Number Systems is a mathematical value used for counting and measuring objects, and for performing arithmetic calculations. It is a system of writing for expressing numbers. It gives a special representation to every number and represents the arithmetic and algebraic form of the number. It allows us to operate arithmetic operations like addition, subtraction, multiplication, and division.
An equation is a statement that connects two algebraic expressions of the same values with the ‘=’ sign. For example: In equation 9x + 4 = 7, 9x + 4 is the left-hand side expression and 7 is the right-hand side expression connected with the ‘=’ sign.
What is a Number?
A word or symbol that indicates a quantity is known as a number. The numbers 2, 4, 6, etc. are even numbers and 1, 3, 5, etc. are odd numbers. A number is a value created by the combination of digits. These numbers are used to represent algebraic quantities. An integer is a sign from a set of 10 characters ranging from 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. Any combination of integers represents a number. The size of a Number depends on the count of digits that are used for its formation. For Example: 126, 128, 0.356, -12, 78, 94 etc.
Types of Numbers
Numbers are of various types depending upon the patterns of digits that are used for their creation. Various symbols and rules are also applied to Numbers which classifies them into a variety of different types,
Integers
Integers are the collection of Whole Numbers plus the negative values of the Natural Numbers. Integers do not include fraction numbers i.e. they can’t be written in a/b form. The range of Integers is from the Infinity at the Negative end and Infinity at the Positive end, including zero. Integers are represented by the symbol Z. Integers are those numbers whose fractional part is 0 like -3, -2, 1, 0, 10, 100.
Natural Numbers
Natural Numbers are numbers that range from 1 to infinity. These numbers are also known as Positive Numbers or Counting Numbers. We can also represent Natural numbers by the symbol N. All the integers which are greater than 0 are natural numbers, Counting numbers like 1, 2, 3, 4, 5, 6.
Whole Numbers
Whole Numbers are the same as Natural Numbers, but they also include ‘zero’. Whole numbers can also be represented by the symbol W. Whole numbers are all natural numbers and 0 (zero).
Prime Numbers and Composite Numbers
All those numbers which are having only two distinct factors, the number itself and 1, are called prime numbers. All the numbers which are not Prime Numbers are termed as Composite Numbers except 0. Zero is nor prime nor a composite number. Some prime numbers are 2, 3, 5, 53, 59, 97, and 191. All numbers greater than 1 are composite numbers. Some composite numbers are 4, 6, 9, 15, 16, and 100.
Fractions
Fractions are the numbers that are written in the form of a/b, where, a belongs to Whole numbers and b belongs to Natural Numbers, i.e., b can never be 0. The upper part of the fraction i.e. a is termed as a Numerator whereas the lower part i.e. b is called the Denominator. Example: -1/5, 0.25, 2/5, 18/4, …
Rational Numbers
Rational numbers are the numbers that can be represented in the fraction form i.e. a/b. Here, a and b both are integers and b≠0. All the fractions are rational numbers but not all the rational numbers are fractions. Example: -2/5, 0.54, 1/5, 13/4, …
Irrational Numbers
Irrational numbers are the numbers that can’t be represented in the form of fractions i.e. they can not be written as a/b. Example: √2, √3, √.434343, Ï€…
Real and Imaginary Numbers
Real numbers are numbers that can be represented in decimal form. These numbers include whole numbers, integers, fractions, etc. All the integers belong to Real numbers but all the real numbers do not belong to the integers. Imaginary Numbers are all those numbers that are not real numbers. These numbers when squared will result in a negative number. The √-1 is represented as i. These numbers are also called complex numbers. Example: √-2, √-5,…
Product of numbers
In mathematics, we get a product as a result of multiplication, or an expression that indicates factors to be multiplied. The order in which real or complex numbers are multiplied and has no bearing on the product, you can multiply numbers in any order and the answer will always be the same this is known as the commutative law of multiplication. The result we get after multiplying numbers is known as the product of numbers. So 14 is the product of 7 and 2, 50 is the product of 5 and 10, and so on…
Algebra
Algebra is a branch of mathematics that deals with number theory, geometry, and analysis. Algebra is the oldest and the most elementary branch of mathematics. It is explained as the scanning of mathematical symbols and laws, also as the operator of these mathematical symbols, according to some definitions. Algebra describes a large range of topics, from solving basic problems to studying abstractions. Algebra equations are explained in several chapters of mathematics that students will study in school. Algebra contains a number of formulas and identities.
The focus of algebra is on variables, which are undefined values. Equations are a critical concept in algebra. To accomplish arithmetic operations, it follows a set of laws. The laws are used to explain data sets that have two or more variables. It covers a variation of topics around us. Basic algebra, abstract algebra, linear algebra, advanced algebra, and commutative algebra are some of the sub-branches of algebra
One number is 3 less than another number if their product is 40, what are the numbers?
Solution:
The numbers are 8 and 5.
Step-by-step explanation
First number: y
Second number: y – 3
Product: 40
Equation: (y)(y – 3) = 40
y² – 3y – 40 = 0
Factor: (y – 8)(y + 5) = 0
y – 8 = 0
y = 8
y + 5 = 0
y = -5
The numbers:
First number: y = 8
Second number: 8 – 3 = 5
The numbers are 8 and 5.
Similar Problems
Question 1: One number is 4 more than other numbers if their product is 60. What are the numbers?
Solution:
The numbers are 6 and 10.
Step-by-step explanation
First number: x
Second number: x + 4
Product: 60
Equation: (x)(x + 4) = 60
x² + 4x – 60 = 0
Factors: (x – 6)(x + 10)=0
x – 6 =0
x = 6
x + 10 =0
x = -10
First number: x = 6
Second number: 6 + 4 = 10
The numbers are 6 and 10.
Question 2: One number is 2 less than another number if their product is 15. What are the numbers?
Solution:
First number: x = 3
Second number: x + 2 = 5
Now the product is 15
Equation: (x)(x + 2) = 15
x² + 2x -15 = 0
Factors: (x + 5) (x – 3) = 0
x + 5 = 0
x = -5
x – 3 = 0
x = 3
First number: x = 3
Second number: x + 2 = 5
The numbers are 3 and 5
Question 3: One number is 5 more than the other number if their product is 36. What are the numbers?
Solution:
The numbers are 4 and 9.
Step-by-step explanation
First number: x
Second number: x + 5
Product: 36
Equation: (x)(x + 5) = 36
x² + 5x – 36 = 0
Factors: (x – 4)(x + 9)=0
x – 4 = 0
x = 4
x + 9 = 0
x = -9
First number: x = 4
Second number: 4 + 5= 9
The numbers are 4 and 9.
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