Numbers and Operations



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Chapter 1 : Numbers and Operations



Numbers arrow_upward



Natural Numbers

  • The natural numbers are the counting numbers.
  • Examples: 1, 2, 3, 4, 5.
  • Whole Numbers

  • The whole numbers are the counting numbers including 0.
  • Examples:  0, 1, 2, 3, 4, 5.
  • Integers

  • Integers include counting numbers, zero and negative numbers.
  • Examples: -3, -2, -1, 0, 1, 2, 3.
  • Rational Number

  • A rational number is a number that can be expressed as a fraction or ratio.
  • The numerator and the denominator of the fraction are both integers.
  • Example: 2/3.
  • Note that 6 can also be written as fraction.
  •     6 = 6/1.

    Irrational Numbers

  • Numbers that cannot be written as fractions.
  • Example: √2 = 1.414213562…..
  • Prime Numbers

  • A prime number is a natural number that has exactly two (distinct) natural number divisors, which are 1 and the number itself.
  • Examples: 2, 3, 5, 7, 11, 13.
  • Complex Numbers

  • Complex number is a number that can be expressed in the form of a + bi where a and b are real numbers and i is the square root of -1).
  •   

  • Example: We can write
  •    

         

        = 3i

    Composite

    Number

  • A Composite Number can be divided evenly by numbers other than 1 or itself.
  • Examples: 4, 6, 8, 9, 10, etc.
  • Multiples

  • If a number is divisible exactly by a second number, then the first number is said to be a multiple of the second number.
  • Example: 15 is a multiple of 5.
  • Factors

  • If one number divides a second number exactly, then the first number is said to be a factor of the second number.
  • Example: 5 is a factor of 15.
  • Even Numbers

  • Numbers divisible by 2 are called even numbers.
  • Examples: 2, 4, 6, 8.
  • Odd

    Numbers

  • Numbers which are NOT divisible by 2 are called odd numbers.
  • Examples: 1, 3, 5, 7.
  • Roman

    Numerals

  • Roman numerals are numeral system of ancient Rome based on the letters of the alphabet, which are combined to signify the sum of their values.

  • I

    1

    V

    5

    X

    10

    L

    50

    C

    100

    D

    500

    M

    1000


       



    Place Value arrow_upward


  • Place value determines the value of a digit in a number, based on the location of the digit.
  • Below image is showing the place value for 25612.5.

  • Addition arrow_upward


  • Addition is finding the total, or sum, by combining two or more numbers.
  • Example: 6 + 3 = 9

  • Subtraction arrow_upward


  • The process of taking one number or amount away from another.
  • Example: 5 – 4 = 1

  • Multiplication arrow_upward


  • It signifies repeated addition.
  • Example: 3 × 4 = 12.
  • It can also be written as:
  •     3 × 4 = 3 + 3 + 3 + 3 = 12

  • Here 3 is the multiplicand, 4 is the multiplier and 12 is the product.

  • Division arrow_upward


  • Division is a way to find out how many times a number is contained in other number.
  • Example: Division of 19 by 4.
  • Here, 4 is Divisor, 19 is Dividend, 4 is quotient, and 3 is remainder.

  • Sign Rules arrow_upward



    Rules for Adding Integers with the Same Signs:

  • Add the numbers together.
  • Give the answer with the same sign.
  • Example: Add (-5) and (-10).
  • Both the numbers are negative. First add the numbers.
  •     5 + 10 = 15.

  • As the numbers are negative so the answer would be – 15.

  • Rules for Adding Integers with Different Signs:

  • Ignore the signs and find the difference.
  • Give the answer with the sign of the larger number.
  • Example: Add (-5) and (10).
  • Ignore the signs and find the difference.
  •     10 – 5 = 5

  • Here the larger number 10 is positive, So the answer is 5.

  • Rules for Subtracting Integers:

  • Change the subtraction sign to addition.
  • Change the sign of the second number to the opposite sign.
  • Follow the rules for adding integers.
  • Example: Subtract: -8 – (-5).
  • Change the subtraction sign to addition and change -5 to 5.
  •     -8 + 5

  •     = -3

  • Rules of Signs for Multiplication:

  • When two numbers with like signs are multiplied, the result is always positive.
  • When two numbers with different signs are multiplied, the result is always negative.
  • Examples:
    • 6 × 8 = 48  
    • (-6) × 8 = -48

    Rules of Signs for Division:

  • When two numbers with like signs are divided, the result is always positive.
  • When two numbers with different signs are divided, the result is always negative.
  • Examples:
    • 36 ÷ 6 = 6
    • (-36) ÷ 6 = -6

    Order of Operations arrow_upward


  • To ensure that anyone evaluating a mathematical expression calculates the same value, one has to follow the established order of operations:

  • 1

    Find whether a bar sign is present in the expression, evaluate it first.

    2

    First perform any calculations inside parentheses ( ), then braces { } and then brackets [ ].

    3

    Next perform any exponent operations.

    4

    Next perform all multiplication and division, working from left to right.

    If the expression contains both operations (multiplication and division), do whichever occurs first, going from left to right.

    5

    Lastly, perform all addition and subtraction, working from left to right.



    Trick for remembering the Order of Operations arrow_upward



    Please

  • First perform any calculations inside parentheses ( ), then braces { } and then brackets [ ].
  • Excuse

  • Next perform any exponent operations.
  • Me

  • Next perform all multiplication working from left to right.
  • Dear

  • Next perform all division, working from left to right.
  • Aunt

  • Next perform all addition working from left to right.
  • Susan

  • Lastly, perform all subtraction, working from left to right.

  •  



    Thank You from Kimavi arrow_upward


  • Please email us at Admin@Kimavi.com and help us improve this tutorial.


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