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Equal Sign Use/Abuse
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Equal Sign Use/Abuse
Real Numbers
Simplify Square Roots
Absolute Value |x|
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Rectangular Coords 1, 2&3D
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Abuse of Equal Sign

The solution of the equation

3
4
x
4

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

3
4
x =3
3     

in place of writing x = 3. While a person who writes 

x =3
3     

 

may mean x = 3, the expression 

x =3
3     

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.
  1. two different symbols (or expressions) have the same value
  2. two different calculations or expressions have the same value,
  3. 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.
4+5
=
7+2
true as both sides have the same value 9
r2
=
r·r
true as the right hand side says how to compute the left-handside.
3x+2
=
x+6
true when and only when x has the value 2.
x+4
=
x+6
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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