Basic Number Properties for Algebra
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Basic Number Properties for Algebra

Hey it’s Professor Dave, let’s learn some
algebraic properties. Remember when we learned about a few different
mathematical properties and how they pertained to numbers? Let’s learn how they will be important in
algebra when manipulating variables. The most important of these properties will
be the distributive property. This told us how a number could be distributed
across a parenthetical sum or difference. This didn’t matter too much for arithmetic,
because four times the quantity of two plus three is certainly equal to four times two
plus four times three, but there was nothing stopping us from adding two and three first,
and then multiplying by four. We should get twenty no matter which method
we choose. But with algebra, there are variables that
must remain as they are, and can’t be combined with numbers, so if we have four times the
quantity of two X plus three, the only other meaningful way to express this is by distributing
the four across the sum. That will give us four times two X plus four
times three, which will simplify to eight X plus twelve. In algebra, we will want to be able to use
the distributive property this way, and we will even want to be able to do it in reverse,
by removing some common factor from a sum or difference. For example, if we have three X squared plus
six X, another way to express this would involve identifying the greatest common factor of
these terms, and factoring it out of the expression. In this case, we can rewrite these as three
times X times X and two times three times X. Taking everything we find in both terms,
the greatest common factor would be three X, and if we pull a three X out of both terms,
meaning we divide each term by three X, we end up with the three X out here, and then
X plus two in parentheses. We can verify that this worked as expected
by then distributing the three X across the sum. Three X times X is three X squared, and three
X times two is six X. So we can use the distributive property in
a variety of ways to generate equivalent expressions. Other properties that also apply include the
commutative property for addition and multiplication. Two plus three is the same as three plus two,
and two times three is the same as three times two. If these become algebraic terms, the commutative
property applies in precisely the same way; the order in which we add or multiply algebraic
terms is irrelevant. Two X plus three is the same as three plus
two X, and two X times three is the same as three times two X. However, let’s recall that the commutative
property does not apply to subtraction or division, and that will be the case in algebra
as well. We also learned about the associative property,
and that will apply in algebra too, which we will find out later when we have to manipulate
equations with lots of terms in them. Changing the way these are grouped will not
matter if we are doing addition or multiplication. To be thorough, let’s also mention some
pretty intuitive properties like the additive identity property. This says that you can add or subtract zero
to any number or algebraic term, and it will retain its identity. Five plus zero is five. Three X minus zero is three X. Seems obvious,
but it will come in handy. The multiplicative identity property works
the same way except with the number one instead of zero. Any number or algebraic term times one will
give you the same term again. Four X times one is four X. And lastly, the inverse property of addition
says that anything plus its additive inverse equals zero, so X plus negative X equals zero,
and the inverse property of multiplication says that anything times its multiplicative
inverse equals one, so X times one over X equals one. That’s pretty much all we need to know in
terms of number properties for algebra, so let’s get to some equations.


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