IB MYP 4-5 Maths-Number systems notation - Study Notes - New Syllabus
IB MYP 4-5 Maths- Number systems notation – Study Notes
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- Number systems notation
IB MYP 4-5 Maths- Number systems notation – Study Notes – All topics
Understanding Number Systems
Understanding Number Systems
A number system is a way to represent and express numbers. Each system is based on a certain base (also called a radix), which tells how many digits are used in that system.
Common Number Systems:
| Number System | Base | Digits Used | Example |
|---|---|---|---|
| Binary | 2 | 0, 1 | \(1011_2\) |
| Octal | 8 | 0–7 | \(157_8\) |
| Decimal | 10 | 0–9 | \(205_{10}\) |
| Hexadecimal | 16 | 0–9, A–F | \(3F_{16}\) |
Base (Radix): The number of digits used in the system. For example, base 10 uses 10 digits (0–9).
Place Value System: Each digit in a number has a place value depending on its position and the base.
For example, in decimal \(235_{10}\):
- 2 is in the hundreds place → \(2 \times 10^2\)
- 3 is in the tens place → \(3 \times 10^1\)
- 5 is in the ones place → \(5 \times 10^0\)
So, \(235 = 2 \times 100 + 3 \times 10 + 5 = 200 + 30 + 5\)
Counting and Converting Between Number Bases
Each number system uses a different base (radix). In base systems other than decimal, numbers are written using only digits less than the base.
Counting in Base 2 (Binary):
Binary only uses digits 0 and 1. After 1 comes 10 (which is 2 in decimal).
- Decimal: 0, 1, 2, 3, 4, 5, 6, 7
- Binary: 0, 1, 10, 11, 100, 101, 110, 111
To Convert from Binary to Decimal: Multiply each binary digit by \(2^n\), where \(n\) is its position from the right (starting from 0).
Example:
Convert \(1011_2\) to decimal.
▶️ Answer/Explanation
Step 1: Write the place values
\(1 \cdot 2^3 + 0 \cdot 2^2 + 1 \cdot 2^1 + 1 \cdot 2^0\)
Step 2: Calculate
\(8 + 0 + 2 + 1 = 11\)
Final Answer:
\(\boxed{11_{10}}\)
To Convert from Decimal to Binary: Keep dividing by 2 and write down the remainders from bottom to top.
Example:
Convert \(13_{10}\) to binary.
▶️ Answer/Explanation
Step 1: Divide by 2 repeatedly
- \(13 \div 2 = 6\) remainder 1
- \(6 \div 2 = 3\) remainder 0
- \(3 \div 2 = 1\) remainder 1
- \(1 \div 2 = 0\) remainder 1
Step 2: Read remainders bottom to top
\(1101_2\)
Final Answer:
\(\boxed{1101_2}\)
Other Bases (e.g. base 5): Use the same rules. Base 5 uses digits 0, 1, 2, 3, 4 only.
Example:
Convert \(132_5\) to decimal.
▶️ Answer/Explanation
Use base-5 place values
\(1 \cdot 5^2 + 3 \cdot 5^1 + 2 \cdot 5^0\)
\(25 + 15 + 2 = 42\)
Final Answer:
\(\boxed{42_{10}}\)
Performing Operations in Different Bases
Just like in base 10, we can do addition, subtraction, multiplication, and division in other bases (like binary or base 5). The key is to follow the place value and carry/borrow rules of that base.
Addition in Binary (Base 2)
Binary Addition Rules:
- \( 0 + 0 = 0 \)
- \( 0 + 1 = 1 \)
- \( 1 + 0 = 1 \)
- \( 1 + 1 = 10 \) → write 0, carry 1
- \( 1 + 1 + 1 = 11 \) → write 1, carry 1
Example:
Add \( 1011_2 + 1101_2 \)
▶️ Answer/Explanation
Binary column addition with carries:
Step-by-step:
- 1 + 1 = 10 → write 0, carry 1
- 1 + 0 + 1 (carry) = 10 → write 0, carry 1
- 0 + 1 + 1 = 10 → write 0, carry 1
- 1 + 1 + 1 = 11 → write 1, carry 1
- Carry 1 to new digit
Final Answer:
\(\boxed{11000_2}\)
Subtraction in Binary (Base 2)
Binary subtraction follows rules similar to base 10 with borrowing:
- \( 0 – 0 = 0 \)
- \( 1 – 0 = 1 \)
- \( 1 – 1 = 0 \)
- \( 0 – 1 = 1 \) (borrow 1 from the left)
Example:
Subtract \( 1010_2 – 0111_2 \)
▶️ Answer/Explanation
Convert both to decimal:
\( 1010_2 = 10_{10} \), \( 0111_2 = 7_{10} \)
Perform subtraction: \( 10 – 7 = 3 \)
Convert back: \( 3 = 0011_2 \)
Final Answer:
\(\boxed{0011_2}\)
Addition in Base 5
In base 5, digits go from 0 to 4. If a sum exceeds 4, carry to the next place value.
Example:
Add \( 243_5 + 134_5 \)
▶️ Answer/Explanation
Start from the right (units place):
- \(3 + 4 = 7\) → 7 in base 5 is 2 with carry 1
- \(4 + 3 + 1 = 8\) → 8 in base 5 is 3 with carry 1
- \(2 + 1 + 1 = 4\) → No carry
Final Answer:
\(\boxed{432_5}\)