Abuse of Equal Sign
The solution of the equation
is x =3. But is an error, a mistake, a major misuse of the equal sign to
insert an = 3 besides the x in the above equation to obtain or write
in place of writing x = 3. While a person who writes
may mean x = 3, the expression
actually means a third of x is 3.
Mathematics and English teachers should mark what is
written, not what might of been meant, so their students learn to write
precisely.
Precision is important. A person who does not write exactly
what he or she means does not know how to read precisely what is written in
their notes and textbooks, and so is easily confused.
Moreover, in mathematics, confusion about notation, what is proper or
not, leads to errors in all calculations and in problem solving.
Ouch!
2.1 Proper Use of the Equal Sign
Here are a few words about the equal sign. The equal sign is used to say two
different items have the same value. So the equal sign = may be used to say or
suggest the following.
- two different symbols (or expressions) have the same value
- two different calculations or expressions have the same value,
- the value of a number or quantity is the same as the value of another
expression.
The suggestion of identical values can be true or false depending on
circumstances. See the next examples.
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true as both sides have the
same value 9 |
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true as the right hand side
says how to compute the left-handside. |
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true when and only when x has
the value 2. |
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always false as x + 6 = (x+4)
+2 is two units more than x+4 |
Here the first equation or equality holds (meaning is true) since both 4+5
and 7+2 are expressions with the same value 9.
The second equation r2
= r·r always holds, no matter what value you give to r. It
tells us how to compute the number or quantity described by the expression r2.
The third equation 3x+2 = x+6 holds (is true) when and only when x
= 2.
When x has a value other than 2, the statement (suggestion or
assertion) that 3x+2 has the value as x+6 is false.
The
fourth statement x+4 = x+6 is always false.
No value given to (or
substituted for) x will make x+4 and x+ 6 have the same value.
Adding 4 and adding 6 to a number gives different results, different
values, no matter what the number is.
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