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OCR GCSE Maths (Higher) Study Notes

1.6.1 Basic Operations

In mathematics, mastering the basic operations of addition, subtraction, multiplication, and division is crucial. These operations are the building blocks for more complex mathematical concepts. This section delves into each operation with a focus on integers, fractions, and decimals, emphasizing the correct ordering of operations and the use of brackets.

Basic operations in Mathematics

Addition

Addition combines two or more numbers into their total or sum.

Addition
  • Properties:
    • Commutative: Order doesn't affect the sum, e.g., 3+4=4+33 + 4 = 4 + 3.
    • Associative: Grouping doesn't affect the sum, e.g., (2+3)+4=2+(3+4)(2 + 3) + 4 = 2 + (3 + 4).

Example: Calculating a Sum

Question: What is the sum of 7878, 45.345.3, and 22-22?

Solution:

Sum=78+45.322=101.3\text{Sum} = 78 + 45.3 - 22 = 101.3

Subtraction

Subtraction finds the difference between two numbers, essentially reversing addition.

Subtraction
  • Symbols and Terms: The - symbol denotes subtraction. The number being subtracted is the subtrahend, from the minuend, to get the difference.
  • Properties:
    • Non-Commutative: The order in subtraction matters, e.g., 52255 - 2 \neq 2 - 5.

Example: Finding a Difference

Question: What is the difference between 5050 and 32.7532.75?

Solution:

Difference=5032.75=17.25\text{Difference} = 50 - 32.75 = 17.25

Multiplication

Multiplication adds a number to itself a specified number of times, streamlining addition.

Multiplication
  • Symbols and Terms: The × or * symbol is for multiplication. The numbers being multiplied are factors, with the result called the product.
  • Properties:
    • Commutative: The order of factors doesn't affect the product, e.g., 3×4=4×33 × 4 = 4 × 3.
    • Associative: Grouping of factors doesn't affect the product, e.g., (2×3)×4=2×(3×4)(2 × 3) × 4 = 2 × (3 × 4).

Example: Multiplying Numbers

Question: Multiply 3.53.5 by 2.42.4.

Solution:

Product=3.5×2.4=8.4\text{Product} = 3.5 × 2.4 = 8.4

Division

Division splits a number into equal parts, the inverse of multiplication.

Division
  • Symbols and Terms: The ÷ or / symbol represents division. The number being divided is the dividend, by the divisor, to get the quotient.
  • Properties:
    • Non-Commutative: Order matters significantly in division.
    • Division by Zero: Undefined.

Example: Dividing Numbers

Question: Divide 5656 by 0.70.7.

Solution:

Quotient=56÷0.7=80\text{Quotient} = 56 ÷ 0.7 = 80

Order of Operations and Bracket Usage

Correct operation ordering and bracket usage are pivotal for solving problems accurately. PEMDAS guides the order: Parentheses, Exponents, Multiplication and Division (left to right), Addition and Subtraction (left to right).

  • Brackets: Operations within are prioritized; nested ones are solved inside out.

Example:

2+3×(4+5)=2+3×9=2+27=292 + 3 × (4 + 5) = 2 + 3 × 9 = 2 + 27 = 29(2+3)×(4+5)=5×9=45(2 + 3) × (4 + 5) = 5 × 9 = 45

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