, are called expressions.

We can evaluate an expression if we know the values of the variables within the expression. However, there are rules about the order in which the arithmetic operations are to be done when the evaluation involves several steps. The rules arise from basic facts about adding and multiplying numbers.

Example 1

Compare these expressions: $2x+3$2x+3 and $2(x+3)$2(x+3).

Both expressions involve an addition and a multiplication. In the first case, the multiplication is done first, followed by the addition. In the second case, the addition is done before the multiplication, because the quantities to be added are inside brackets.

You should check that the two are different by choosing a value for $x$x. For example, if $x=5$x=5, we have 

$2x+3$2x+3	$=$=	$2\times5+3$2×5+3
	$=$=	$10+3$10+3
	$=$=	$13$13

 On the other hand,

$2(x+3)$2(x+3)	$=$=	$2\times(5+3)$2×(5+3)
	$=$=	$2\times8$2×8
	$=$=	$16$16

 

In fact, $2(x+3)$2(x+3) is the same as $2x+6$2x+6. Everything inside the bracket has to be multiplied by the number that multiplies the bracket. Again, if $x=5$x=5, then $2x+6=16$2x+6=16, as expected.

 

Example 2

Compare the expressions $\sqrt{x+1}$√x+1 and $\sqrt{x}+\sqrt{1}$√x+√1 by substituting the values $x=0$x=0, $x=3$x=3 and $x=48$x=48. What general rule should be used in this case?

We could make a table to include the possibilities.

$x$x	$\sqrt{x+1}$√x+1	$\sqrt{x}+\sqrt{1}$√x+√1
$0$0	$\sqrt{0+1}=\sqrt{1}=1$√0+1=√1=1	$\sqrt{0}+\sqrt{1}=0+1=1$√0+√1=0+1=1
$3$3	$\sqrt{3+1}=\sqrt{4}=2$√3+1=√4=2	$\sqrt{3}+\sqrt{1}\approx1.732+1=2.732$√3+√1≈1.732+1=2.732
$48$48	$\sqrt{48+1}=\sqrt{49}=7$√48+1=√49=7	$\sqrt{48}+\sqrt{1}\approx6.928+1=7.928$√48+√1≈6.928+1=7.928

If the square roots are done before the addition, the answer will nearly always be too large. The general rule to adopt is that the operations under the square root sign should be done first.

 

Example 3

Verify that the virgule or fraction line is to be treated in much the same way as brackets so that the numerator and denominator are evaluated separately, before the division.

The fraction $\frac{2+3}{5+1}$2+35+1​ is the same as $\left(2+3\right)\div\left(5+1\right)$(2+3)÷(5+1).

But could it be the same as $\frac{2}{5+1}+\frac{3}{5+1}$25+1​+35+1​ or possibly $\frac{2}{5}+\frac{3}{1}$25​+31​ or maybe, $\frac{2+3}{5}+\frac{2+3}{1}$2+35​+2+31​? 

Here are the four cases. The first one is known to be correct.

$\frac{2+3}{5+1}$2+35+1​	$=$=	$\frac{5}{6}$56​
$\frac{2}{5+1}+\frac{3}{5+1}$25+1​+35+1​	$=$=	$\frac{2}{6}+\frac{3}{6}=\frac{5}{6}$26​+36​=56​
$\frac{2}{5}+\frac{3}{1}$25​+31​	$=$=	$\frac{2}{5}+\frac{15}{5}=\frac{17}{5}$25​+155​=175​
$\frac{2+3}{5}+\frac{2+3}{1}$2+35​+2+31​	$=$=	$\frac{5}{5}+\frac{5}{1}=6$55​+51​=6

Clearly, only the second option is equivalent to the first. So the general rule when dealing with fractions is either evaluate the numerator and denominator separately or use the equivalence $\frac{a+b}{x+y}\equiv\frac{a}{x+y}+\frac{b}{x+y}$a+bx+y​≡ax+y​+bx+y​.

 

Example 4

Show by means of a counterexample that $x^2+y^2$x2+y2 is not generally equal to $\left(x+y\right)^2$(x+y)2.

We pick any two numbers for $x$x and $y$y. Say, $x=5$x=5 and $y=4$y=4. So,

$x^2+y^2$x2+y2	$=$=	$5^2+4^2$52+42
	$=$=	$25+16$25+16
	$=$=	$41$41

But, 

$\left(x+y\right)^2$(x+y)2	$=$=	$\left(5+4\right)^2$(5+4)2
	$=$=	$9^2$92
	$=$=	$81$81

 

In an expression like $\left(x+y\right)^2$(x+y)2, we must do the addition first and then square the result. Alternatively, everything in the first bracket must be multiplied by everything in the second bracket.

$\left(x+y\right)^2$(x+y)2	$=$=	$\left(x+y\right)\left(x+y\right)$(x+y)(x+y)
	$=$=	$x^2+xy+yx+y^2$x2+xy+yx+y2
	$=$=	$x^2+2xy+y^2$x2+2xy+y2

When $x$x and $y$y are positive numbers, $\left(x+y\right)^2$(x+y)2 must be greater than $x^2+y^2$x2+y2 by the amount $2xy$2xy.

 

More Worked Examples

Question 1

Question 2

Question 3