Mathematics is a language of science and a universal language. Algebra is, in effect, the language of mathematics because of which every other branch revolves. To think of it, without algebra, there wouldn’t be any other form of mathematics. Algebra is the study of symbols and the rules around their manipulation. Algebra unifies all of mathematics.
Algebra consists of various sets of numbers. The two sets that are opposite but combine to form all the numbers are genuine and imaginary. The actual number is any number that can be plotted on the number line. We know real numbers to be rational and irrational, and their only qualification criterion is the ability to plot them on the number line.
Several distinct sets of numbers are all part of real numbers. These are natural, whole, integers, rational, and irrational numbers. These aren’t mutually exclusive as several of these numbers overlap. Let’s discuss them in detail.
Another way of defining real numbers could be that they are the rational numbers with denominator ≠ 0. Thus, all the following can be a part of real numbers set: -3, 1, 0, 1 1/3, 2/-3, -4/9, 0.43, 999.19, -2 5/9, and so on. Do note; all rational numbers can also be represented in their decimal form, as with 0.43 and 999.19.
1. Natural Numbers
The counting numbers are the natural numbers. So, the number ‘1’ starts with the natural numbers. The set of natural numbers can be defined as:
N = {1,2,3, 4,}
Where N is an infinite set of numbers that starts with 1 and can go up to infinity. Natural numbers are among the first to have arrived, even before mathematics existed. Often, 0 is included in the set of natural numbers. Such a set is referred to as the following:
N0 = N0 = {0,1,2,3, 4,}
The set of natural numbers, whether 0 is included or not, is an open set. Thus, it is an infinite set because there’s no upper limit to the number that can be contained in this set.
2. Whole Numbers
Whole numbers are the same as N0 or N0. Whole numbers are natural numbers plus the number 0. The set of whole numbers can be written as:
W = {0,1,2,3,4,}
Whole number set carries the same properties as one of Natural Numbers. It is an infinite set because of a lack of an upper limit, and hence it constitutes to be an open set as well. So, natural numbers set N is a subset of whole numbers set W.
3. Integers
Integers are all the numbers that do not have a value after the decimal. Integers can all be written as a rational number, but the denominator will be 1. Thus, comes the lack of any number after the decimal (or it can be written as .00, for example, 1 = 1.00).
I = {,-4,-3,-2,-1,0,1,2,3,4,}
The set of integers is an infinite set that is open at both ends. The numbers can proceed to infinity either in their positive or negative form. Integers are often divided into two or three subsets, on the basis of classification. These are:
· Integer is a combination of Natural Numbers, 0, and Negative Natural Numbers also called as positive integers, 0, and negative integers. The set of positive integers can be determined as:
I+ = {1,2,3,4,}
The set of negative integers can be determined as:
I- = {-1,-2,-3,-4,} or {,-4,-3,-2,-1}
Integers can also be called as the sum of negative natural numbers and whole numbers.
All the above numbers are non-fractional numbers or numbers that do not have a fractional part. That is, their denominator is equal to 1 when expressed in rational number format. Any number can automatically be given a denominator as 1 unless otherwise written. So, 99 = 99/1, and -1029 = -1029/1.
4. Rational Numbers
Rational, in mathematics, means ratio-like. A rational number is a number that can be represented in the form of a ratio of two integers, like a/b. All integers are rational numbers as they can be represented in the form of a/b, with a denominator value of 1. Like 3 = 3/1, and so on.
Two types of rational numbers exist:
a) Terminating Decimal: Where the value after decimal point terminates to represent a full number. Like 1/25 = 0.04 and 86/8 = 10.75.
b) Repeating Decimal: Where the value after the decimal point keeps repeating. The simplest example here is 1/3 = 0.3333333333333333 (here 3 keeps repeating)
5. Irrational Numbers
A number that can not be presented as a ratio is an irrational number. Irrational numbers are also real numbers, although they can’t be put in the form of an a/b ration. It is often possible to find such numbers in existence, and hence they are real. For example, a hypotenuse may come as √3. Here, it can’t be further presented as a ratio, although the number can be simplified and put in the decimal form. Thus, not all numbers that can be presented as a decimal are rational numbers.
Irrational numbers are represented as:
{h | h is not a rational number}
Let’s have an exercise to find out if the number is rational or irrational. Let’s take examples of the following numbers:
- a) √64: The number comes out to be 8 when solved, which can be presented as 8/1, and hence it is a rational number.
- b) 37/9: The number comes out to be 4.111111111111, and is a repeating decimal form of a rational number.
- c) √7: The number cannot be simplified further and is an irrational number.
- d) 18/4: The number comes out to be 4.5 and is a terminating decimal form of a rational number.
- e) 0.3033033303333…: The number isn’t a terminating decimal or repeating decimal. The repeat pattern is uneven. It cannot be represented in the form of a ratio, and hence constitutes an irrational number.
- f) Î : Pi comes in the form of a decimal that isn’t terminating, nor repeating, and makes up to be an irrational number.
6. Calculations in Real Numbers
Real numbers can be expressed in normal calculations following the PEMDAS order of operations. PEMDAS refers to:
- P(parentheses)
- E(exponents)
- M(multiplication) and D(ivision)
- A(ddition) and S(subtraction)
As can be seen, parentheses refer to brackets in the order of first to last, the given sequence being the case: [] > {} > (). Brackets are first to be calculated even as per the older format (which is basically the same – BODMAS – Bracket, Off, Divide, Multiply, Add, Subtract).
Exponents refer to any number raised to the power of another number. For example, an, an refers to a x a x a x a x…. (n times).
So, 22 is an exponential number that means 2 x 2.
Multiplication and Division go the same way. So, a/b and a*b are the signs of divide and multiply operations. In operation involving both multiply and divide, the result will be the same whichever way it is performed. For example, 3 / 5 * 3 = 0.6 * 3 = 1.8. It is the same if we perform the multiplication first, so 3 / 5 * 3 = 9 / 5 = 1.8. So, even though division takes precedence, it doesn’t make much difference.
Addition and Subtraction are at the lowest rung in order of calculations. Similar to multiply and divide, in a problem involving both + and -, the result will be the same.
For example, 7 + 11 – 20 = 18 – 20 = -2. The result is the same if subtraction is performed first. So, 7 + 11 – 20 = 7 – 9 = -2.
The Number Line
The number line is the essential factor that determines whether a number is a real number or not. The number line is a straight line with integers marked on either side, negative integers on left and positive integers on the right, 0 lying between them. Any real number can be plotted on the number line.
Conclusion
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