What MATHEMATICIANS say answer of 6/2(2+1) =...?
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Answer
1
Explanation
Let us solve it by BODMAS
6/2(2+1)
= 6/2(3)
Here the parentheses still exist so solve it first and don't try to convert it into multiplication sign 'x'. The problem is that people do not finish the parentheses step before jumping to the multiplication/division step and move toward wrong answer.
= 6/6
= 1
The problem is that some solve it as 6/2(2+1)= 6/2(3)= 6/2*3=3*3=9 which is wrong.
Take a look on following images which are taken from different books. How mathematicians define the factor, division and coefficient. How math expert solve 1/2a, 1/ab or expplain it.
Factors
This brings us to factors, such as 5ab = b5a = ba5=ab5. Factors can be rearranged in any order because their product result remains the same. (means the quantity doesn’t change).
Because the quantity is the same therefore there is no need of parenthesis. These rules was very clear in past because people were actually taught basics of mathematics but these problems appears in today, when more and more people are relying on calculators, most of the programmings does not follow the above rules, they always tell you that you should use parentheses when entering the data.
Division
Let us also see some division rules.
Coefficient
There is also the coefficient rule that how coefficients are “attached” to their variable to make a single quantity?
The above images are taken from following books
1) Bonnycastle's introduction to Algebra by John Bonnycastle, James Ryan, John Francis Jenkins
2) Algebra for high schools and colleges: containing a systematic exposition By James B. Dodd
Some more links
Introduction to Real Analysis by Bartle and Sherbert
Page 42: written as
(x²-2)/2x
Page 53: written as
1/2n = 1/(2n)
Page 350: written as
e/2M
Page 363: written as
1/n(n+1) < 1/n² <= 1/n and
n/(n-2)(n-1)
Introduction to Real Analysis by William F. Trench
Page 123:
k(ε/2k) = ε/2
An Introduction to Real Analysis: Richard Wheeden, Antoni Zygmund 1977
Page # 31:
2Mε/4(k + 1)M = ε/2(k + 1)
Introduction to Real Analysis by William F. Trench
Page # 123:
k(ε/2k) = ε/2
The Everything Guide to Algebra by Christopher Monahan
Page # 46 shows: 6x³÷3x=2x²
(That's because "3x" is a single quantity/operand, and is a product of both factors: 3 and x)
He also describes "PEMDAS" in detail.
Basic Algebra I: Second Edition by Nathan Jacobson
Page # 116:
a/b + c/d = (ad +bc)/bd
Introduction to Algebra by Peter Jephson Cameron
Page # 17
"Since mx/nx = m/n" [Notice how it is NOT m * x / n * x = mx²/n ???]
Other links for correct solution
http://www.jstor.org/stable/pdf/2972726.pdf
http://ramanujan.math.trinity.edu/wtrench/texts/TRENCH_REAL_ANALYSIS.PDF
http://www.slate.com/articles/health_and_science/science/2013/03/facebook_math_problem_why_pemdas_doesn_t_always_give_a_clear_answer.html
http://www.purplemath.com/modules/orderops2.htm
https://www.onlinemathlearning.com/multiply-divide-expressions.html
http://www.askamathematician.com/2011/04/q-how-do-you-calculate-6212-or-48293-whats-the-deal-with-this-orders-of-operation-business/
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ORDER OF OPERATION ([{(-)}], ()^2 √ , ÷, ×, +, -)
What is the correct answer of 6/2(2+1)=? (1 or 9)
How MATHEMATICIANS explain answer of 6/2(2+1)=? (1 or 9)
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