When studying algebra, students must understand the field in which they find themselves. After all, one can easily get lost among all the mathematical formulas, equations, variables, and symbolism. Real numbers are those entities that play a central role in algebra. Here we look at some of the most basic and foundational properties to make this topic more meaningful to the student.
Real numbers, those that comprise whole numbers, fractions, and decimals that do not repeat or terminate, are the key players in algebra. True, complex numbers — those of the form un + bisuch that has Y b are real numbers and i^2 = -1 — are studied in algebra and indeed have important applications in various real-world sciences, however, real numbers play the predominant role. The real ones behave in a predictable way. 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, the closure is the property that ensures that every time we operate on the elements of the set, we get a member of the set. In simple terms, if we have a set of green apples and add two of them, 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 are not done with red apples or any other type of apple. As far as the set of real numbers is concerned, this property states that when we add or multiply real numbers, we end up with… yes, a real number. We don’t end up with a number that isn’t real. Specifically, if we add has Y band both has Y b are real numbers, so the sum a+b is also a real number.
commutative properties
The set of real numbers is also commutative in the operations of addition and multiplication. Commutativity implies that the order of performing the operation on the two real numbers has Y b no matter. For example, 3 + 4 = 4 + 3; 5×8 = 8×5. It should be noted that division and subtraction are not commutative, such as 3 – 1 is not the same as 1 – 3.
Associative properties
When performing the addition or multiplication operation on groups of three numbers, we can group the numbers however we want and still get 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 addition operation and 1 for the multiplication. These numbers are called identities because when operating 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 his identity. In 8×1 = 1×8 = 8, 8 has neither changed value nor lost its identity.
Inverse properties
Completely analogous to the two elements of identity, the real numbers have two inverse elements. For addition, the inverse element is the negative of the given number. Thus, the additive inverse of 8 is -8. Note that when we add a number to its inverse, as in 8 + -8, we always get 0, the identity by addition. For multiplication, the inverse element is the reciprocal. Therefore, 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. Note also that a number times its reciprocal as in 2(1/2) always produces 1, the identity for multiplication.
Distributive property
The distributive property allows us to multiply a real number on 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 flash multiplication with this property and also perform algebraic FOIL (First Outer Inner Last) quite easily. For example, this property allows us to divide 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 this to be equal to x(x + 3) + 2(x + 3). Taking the pieces apart and adding, we get x^2 +5x + 6.
As you can see from the above, mastering these properties will not only give you more confidence to tackle algebra, or any math course, but it will also give you a much better understanding of your teacher. After all, if you don’t speak the language, you can’t understand what is being said. Plain and simple.
