Numbers & Number System

Numbers are mathematical objects used to measure count and label. Numbers can be classified into sets called Number Systems such as natural numbers and real numbers. These are major categories –

Natural Numbers (N): Counting numbers starting from 1, 2, 3, and so on.
Whole Numbers (W): Natural numbers including zero, i.e., 0, 1, 2, 3, etc.
Integers (Z): Whole numbers and their negative counterparts, i.e., -3, -2, -1, 0, 1, 2, 3, etc.
Rational Numbers (Q): Numbers that can be expressed as a fraction of two integers, where the denominator is not zero. For example, 1/2, 3/4, and 5/1. Every rational number is a real number but every real number may not be rational number.
Irrational Numbers (P): Numbers that cannot be expressed as fractions. They have non-repeating, non-terminating decimal expansions such as √2, π.
Real Numbers (R): All rational and irrational numbers.
Complex Numbers (C): Numbers in the form of a + bi, where a and b are real numbers and i is the imaginary unit (i² = -1). If the real part of the complex number is 0, then the number is called imaginary number. If the real and imaginary parts of the complex numbers are both integers, then the number is called a Gaussian integer.

Each type of number system is a subset of another number system so mathematically can be shown as N⊂Z⊂Q⊂R⊂C.

Number Systems

Decimal Number System (Base 10):
Digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9
Place Value: Each position is a power of 10. For example, 123.45 can be expressed as 1×10²+2×10¹+3×10⁰+4×10⁻¹+5×10⁻²

Binary Number System (Base 2):
Digits: 0, 1
Place Value: Each position is a power of 2. For example, 1011 in binary can be expressed as 1×2³+0×2²+1×2¹+1×2⁰=11 in decimal.

Octal Number System (Base 8):
Digits: 0, 1, 2, 3, 4, 5, 6, 7
Place Value: Each position is a power of 8. For example, 157 in octal can be expressed as 1×8²+5×8¹+7×8⁰=111 in decimal.

Hexadecimal Number System (Base 16):
Digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F (A = 10, B = 11, …, F = 15)
Place Value: Each position is a power of 16. For example, 1A3 in hexadecimal can be expressed as 1×16²+A×16¹+3×16⁰=1×256+10×16+3=419 in decimal.
Similarly, we can convert decimals to binary. Let us convert decimal e.g., 13 to binary.
Divide by 2 and write down the remainder.
13 ÷ 2 = 6 remainder 1
6 ÷ 2 = 3 remainder 0
3 ÷ 2 = 1 remainder 1
1 ÷ 2 = 0 remainder 1
Now, read the remainders from bottom to top & it will be 1101.
Let us convert binary e.g., 1011 to decimal.
1×2³+1×2²+0×2¹+1×2⁰ = 8+4+0+1=13

Let us convert decimal e.g., 255 to hexadecimal. Divide by 16 and write down the remainder.
255 ÷ 16 = 15 remainder 15 (F)
15 ÷ 16 = 0 remainder 15 (F)
Now, read the remainders from bottom to top and it will FF i.e., 1515.
Let us convert hexadecimal e.g., 1A3 to decimal. 1×16²+10×16¹+3×16⁰ =256+160+3=419