Every aspect of our daily life involves numbers, whether it’s the number of rounds we run around the race track or the number of hours we sleep at night. In mathematics, numbers can be even or odd, prime or composite, decimal, fractional, natural or integer, real or whole, rational or irrational, or any combination of these. In this article, we explain how to classify numbers and how number groups can be derived.
Why do we classify numbers?
Knowing whether a numerical unit is a whole number, a rational number, or even a real number, can be crucial to derivations and other mathematical operations. When counting discrete or indivisible items, whole numbers are utilized. When searching for objects that can be divided into fractions, but for which it is difficult or impossible to divide into non-rational portions, solutions with rational numbers are used.
Rationals are answers to linear equations like y = mx + b that arise in situations like figuring out the position of a moving object that isn’t speeding, figuring out sales tax, and dividing goods among multiple components, like halving a recipe or distributing prizes among numerous players.
When a solution with a tangible existence is required, real numbers are utilized. Real numbers frequently appear as the solution to polynomials, although only some polynomials have real solutions. Additionally, they are widely used as solutions to exponentials, such as half-life decay rates (e.g,. the natural log of 2). Complex numbers make up the rest. They do not have to be algebraic or logical.
What are the various types of numbers?
- Ordinal or counting numbers: 1, 2, 3, 4,…
- Whole numbers: 0, 4, 8, 100 …
- Integers: -3, -2, -1, 0, 1, 2, 3…
- Negative numbers: -3, -2, and -1
- Positive numbers: 1, 2, 3,…
In addition, fractions, such as ½ or ¾, are rational numbers. All integers are rational numbers. All rational numbers can be expressed as fractions, where a and b are integers and natural numbers, respectively. Since they may both be expressed as fractions in this way, terminating decimals, like 0.5, and repeating decimals like 0.333 are rational numbers as well.
Finally, we come to irrational numbers. Examples of these include numbers where a is an integer and b is a natural number that cannot be expressed as a fraction (a/b). Irrational numbers also include roots and ratios such as pi.
A complex number is created by adding a fictitious number to a real one. The complex number is represented as a + bi, where a represents the real part and b the complex part. Complex numbers are found on a two-dimensional flat plane, whereas real numbers are found on a number line.
How can number groups be derived?
Combining like terms with integer coefficients
Simplify the following expression by combining like terms:
3x + 4y – 7x – 5x2 – 6y + 2x2
3x and -7x are like terms, as they both contain the component x. Equally, 4y and 6y are like terms as are + 2x2 and – 5x2. It is important to understand how to group these as many people will forget to keep the arithmetic symbol with the associated number. 3x + 4y – 7x means that 3x and 4y are positive, but 7x is a negative integer. As such:
3x – 7x = -4x
4y + 6y = 10y
– 5x2 + 2x2 = 3x2
Therefore, the simplified expression is:
-4x + 10y – 3x2
Classify a real number as a rational number
Express the following real numbers as rational numbers:
7 = 7/1
6.25 = 6 ¼ or 25/4 (depending on whether it asks to express fractions as mixed numbers)
If a real number cannot be expressed as a fraction, classify it as an irrational number. Examples include root 2, pi, and e. If a real number has no fractional component, classify it as an integer. Examples are -2, 7, 0, and 24.
If a real number is 0 or a positive integer, classify it as a whole number, as in 0, 1, 3, and 11.
If a real number is a positive integer, classify it as a natural number or counting number. Examples are 1, 9, and 42.
Establish the presence of irrational numbers
Is √144 an irrational number?
A real number that cannot be stated as a ratio of integers is said to be irrational. An example of this is the number √2. With that in mind, we must determine if the square root of 144 is a whole number. To work this out we can use prime factorization.
Prime factorization allows us to find the factors of the number, which we can then multiply to find the square root. To do this, we take our original number (144) and keep dividing it until we reach a prime number:
144
2 72
2 36
2 18
2 9
3 3
3 1
In this way, 144 = 2 x 2 x 2 x 2 x 3 x 3
We can then combine the common factors:
144 = 2 x 2 x 2 x 2 x 3 x 3
144 = 2 x 2 x 3
144 = 12
The square root of 144 is 12, which is a whole number, therefore √144 is not an irrational number!
How to solve a problem in mathematics
Math tutoring services
Students often find math difficult. Fortunately, Tutorax offers online and in-person tutoring for students of all ages. Our tutors can assist with homework, test preparation, and in-class support. If your child is having difficulty with math, tutoring assistance can greatly enhance his performance and boost this self-esteem.