Types of Number | Mathematics for GCSE/IGCSE - Year 11 PDF Download

Types of Number

Understanding terms like real numbers, integers, natural numbers, indices, factors, multiples, prime, square and cube numbers, reciprocals, rational, and irrational numbers is crucial at the GCSE level. It's essential to grasp the meanings of these concepts.

What are real numbers, integers and natural numbers?

• Real numbers encompass all numbers, including integers, fractions, rational, and irrational numbers.
  • Numbers studied at the GCSE level are considered real numbers.
  • The symbol ℝ is often used to represent real numbers.

Integers

• Integers are whole numbers that can be positive, negative, or zero.
  • The symbol ℤ is used to denote integers.
  • Examples of integers include ..., -3, -2, -1, 0, 1, 2, 3, ...

Natural Numbers

• Natural numbers are positive integers, often referred to as counting numbers.
  • The symbol ℕ is used to represent natural numbers.
  • Examples of natural numbers are 1, 2, 3, ...

What is a rational number?

• A rational number is a number that can be expressed as a fraction in its simplest form.
  • This fraction can be written in the form a/b, where a and b are whole numbers.
  • This definition encompasses all terminating and recurring decimals.

What are factors, multiples and prime numbers?

Factors

• Factors divide into numbers without remainders.
  • For instance, the factors of 18 are 1, 2, 3, 6, 9, and 18.
  • Every number has at least two factors: the number itself and 1.

Multiples

• A multiple is a number found in the times table of another number.
  • For example, multiples of 3 include 3, 6, 9, 12, 15, 18, and so on.
  • Every non-zero number has an infinite number of multiples.

Prime Numbers

• A prime number has exactly two factors: 1 and itself.
  • 1 is not a prime number because it has only one factor.
  • The first ten prime numbers to remember are 2, 3, 5, 7, 11, 13, 17, 19, 23, and 29.

What are squares, cubes and indices?

Square Numbers

• A square number is obtained by multiplying a number by itself.
• For instance, 3 multiplied by 3 equals 9, making 9 a square number.
• Mathematically, a × a can be represented as a².
• It is important to memorize the first fifteen square numbers: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225.

Cube Numbers

• A cube number results from multiplying a number by itself twice.
• For example, 3 multiplied by 3 multiplied by 3 equals 27, making 27 a cube number.
• Symbolically, a × a × a can be denoted as a³.
• One should memorize the first five cube numbers: 1, 8, 27, 64, 125.
• Additionally, it's crucial to remember that 10³ equals 1000.

Indices

• An index (plural: indices) is a method of neatly representing a series of multiplications of the same number.
• These are commonly known as powers or exponents.
• For example, 3 multiplied by itself 4 times can be written as 3⁴.
• Similarly, a × a × a × a × b × b × b × b × b can be expressed in index form as a⁴ × b⁵.

What is a reciprocal?

• Definition: The reciprocal of a number is the result of dividing 1 by that number. 
  • Multiplying a number by its reciprocal always equals 1.
• The reciprocal of a is 1/a 
  • The reciprocal of 1 /a  is a 
• The reciprocal of a number, n, may also be written as n^-1.

Irrational Numbers

What is an irrational number?

• An irrational number is a number that cannot be expressed as a/b where a and b are whole numbers (or integers).
  • Any non-terminating and non-repeating decimal is considered an irrational number.
  • Numbers like √n, where n is not a perfect square, fall under irrational numbers.
• The square root of a non-square integer is also known as a surd.
  • On many calculators, irrational numbers are typically displayed as surds.

What irrational numbers should I know?

• You might encounter tasks where you're required to distinguish an irrational number from a given list. 
• Some common irrational numbers you should be familiar with are π, √2, √3, and √5. 
  • Any multiple of these numbers is also irrational. 
    • For instance, π/2, 3√2, and 3√5 are all irrational. 
• Modern calculators typically display irrational numbers in their exact form, either as a multiple of π or as √n, where n is not a perfect square. 
  • If the calculator cannot display the exact form, it will round the number to 9 or 10 decimal places.

Negative Numbers

What are negative numbers?

• Negative numbers are any numbers less than zero and are commonly encountered in various mathematical contexts, including algebraic equations and real-life situations like temperature readings or financial debts.

What are the rules for working with negative numbers?

• When multiplying and dividing with negative numbers, the rules are as follows:
  • When two numbers with the same sign are multiplied or divided, the result is positive. For example, (-12) ÷ (-4) = 3 and (-6) × (-4) = 24.
  • When two numbers with different signs are multiplied or divided, the result is negative. For example, (-12) ÷ 4 = -3 and 6 × (-4) = -24.
  • For multiplication and division, it's often simpler to perform the calculation without considering the signs initially, and then determine whether the result should be positive or negative based on the signs of the original numbers.
• When adding and subtracting with negative numbers:
  • Subtracting a negative number is equivalent to adding the positive number. For example, 5 - (-3) = 5 + 3 = 8.
  • Adding a negative number is equivalent to subtracting the positive number. For example, 7 + (-3) = 7 - 3 = 4.