The Reals
When studying algebra, students need to understand the realm in which they find themselves. After all, one can easily get lost amidst all the formulas, equations, variables, and mathematical symbolism. The real numbers are those entities which play the pivotal role in algebra. Here we look at some of the most basic and fundamental properties so that this subject becomes more meaningful for the student.
The real numbers — those comprising the integers, fractions, and non-repeating, non-terminating decimals— are the key players in algebra. True, the complex numbers — those of the form a + bi, such that a and b are real numbers and i2 = -1 — are studied in algebra and do indeed have important applications in various real world sciences, yet the real numbers are the ones that have the predominant role. Reals behave in predictable ways. By mastering the basic properties of this set, you will be in a much stronger position to master algebra.
Closure Property
Closure is a very important property in mathematics. When we talk about sets, closure is the property that insures that whenever we operate on the elements of the set, then we obtain a member of the set. In layman’s terms, if we have a set of green apples and we add two of them together we end up with a new number of green apples. Notice that the word green has been emphasized. This is to point out that we do not end up with red apples or any other type of apple. Insofar as the set of real numbers goes, this property states that when we add or multiply real numbers, we end up with…yes, a real number. We do not end up with a number that is not real. Specifically, if we add a and b, and both a and b are real numbers, then the sum a + b is also a real number.
Commutative Properties
The set of real numbers is commutative under the operations of addition and multiplication as well. Commutativity implies that the order of performing the operation on the two real numbers a and b does not matter. For example, 3 + 4 = 4 + 3; 5×8 = 8×5. It should be pointed out that division and subtraction are not commutative, as for example 3 – 1 is not the same as
1 – 3.
Associative Properties
When performing the operation of addition or multiplication on groups of three numbers, we can group the numbers as we like and still obtain the same result. For example, (7 + 4) + 5 = 7 + (4 +5); 3x(4×7) = (3×4)x7.
Identity Property
The set of real numbers has two identity elements, one for addition and one for multiplication. These elements are 0 and 1, respectively. Zero is the identity for the operation of addition and 1 that for multiplication. These numbers are called identities because when operated on with other real numbers, the values of the latter remain unchanged. For example 0 + 6 = 6 + 0 = 6. Here 6 has not changed value or lost its identity. In 8×1 = 1×8 = 8, 8 has not changed value or lost its identity.
Inverse Properties
Completely analogous to the two identity elements, the real numbers has two inverse elements. For addition, the inverse element is the negative of the given number. Thus the additive inverse of 8 is -8. Notice that when we add a number to its inverse, as in 8 + -8, we always obtain 0, the identity for addition. For multiplication, the inverse element is the reciprocal. Thus the multiplicative inverse of 2 is 1/2. Note that the only number that does not have a multiplicative inverse is 0, since division by 0 is not allowed. Notice as well, that a number times its reciprocal as in 2(1/2) always yields 1, the identity for multiplication.
Distributive Property
The distributive property allows us to multiply one real number over the sum of two others, as in 2x(2 + 5) to get 2×2 + 2×5. This property is very powerful and very important to understand. We can do lightning multiplications with this property and also perform the algebraic FOIL (First Outer Inner Last) quite easily. For example, this property allows us to split the multiplication 8×14 as 8x(10 + 4) = 8×10 + 8×4 = 80 + 32 = 112. When we do an algebraic FOIL as in (x + 2)(x + 3), we can apply the distributive property twice to get that this is equal to x(x + 3) + 2(x + 3). By separating the pieces and adding, we obtain x2 +5x + 6.
As you can see from the above, mastering these properties will not only give you more confidence in approaching algebra — or any math course for that matter—but also allow you to understand your teacher much better. After all, if you don’t speak the language, you cannot understand what’s being said. Plain and simple.