History of Numbers:
The origin of numbers dates back to early human civilization. Before the invention of written symbols, humans used marks, stones, bones, and fingers to count objects. Archaeological findings such as the Lebombo Bone and the Ishango Bone provide evidence that counting existed as early as 20,000 – 35,000 years ago. These discoveries show that the concept of numbers developed gradually to meet practical needs.Concept of ono to one correspondence:
One-to-one correspondence is one of the earliest methods of counting. In this method, each object in one group is paired with exactly one object in another group.
In the absence of number symbols, early humans used this technique to compare quantities.
In the absence of number symbols, early humans used this technique to compare quantities.
Example:
\(1\) cow \(→\) \(1\) pebble
\(1\) student \(→\) \(1\) chair
Natural Numbers \(\mathbb{N}\):
Natural Numbers are the counting numbers.
\(\mathbb{N}\) \(=\) \({1, 2, 3, ... }\).
They are used for counting objects in everyday life.
Concept of zero:
Zero is a fundamental number that represents the absence of quantity.
1. As a Number: It indicates “nothing.”
Example:
\(0\) apples means no apples.
Example:
\(10\), \(100\), \(5034\),..
Brahmagupta’s Rules for Zero:
• When zero is added to any number \(x\), the number remains unchanged. (i.e.) \(x + 0 = x\).
• When zero is subtracted from any number \(x\), the number remains unchanged. (i.e.) \(x – 0 = x\).
• When any number \(x\) is multiplied by zero, the result is zero. (i.e.) \(x × 0 = 0\).
Whole number \(\mathbb{W}\):
The set of all natural numbers including \(0\) are called whole numbers.
That is, \(\mathbb{W}\) \(=\) \({0,1,2,3,...}\).
Integers \(\mathbb{Z}\):
Integers are the set of positive numbers, negative numbers and zero.
That is, \(\mathbb{Z}\) \(=\) \({...,-3,-2,-1,0,1,2,3,...}\).
Arithmetic operation on integers:
Brahmagupta termed the positive numbers as fortunes and negative numbers as debts.
Rules:
1. A fortune plus a fortune is a fortune: \(1 + 2 = 3\).
2. A debt plus a debt is a debt: \((– 3) + (– 2) = –5\).
3. A fortune minus zero is a fortune, a debt minus zero is a debt: \(5 – 0 = 5\), and \(– 7 – 0 = – 7\).
4. The product of a debt and a fortune is a debt: \((– 5) × 4 = –20\).
5. The product of two debts is a fortune: \((–1) × (–8) = 8\).
Rational numbers:
A rational number is a number that can be expressed in the form of \(\frac{p}{q}\) where \(p\) and \(q\) are integer and \(q\neq 0\).
Example:
Equivalent rational numbers:
Two rational numbers are said to be equivalent if they represent the same value, even though their forms may differ.
This can be obtained by multiplying or dividing the numerator and denominator by the same non-zero number.
This can be obtained by multiplying or dividing the numerator and denominator by the same non-zero number.
Example:
\(\frac{1}{2}\) and \(\frac{2}{4}\) are equivalent.
Where, \(\frac{1 \times 2}{2 \times 2} = \frac{2}{4}\)