Basic Arithmetic

What are Basic Arithmetic Operations

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Table of Contents: What is Arithmetic

  1. Basic Arithmetic Principles

  2. Basic Arithmetic Operations

  3. Factors and Multiples

  4. Even and Odd Numbers

  5. Order of Operations (PEMDAS)

  6. Integers

  7. Properties of Zero

  8. Properties of 1

What is Arithmetic?

Definition of Arithmetic

The following are some key points, which are phrased here to refresh your knowledge of basic Arithmetic principles.

Basic Arithmetic:

  • For any number x, exactly one of the following is true:
  1. x is negative
  2. x is zero
  3. x is positive
  • The only number that is equal to its opposite is 0 (e.g. x = – x only if x = 0
  • If 0 is multiplied to any other number, it will make it zero (x × 0 = 0)
  • Product or quotient of two numbers of the same sign are always positive and of a different sign are always negative. e.g. if a positive number is multiplied to a negative number the result will be negative and if a negative number is divided by another negative number the result will be positive.

See the following tables for Basic Arithmetic Operations.

Basic Arithmetic
  • The sum of two positive numbers is always positive.
  • The sum of two negative numbers is always negative.
  • Subtracting number from another is the same as adding its opposite x – y = x + (- y)
  • The reciprocal of a number x is 1/x
  • The product of a number and its reciprocal is always one ax ×1/a = 1
  • Dividing by a number is the same as multiplying by is reciprocal x ÷ y = x × 1/y
  • Every integer has a finite set of factors (divisors) and an infinite set of multipliers.

If x and y are two integers, the following four terms are synonyms.

  1. x is a divisor of y
  2. x is a factor of y
  3. y is a divisible by x
  4. y is a multiple of x

They all mean that when x is divided by y there is no remainder.

  • Positive integers, other than 1, have at least two positive factors.
  • Positive integers, other than 1, which have exactly two factors, are known as prime numbers.
  • Every integer greater than 1 that is not a prime can be written as a product of primes.
  • To find the prime factorization of an integer, find any two factors of that number, if both are primes, you are done; if not, continue factorization until each factor is a prime. e.g. to find the prime factorization of 24, two factors are 6 and 4. Both of them are not prime numbers, so continue to factor them.
  • Factors of 4 are 2 and 2(2 × 2)
  • Factors of 6 are 3 and 2(3 × 2)
  • So the number 24 can be written as 2 × 2 × 2 × 3
  • The Least Common Multiple (L.C.M) of two integers x and y is the smallest integer which is divisible by both x and y, e.g. the L.C.M of 15 and 25 is 75.
  • The Highest Common Factor (H.C.F) of two integers x and y is the largest integer which divides both x and y, e.g. the H.C.F of 15 and 25 is 5.
  • The product of HCF and LCM of two integers is equal to the products of numbers itself. e.g. 15 ×  25 = 375
  • 5 × 75 = 375 (where 5 is H.C.F and 75 is LCM of 15 and 25).
  • Even numbers are all the multiples of 2e .g.(….,-4,-2,0,2,4,……)
  • Odd numbers are all integers not divisible by 2 (…..,-5,-3,1, 1, 3, 5,…….)
  • If two integers are both even or both odd, their sum and difference are even.
  • If one integer is even and the other is odd, their sum and difference are odd.
  • The product of two integers is even unless both of them are odd.
What is the correct order of Math Operations?

Following formula is used to operate correctly the mathematical operations of addition, subtraction, division and multiplication.

PEMDAS

When an equation involves more than one operation, it is important to carry them out in the correct order. The correct order is Parentheses, Exponents, Multiplication and Division, Addition and Subtraction, or just the first letters PEMDAS to remember the proper order.

What are integers and their properties
Integers

The set of numbers which consists of whole numbers and negative numbers is known as integers. It is denoted by Z.

Z = {……..-4-3,-2,-1, 0, 1, 2,3,4,5……)

Positive Integers

The set Z ₊ = (1, 2, 3, 4 …) is the set of all positive integers. It is clear that positive integers and natural numbers are synonyms.

Negative Integers

The set Z₋ = (-1,-2,-3 …) is the set of all negative integers.

Remember: "0" is neither positive nor negative.
Even Numbers

Even numbers are all the multiple of 2 the set of even numbers is (……. 4, 2, 0, 2, 4……..)

Properties of Even Numbers
  1. If an even number is multiplied by any integer will yield an even number

Example: here 2 is even integer      2 × 5 = 10,      2 × 6 = 12           

  1. Two even numbers added together will yield an even number.

Example: 4 ₊ 6  = 10,      8 ₊ 6 = 14

  1. Zero would be considered an even number for practical purposes.

Odd Numbers:

The numbers which are not divisible by 2 are called Odd Numbers. The set of odd numbers is (-5-3-1, 1,3,7,9 …)

Properties of Odd Numbers
  1. If an odd number is multiplied by an odd number will yield an odd number.

Example 1:            5 × 5 = 25,          7 × 7 = 49

  1. Two odd numbers added together will yield an even number.

Example: 5 ₊ 3  = 8,         5 ₊ 7= 12

=

Is equal to

> 

Larger than

< 

Is less than

≥

Is greater than or equal to

≤

Is less than or equal to

Properties of Zero
  1. 0 is neither positive nor negative.
  2. 0 is smaller than every positive number.
  3. 0 is greater than every negative number.
  4. 0 is an even integer.
  5. For any integer a; a × 0 = 0
  6. For any integer a (including 0) a ÷ 0 = undefined.
  7. For any positive integer a; 0 ÷ a; 0/a = 0.
  8. For every integer a; a + 0 and a ₋ 0 = a.
  9. If the product of two or more numbers is 0, than at least one of them is 0.
Properties of 1
  1. For any number a: 1 x a = a and a/1 = a
  2. For any integer n: 1n = 1.
  3. 1 is a divisor of every integer.
  4. 1 is the smallest positive integer.
  5. 1 is an odd integer.
  6. 1 is not a prime.