Binary, Decimal, Octal and Hexadecimal Basics

What you will learn#

• Why numbering systems use a base (or radix).
• Which digits belong to binary, octal, decimal and hexadecimal.
• What place value means in positional notation.
• Why computers use binary while people often read decimal or hexadecimal.

Why this topic matters#

Exam questions may ask which number contains an invalid digit, identify the base of a numeral, or ask how many bits correspond to one hexadecimal digit. These facts become easy when you understand what symbols a numbering system allows.

A number's meaning depends on its base#

Most everyday prices and quantities use the decimal system, whose base is 10. Decimal has ten distinct digit symbols, 0 through 9. A computer, however, commonly represents its internal information using two distinguishable states represented by the digits 0 and 1—the binary system.

In a positional system, each position represents a power of the base. Thus, 314 in decimal means three hundreds, one ten and four ones:

314₁₀ = 3 × 10² + 1 × 10¹ + 4 × 10⁰.

The rightmost place has weight base⁰ = 1. Each step to the left multiplies the weight by the base. A numeral is not just a string of symbols; its base determines its numeric value.

The subscript indicates the base; it is not part of the represented number's digits. You may also see programming-oriented notations such as 0b1011 (binary) and 0xF (hexadecimal), but notation rules differ across languages.

Binary: base 2#

Binary uses two digits. Every position to the left is worth twice the position to its right:

So 1011₂ = 8 + 0 + 2 + 1 = 11₁₀. Each individual binary digit is a bit. In the usual computing sense, eight bits make one byte (see CA-02-07). A bit is a unit of information, not a literal physical switch by definition; hardware uses physical states to represent it.

Concept to understand: Binary is useful in digital systems because engineering two distinguishable states can be comparatively robust. That does not mean every part of a computer contains only two kinds of electrical voltage or that real hardware is free of analogue effects.

Octal: base 8#

Octal allows digits 0 to 7. After 7₈, the next numeral is 10₈, meaning eight in decimal, not ten in decimal. A numeral such as 18₈ is invalid, because octal has no digit 8.

Octal is historically useful as a shorthand for binary. One octal digit corresponds to three binary bits, because 2³ = 8. More about grouping appears in CA-04-02.

Hexadecimal: base 16#

Hexadecimal needs sixteen symbols: 0–9, then letters A–F for ten through fifteen.

Therefore 1A₁₆ = 1 × 16 + 10 = 26₁₀. A single hexadecimal digit corresponds to four binary bits since 2⁴ = 16. Hexadecimal makes long bit patterns easier to read, and is common in contexts such as memory addresses, color codes and technical identifiers. An address written in hexadecimal is still an address: hexadecimal is a notation, not a memory technology.

Four representations, one value#

The number 15 can be written as 1111₂, 17₈, 15₁₀ or F₁₆. These numerals look different but represent the same value.

Avoid interpreting 10 without the base: 10₂ means two, 10₈ means eight, 10₁₀ means ten and 10₁₆ means sixteen.

Worked examples#

Example 1 — Valid binary: Is 1021₂ valid? No. Binary permits only 0 and 1; digit 2 is invalid.

Example 2 — Value of 23₈: Compute 2 × 8 + 3 = 19₁₀.

Example 3 — Value of 2F₁₆: 2 × 16 + 15 = 47₁₀.

Example 4 — Representing a bit pattern: The eight-bit pattern 11110000 can be split 1111 0000 and written as F0₁₆. Grouping is a shorthand; it does not change the represented bits.

Important terms#

Common confusions and exam traps#

• Base versus number of digits: Binary has two available symbols, but a binary numeral may have any number of positions.
• 10 is not always ten: Its value changes with the base.
• A in hexadecimal is not the letter A as ordinary text: It is the digit for decimal ten in that numeral.
• Octal versus decimal: A valid decimal numeral may be invalid in octal (89).
• Hexadecimal versus binary storage: A hex numeral is a more compact way to write a value; it is not a new underlying physical state.

Objective-exam facts to remember#

Quick revision#

1. A number's base determines its legal digits and positional weights.
2. Binary uses 0, 1; octal uses 0–7; hexadecimal uses 0–9, A–F.
3. 1111₂ = 17₈ = 15₁₀ = F₁₆.
4. A hexadecimal digit groups four bits; an octal digit groups three.
5. Do not compare visually identical numerals without checking the base.

Practice questions — CA-04-01#

1. What is the base (radix) of the binary number system?
A. 2
B. 8
C. 10
D. 16

2. Which of the following is not a valid octal numeral?
A. 17₈
B. 70₈
C. 28₈
D. 106₈

3. In hexadecimal notation, the digit B means which decimal value?
A. 10
B. 11
C. 12
D. 16

4. How many binary bits are represented by one hexadecimal digit?
A. 2
B. 3
C. 4
D. 8

5. What does 10₂ represent in decimal?
A. 1
B. 2
C. 8
D. 10

6. Which set contains only valid binary digit symbols?
A. 0, 1, 2
B. 0, 1
C. 0–7
D. 0–9

7. What is the largest single digit of the hexadecimal system?
A. 9
B. D
C. E
D. F

8. In a positional number system, the rightmost whole-number digit has which place value?
A. Base²
B. Base¹
C. Base⁰
D. Always ten

9. How many bits correspond to one octal digit?
A. 2
B. 3
C. 4
D. 8

10. Which numeral equals decimal fifteen?
A. 1111₂
B. 1111₈
C. 1111₁₆
D. 1000₂

11. What is the decimal value of 1A₁₆?
A. 16
B. 20
C. 26
D. 110

12. What is the main reason hexadecimal is used to display many low-level bit patterns?
A. It creates extra storage space
B. It gives a compact, readable representation of bits
C. It eliminates binary hardware
D. It encrypts the information

Answers and explanations — CA-04-01#