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Number Bases

Number Bases - Pandai Notes
MATHEMATICS • Form 4 • Chapter 2: Number Bases

2.1 Number Bases

Number bases are mathematical systems that represent numbers using specific sets of digits. While the decimal system (base ten) is used in daily life, other bases such as base two, base eight, and base sixteen are fundamental in computer science and digital technology.

Learning Objectives

  • Represent and explain numbers in base two up to base ten in terms of numerals, place values, digit values, and number values based on the collection process.
  • Convert numbers from one base to another using various methods.
  • Perform calculations involving addition and subtraction of numbers in various bases.
  • Solve routine and non-routine problems involving number bases.

Conversion Routes and Grouping Between Number Bases

See the standard conversion bridge through base ten, and the shortcut grouping method between base two and base eight.

Base \(p\)
Any base from 2 to 9
→
Base 10
Sum of digit values
→
Base \(q\)
Repeated division by \(q\)
Direct 3-bit binary grouping bridge
\(110_2\)
Group of three binary digits
↔
\(6_8\)
\(111_2\)
Group of three binary digits
↔
\(7_8\)

Brief Explanation

What is a number base?

A number base defines the maximum number of unique digits used in a positional numeral system. In any base \(a\), valid digits range from \(0\) up to \(a - 1\).

How is a number value determined?

Each position represents a power of the base, beginning with \(a^0 = 1\) on the far right. The value of any digit is its face value multiplied by its positional place value, and the sum of all digit values yields the total number value in base ten.

Base Digits used Example
20, 1\(101_2\)
30, 1, 2\(210_3\)
40, 1, 2, 3\(123_4\)
50, 1, 2, 3, 4\(340_5\)
60, 1, 2, 3, 4, 5\(451_6\)
70, 1, 2, 3, 4, 5, 6\(341_7\)
80, 1, 2, 3, 4, 5, 6, 7\(341_8\)
90, 1, 2, 3, 4, 5, 6, 7, 8\(5037_9\)
100, 1, 2, 3, 4, 5, 6, 7, 8, 9\(95_{10}\)
1
Identify the base
Check the subscript next to the number. For example, \(341_8\) is in base eight.
2
Find each digit value
Multiply each digit by its place value. In \(341_8\), the digit value of \(3\) is \(3 \times 8^2 = 192\).
3
Add all digit values
Sum all digit values to get the number value in base ten.
4
Convert to another base if needed
Use repeated division by the new base, reading remainders from bottom upwards.

Try Answering First

Read the question, answer it in your mind, then press “Check Answer”.

1

State the digit value of the underlined digit in \(5\underline{0}37_9\).

Answer: The place value of the underlined digit is \(9^2 = 81\). The digit value is \(0 \times 9^2 = 0\).
2

Convert \(110111_2\) directly to a number in base eight.

Answer: Group into sets of three digits from right to left: \((110)(111)_2\). For the left group: \(1(2^2) + 1(2^1) + 0(2^0) = 4 + 2 + 0 = 6\). For the right group: \(1(2^2) + 1(2^1) + 1(2^0) = 4 + 2 + 1 = 7\). Combining gives \(67_8\).
3

Calculate \(110_2 + 111_2\) in vertical form.

Answer: From right to left: \(0 + 1 = 1_2\); \(1 + 1 = 2_{10} = 10_2\) (write 0, carry 1); \(1 + 1 + 1 = 3_{10} = 11_2\). Result is \(1101_2\).

Common Mistakes

  • ! Using a digit that is equal to or greater than the base, for example writing the digit 5 in a base five number or digit 2 in binary.
  • ! Forgetting that the rightmost place value is always \(a^0 = 1\), not \(a^1\).
  • ! Borrowing 10 instead of the base value during vertical subtraction, for example borrowing 6 in base six, or 8 in base eight, not 10.
  • ! Carrying groups of 10 instead of groups equal to the base during vertical addition.
  • ! Reading the remainders from top to bottom instead of bottom upwards during repeated division.
  • ! Grouping binary digits from left to right instead of right to left when converting to base eight.

Misconceptions

Misconception

A number written with larger digits always has a greater value than a number written with smaller digits.

