Lesson EXPLANATION of distributive property
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Algebra: Distributive, associative, commutative properties, FOIL
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Distributive property is a property of numbers that ties the operation of addition (or subtraction) and mutiplication together. It says that for ANY numbers a, b and c, {{{a*(b+c) = a*b+a*c}}}. For those used to multiplications without the multiplication sign, {{{a(b+c) = ab+ac}}}. The same property applies when there is subtraction instead of addition: a*(b-c) = a*b - a*c. In the <A HREF=Solvers.html>Solvers Section</A>, there are two solvers that show CARTOONS of distributive property, for any numbers that you supply. Try them out, they are fun. Here's a sample cartoon that shows how distributive property works: <CENTER> {{{ cartoon( 2(3+4), 2*3+2*4, 2*3+highlight(2*4), 2*3+highlight(8), 2*3+8, highlight(2*3)+8, highlight(6)+8, 6+8, 14 ) }}} </CENTER> <H4>How it is used</H4> The distributive property is used when something in paretheses is multiplied by something, or, in reverse, when you need to take some common multiplier OUT of the parentheses. Example: Taking x INTO parentheses: {{{ cartoon( x(2y-3), x*2y-x*3, 2xy-3x ) }}} Taking <B>a</B> OUT of parentheses: {{{ cartoon( ax+ay, a(x+y) ) }}} <H4> Why is the distributive property true?</H4> Check out a <A HREF=proof-of-distributive-property.lesson>separate lesson</A> that I wrote that proves distributive property for integer numbers. <H4>Addition vs. subtraction inside parentheses</H4> As you know from studying addition of signed numbers (there is a module about this on this site), a subtraction is nothing more than addition of a number that's opposite to the one being subtracted. So, to work it out with an example: a*(b-c) = a*(b + (-c)). The right part of the formula is a regular subtraction. Here's the cartoon for it: <CENTER>{{{ cartoon( a*(b-c), a*(b-c) = a*(b + (-c)), a*(b-c) = a*b + a*(-c), a*(b-c) = a*b - a*c ) }}} </CENTER>