Skip to main content

Common math Errors

Error in mathematics

Today I will show you Common mistake made in Algebraic. and ways to go about them

Error    Correct/Justification/Example

  1. $$ \frac{2}{0} \neq 0 \text{ and } \frac{2}{0} \neq 2 $$ Division by zero is undefined
  2. \begin{align}  -3^2 & \neq  9 \\ \text{But} \\ (-3)^2 & = 9 \\ \text{ and} \\ -3^2 & = -9 \end{align} Watch parenthesis!
  3. \begin{align} (x^2)^3 & \neq  x^5 \quad \text{but} \\ (x^2)^3 & = x^2x^2x^2 \\ & = x^6 \end{align}
  4. \begin{align} \frac{a}{b+c} \neq & \frac{a}{b} + \frac{a}{c} \\ _ \text{Clear Example below} \\ \rightarrow\ \frac{1}{2} & = \frac{1}{1+1} \\ \text{but} \quad \frac{1}{1+1} & \neq \frac{1}{1} + \frac{1}{1} \\ \frac{1}{1} + \frac{1}{1} & = 2 \end{align}
  5. $$ \frac{1}{x^2+x^3} \neq x^{-2} + x^{-3} $$ A more complex of the previous error
  6. $$ -a(x-1) \neq -ax-a \quad \text{but} \\ {-a(x-1) = -ax+a } $$ Make sure you distribute the "-"!
  7. \begin{align} (x+a)^2 & \neq x^2+a^2 \quad \text{but} \\ (x+a)^2 & = (x+a)(x+a) \\  & = x^2+2xa+a2 \end{align}
  8. \begin{align} \require{cancel} \frac{\cancel{a}+bx}{\cancel{a}} & \neq -ax-b \quad \text{but} \\ \frac{a+bx}{a} & = \frac{a}{a}+\frac{bx}{a} \\  & = 1 + \frac{bx}{a} \end{align} Beware of incorrect canceling
  9. \begin{align} \sqrt{x^2 + a^2} & \neq  x + a \\ _ \text{Proof} \\ 5 & = \sqrt{25} \\ \sqrt{3^2 + 4^2} & = \sqrt{3^2} + \sqrt{4^2} \\ & = 3+4 \\ & = 7 \\ \text{and } 5 & \neq 7 \end{align}
  10. $$ \sqrt{x+a} \neq \sqrt{x}+ \sqrt{a} $$ See previous error
  11. $$ (x+a)^n \neq x^n+a^n  \text{   and   } \\ {\sqrt[n]{x+a} \neq \sqrt[n]{x} + \sqrt[n]{a}} $$ More general versions of previous three errors
  12. \begin{align} 2(x+1)^2  & \neq   (2x+2)^2 \\ _ \text{Proof} \\ 2(x+1)^2 & = 2(x^2+2x+1) \\ & = 2x^2+4x+2 \end{align} Square first then distribute!
  13. $$ (2x+2)^2 \neq 2(x+1)^2 $$ See the previous example. you can not factor out a constant if there is a power on the parenthesis!
  14. \begin{align} \sqrt{-x^2 + a^2} & \neq -\sqrt{x^2 + a^2} \\ _ \text{But} \\ \sqrt{-x^2 + a^2} & = (-x^2 + a^2)^{\frac{1}{2}} \end{align} Now see the previous error
  15. \begin{align} \frac{a}{\left (\frac{b}{c} \right)} & \neq \frac{ab}{c} \\ _ \text{but} \\ \frac{a}{ \left (\frac{b}{c} \right) } & =\left(\frac{a}{b} \right)\left(\frac{1}{c} \right) \\ & = \frac{ac}{b} \end{align}
  16. \begin{align} \frac{\left (\frac{a}{b} \right)}{c} & \neq \frac{ac}{b} \\ _ \text{but} \\ \frac{\left (\frac{a}{b} \right)}{c} & =\left(\frac{a}{b} \right)\left(\frac{1}{c} \right) \\ & = \frac{a}{bc} \end{align}

Question misinterpret based on the way they look

Mostly question posted on the Internet is been misunderstood.
1.  Imaging someone post this \( 3\sqrt{x}\) as 3rootx
3rootx can be  \( 3\sqrt{x}\) or \( \sqrt[3]{x}\)
 $$ 3\sqrt{x} = 9^{\frac{1}{2}} x^{\frac{1}{2}} = \left(9x \right)^{\frac{1}{2}} = \sqrt{9x} $$
and $$ \sqrt[3]{x} = x^{ \frac{1}{3} } $$
2.  Real question \( \frac{a+2}{a-2} \) posted as a+2/a-2
to me this a+2/a-2 mean \( a + \frac{2}{a} - 2\) using parentheses ( ) will give make it easy to understand. Eg ( a+2 )/( a-2 )

Nb:

1.  Order of Operation

  • Do things in parentheses first
  • Exponents
  • Multiplication and Division
  • Addition and Subtraction

2.   Master the Key Concepts

  • Do not try to memorize the processes
  • Understand the process and logic that is involved
Labels:

Comments

Post a Comment

Popular Posts

Commutative Law of Addition

TOPIC OVERVIEW In this article, I will explain the use of the Commutative Law of Addition , and suggest a better way of thinking about the topic. Please add your ideas in the comments. THE COMMUTATIVE LAW ("change" the order of the numbers or letters) Over the years, people have found that when we add or multiply, the order of the numbers will  not affect the outcome. " Switching " or " changing " the order of numbers is called " commuting ". When we change the order of the numbers, we have applied the " Commutative Law ". In an addition problem, it is referred to as the " Commutative Law of Addition " such that Addition on the Real Number ( 2, -5, and $ \frac{2}{7} $  )  S , T it is true  that    S + T = T + S THE SUM OF 2 REAL NUMBERS Example Real Numbers 2 & 3 Let make S=2 & T=3 Solution S+T = T + S 2+3 = 3 + 2 5 = 5 Note: - This is the sum of 2 positive real numbers.  S...
Post a Comment
Read more

Index

Topic Overview The index of a number says how many times to use in a multiplication. it is written as a small number to the right and above the base number The plural of index is indices (other names for index are exponent or power) Positive Exponents Positive Exponents is nothing but the power of a number is to be positive. Example: 5 2 = 5*5*5 = 125 2 4 = 2*2*2*2 = 16 you can multiply any number by itself, as many as you want using exponents. [exponent] make it easier to write 8 4 is easier to write and to read than 8*8*8*8 In General a n tells you to multiply a by itself so there are n of those a ’s: Negative Exponents A Negative Exponent  is nothing but the power of a number is to be , what could the opposite of multiply?      Dividing. A Negative exponent means how many times to divide one by the number Example 4 -1 =1 ÷4 = 0.25 you can have many divides 5 -3 = 1 ÷5÷5÷5 =0.008 ...
Post a Comment
Read more