Number representation and number systems

IB Mathematics: Analysis and Approaches HL· 1.1 Number representation· 15 min read

1. Positional Number Systems★☆☆☆☆⏱ 5 min

📘 Definition

Positional Number System

$(d_{n} d_{n - 1} ... d_{0})_{b}$

A number system where the value of a digit depends on its value and its position. The total value is the sum of each digit multiplied by the base raised to the power of its position index.

Example:

$12 3_{10} = 1 \times 1 0^{2} + 2 \times 1 0^{1} + 3 \times 1 0^{0} = 123$

For any base $b$ , there are exactly $b$ distinct digits, ranging from $0$ to $b - 1$ . All number systems studied in IB AA HL are positional, unlike older non-positional systems such as Roman numerals.

✓ Quick check

Test your understanding of base rules

What is the largest possible digit in a base 7 number system?
- 5
- 6
- 7
- 8
Reveal answer
1 —
Correct! The maximum digit in any base $b$ is always $b - 1$ . If you chose 7, you confused the base value itself with the maximum digit.

2. Conversion Between Bases★★☆☆☆⏱ 5 min

There are two core conversion processes: converting from any base to base 10, and converting from base 10 to any other base. Both follow straightforward step-by-step rules.

To convert $(d_{n} ... d_{0})_{b}$ to base 10: Expand the positional sum $\sum_{i = 0}^{n} d_{i} \times b^{i}$ and evaluate in base 10.
To convert a base 10 integer to base $b$ : Repeatedly divide by $b$ , collect remainders, then reverse the list of remainders to get the final number.

📐 Worked Example

Convert $21 3_{4}$ to base 10, then convert the result to base 5.

1
Step 1: Convert $21 3_{4}$ to base 10 using positional expansion:
2
$2 \times 4^2 + 1 \times 4^1 + 3 \times 4^0 = 2(16) + 4 + 3 = 39$
3
Step 2: Convert $3 9_{10}$ to base 5 via repeated division:
4
• $39 \div 5 = 7$ , remainder = $4$ (least significant digit)
5
• $7 \div 5 = 1$ , remainder = $2$
6
• $1 \div 5 = 0$ , remainder = $1$ (most significant digit)
7
Step 3: Reverse the order of remainders to get the final result:
8
$124_5$

3. Binary and Hexadecimal★★☆☆☆⏱ 3 min

📘 Definition

Hexadecimal

Base 16 number system, which uses digits 0-9 for values 0-9, and letters A-F for values 10-15 respectively. It is used to compactly represent binary data in computing.

Since $16 = 2^{4}$ , each hexadecimal digit corresponds directly to 4 binary digits (bits). This gives a shortcut for converting between binary and hexadecimal that avoids converting through base 10.

📐 Worked Example

Convert $101101 0_{2}$ directly to hexadecimal.

1
Step 1: Group binary digits into groups of 4 starting from the right, add leading zeros to complete the first group:
2
$01011010$
3
Step 2: Convert each 4-bit group to its equivalent hexadecimal value:
4
$010 1_{2} = 5$ , $101 0_{2} = 10 = A$
5
Step 3: Combine the digits to get the final result:
6
$5A_{16}$

4. Basic Binary Arithmetic★★★☆☆⏱ 4 min

Addition in binary follows the same process as addition in decimal, except you carry over to the next position when the sum of digits is equal to or greater than 2 (the base), instead of 10.

📐 Worked Example

Calculate $101 1_{2} + 110 1_{2}$ .

1
Step 1: Align numbers by the right, add starting from the rightmost digit:
2
1 + 1 = 2 = $1 \times 2^{1} + 0$ , write 0, carry 1
3
Step 2: Next digit: 1 + 0 + 1 (carry) = 2, write 0, carry 1
4
Step 3: Next digit: 0 + 1 + 1 (carry) = 2, write 0, carry 1
5
Step 4: Leftmost digit: 1 + 1 + 1 (carry) = 3 = $1 \times 2^{2} + 1$ , write 1, carry 1
6
Step 5: Add the final carry as a new leftmost digit:
7
$11000_2$

5. Common Pitfalls

Wrong move:

Writing remainders in the order they are calculated when converting base 10 to a new base.

Why:

The first remainder calculated is the least significant (rightmost) digit, not the most significant.

Correct move:

Reverse the order of collected remainders to get the correct final number, with the last remainder as the leftmost digit.

Wrong move:

Writing 10 as two digits 10 instead of one digit A in hexadecimal.

Why:

This adds an extra digit that shifts all place values, leading to an incorrect result.

Correct move:

Always represent values 10-15 as single digits A-F when working in hexadecimal.

Wrong move:

Grouping binary digits from the left when converting to hexadecimal.

Why:

Grouping from the left shifts all bits to incorrect place values.

Correct move:

Always start grouping from the right (the least significant end) for integer binary numbers.

Wrong move:

Starting the exponent count at 1 for the rightmost integer digit in positional expansion.

Why:

This shifts all exponents by 1, leading to an incorrect value when converting to base 10.

Correct move:

The rightmost integer digit always has an exponent of 0, regardless of base.

6. Quick Reference Cheatsheet

Concept	Key Rule
Max digit in base $b$	$b - 1$
Convert to base 10	Expand $d_{n} ... d_{0} = \sum_{i = 0}^{n} d_{i} b^{i}$
Convert base 10 → new base	Repeated division, reverse remainders
Binary → hexadecimal	Group 4 bits from right, 1 hex digit per group
Binary addition	Carry 1 when sum ≥ 2

7. Frequently Asked

Why do we study different number bases?

Different bases are useful for different applications: binary is used in all computing hardware, hexadecimal is a compact way to represent binary data, and decimal is the standard for everyday use. Base conversion also builds core number reasoning skills for IB Mathematics.

When this came up on past exams

AI-estimated based on syllabus patterns — cross-check with official past papers for accuracy. Use only as revision-focus signals.

2021 · 1
Base conversion question
2023 · 1
Binary arithmetic problem

What's Next

Number representation is the foundational topic for all work in discrete mathematics and algebra in IB AA HL. The logical thinking skills you practice here will help you solve more complex problems across all units of the syllabus, including sequences, series, and calculus. Mastery of binary and base conversion also prepares you for topics like logic, set theory, and counting principles that appear frequently in Paper 1. This topic also forms the base for any further study of computer science or quantitative mathematics.