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for all real numbers a and b, a*b is equivalent to b*a. is the statement true or false ?

for all real numbers a and b, a(b*c) = ab *ac

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This statement is true...

ab = ba

use 2=a and 3=b

2 * 3 = 3 * 2

Not trying to troll, but in mathematics isn't proved by example,. Just using one example, or even a million of them isn't enouth, unless the case is a there exists statement.

We have to base the mathematics we teach laterally from the ground up, from the axioms and definitions that are made to the conclusions that arise from them.


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5 Answers

This statement is true for all real numbers and is known as the Commutative Property. It not only applies to multiplication, but to addition as well: a + b = b + a

The other case you mentioned a(b*c) = ab*ac is false.

There are certain other properties that are central to understanding algebra and working with real numbers that you may also learn depending on your level:

Assiociative property: Given real numbers a, b, and c: (a +b) + c = a + (b +c) also, (a*b)*c = a*(b*c)

Distributive property: Given real numbers a, b, and c: a*(b + c) = a*b + a*c
This property can be shown by applying the definition/concept of multiplication:
a*(b + c) means that we have "a" groups of (b + c) hence,
a*(b + c) = (b + c) + ...  + (b + c) <- a times, we are adding an "a" number of (b + c)'s.
Next we ask the questions how many b's do we have and how many c's do we have? Well applying basic counting we know that there an "a" number of b's, and an "a" number of c's. in other words,

a*(b + c) = (b + c) + ... + (b + c) [a times] = a*b + a*c


Identity: Pretty states that there is a real number for addition and there is a real number for multiplication that doesn't interact with any other number for the respective operation. For addition the number is 0. 0 is know as the additive identity. For multiplication the number is 1, 1 is known as the multiplicative identity.

Additive inverse: The number that you add to a given real number to get the additive identity 0. For example -1 is the additive inverse of 1, and 5/2 is the additive inverse of -5/2. 0 is the additive inverse of 0.

Multiplicative inverse
: The number that you multiply to a given real number to get the multiplicative identity. In almost every case it is the reciprical of that number. For example:
1/2 is the multiplicative inverse of 2, -3 is the multiplicative inverse of -1/3, as before 1 is the multiplicative inverse of 1. HOWEVER, 0 is the only real number with out a multiplicative inverse. There is no real number you can multiply by 0 to get 1. This is why we can't divide by 0, and this is why we say once we understand left and right cancelation that when zero is in the denominator the fraction is "undefined".

Left Cancelation: Given that an operation of addition or multiplication is taking place you can "undo" by using its inverse.
Examples: 

a + 4 = 10
a + 4 + -4 = 10 + -4
a + 0 = 6
a = 6

b * 7 = 21
b * 7 * (1/7) = 21 * (1/7)
b * 1 = 3
b = 3

The same can pretty much be said without lost of generality for right cancelation when we are subtracting and dividing on both sides we are really adding and dividing by the additive and multiplicative inverse. 

Trichotemy Law: given two arbitrary real numbers a and b, one out of the following three relationships must take place: either a < b, a = b, a > b.

If you are not majoring in mathematics, the first five is pretty much all you need to know.

Plug in numbers for a, b, c. Let a = 2, b = 1, and c = 3 and see for yourself! When you are trying to determine the validity of a mathematical proposition, create a few concrete examples as a check. Use the simplest numbers whenever possible.

The commutative property of multiplication does state that a*b = b*a for all real numbers. 

However, a(b*c) is not equal to ab * ac for all real numbers. This is not an example of the distributive property.  Distributive property a(b+c) = ab + ac  or  a(b-c) = ab - ac.

a(b*c) = a*b*c which is not equal to a*b*a*c which is what ab * ac represents. This is not true for all real numbers. I say this because it is true for certain values of a, b, c, but not all values.

True. All real numbers is any positive or negative number, it just can't be a number with the i  in it (imaginary number).

a has to be the same on both sides of the equation, so does b and c

both sides are the same, just written differently. If you distribute the left side you get the right side.

So, if the numbers are the same on both sides, and the equations are the same on both sides, the answer will always be true

Ror all real numbers a and b, a*b is equivalent to b*a.  This is indeed TRUE!  Google Commutative Property for more details, but the basics are this:

Commutative property: When two numbers are multiplied together, the product is the same regardless of the order of the multiplicands. For example 4 * 2 = 2 * 4

As for the sub question for all real numbers a and b, a(b*c) = ab *ac, this is also TRUE.  Google Distributive property for more details, again the basics are as follows:

Distributive property: The sum of two numbers times a third number is equal to the sum of each addend times the third number. For example 4 * (6 + 3) = 4*6 + 4*3

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Lois, you write: "As for the sub question for all real numbers a and b, a(b*c) = ab *ac, this is also TRUE."

This is actually FALSE, as many others have pointed out. The distributive property for the real numbers holds only for addition (i.e. a(b+c)=ab+ac), not for multiplication. Be careful!

Matt

B.S. mathematics, MIT

 

Yes, I see that you are right. I will edit my answer so as not to confuse anyone.  Thank you.

The system will not allow me to edit my response.

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