Correct Fact

The value depends entirely on the base. For example, \(100_2 = 4_{10}\), while \(21_3 = 7_{10}\).

Misconception

Base ten numbers must always be written with a subscript 10.

Correct Fact

Writing the base subscript for base ten is optional and can be left out. Subscripts are required for other bases to avoid ambiguity.

Misconception

Direct grouping of three digits can be used between any two bases.

Correct Fact

Direct grouping of three digits is specifically applicable between base two and base eight because \(2^3 = 8\). It cannot be used directly between arbitrary bases such as base five and base seven.

Summary

  • A number in base \(a\) only uses digits from \(0\) to \(a - 1\).
  • Place values are powers of the base: \(a^n, \dots, a^2, a^1, a^0\).
  • Digit value = \(\text{digit} \times \text{place value}\).
  • Number value in base ten = sum of all digit values.
  • Conversion from base ten to another base uses repeated division by the base value, with remainders read bottom-up.
  • Conversion between non-decimal bases goes through base ten: \(\text{Base } p \rightarrow \text{Base } 10 \rightarrow \text{Base } q\).
  • Base 2 to base 8 conversion groups binary digits into 3-digit clusters from right to left; base 8 to base 2 expands each octal digit into 3 binary digits.
  • Addition and subtraction in base \(a\) follow standard arithmetic, where carries and borrows operate in groups of \(a\).

Short Activity

Complete all parts first, then press the check answer button.

A. Objective Quiz

1 Which of the following numbers is an invalid representation for a number in base six?

2 What is the value of the number \(341_7\) when converted to base ten?

B. Fill in the Blanks

3 In base eight, the place value of the third digit from the right is 8 to the power of .

4 In vertical subtraction of numbers in base seven, when 1 is borrowed from the next higher place value, the value transferred to the current position is equal to .

C. Matching: Octal Digit to Binary Equivalent

Match each octal digit (base 8) to its correct 3-digit binary equivalent (base 2). On a phone, tap the answer choice first, then tap the matching box.

Octal Digit (Base 8)
\(2_8\)
\(5_8\)
\(6_8\)
\(7_8\)
Binary Representation (Base 2)
\(110_2\)CorrectTry againEmpty
\(010_2\)CorrectTry againEmpty
\(111_2\)CorrectTry againEmpty
\(101_2\)CorrectTry againEmpty
 

Key Terms

Number base Place value Digit value Number value Decimal system Binary system Octal system Repeated division
The views expressed in the notes here are those of the contributor and don’t necessarily represent the views of Pandai.

Number Bases

Number Bases - Pandai Notes
MATHEMATICS • Form 4 • Chapter 2: Number Bases

2.1 Number Bases

Number bases are mathematical systems that represent numbers using specific sets of digits. While the decimal system (base ten) is used in daily life, other bases such as base two, base eight, and base sixteen are fundamental in computer science and digital technology.

Learning Objectives

  • Represent and explain numbers in base two up to base ten in terms of numerals, place values, digit values, and number values based on the collection process.
  • Convert numbers from one base to another using various methods.
  • Perform calculations involving addition and subtraction of numbers in various bases.
  • Solve routine and non-routine problems involving number bases.

Conversion Routes and Grouping Between Number Bases

See the standard conversion bridge through base ten, and the shortcut grouping method between base two and base eight.

Base \(p\)
Any base from 2 to 9
→
Base 10
Sum of digit values
→
Base \(q\)
Repeated division by \(q\)
Direct 3-bit binary grouping bridge
\(110_2\)
Group of three binary digits
↔
\(6_8\)
\(111_2\)
Group of three binary digits
↔
\(7_8\)

Brief Explanation

What is a number base?

A number base defines the maximum number of unique digits used in a positional numeral system. In any base \(a\), valid digits range from \(0\) up to \(a - 1\).

How is a number value determined?

Each position represents a power of the base, beginning with \(a^0 = 1\) on the far right. The value of any digit is its face value multiplied by its positional place value, and the sum of all digit values yields the total number value in base ten.

Base Digits used Example
20, 1\(101_2\)
30, 1, 2\(210_3\)
40, 1, 2, 3\(123_4\)
50, 1, 2, 3, 4\(340_5\)
60, 1, 2, 3, 4, 5\(451_6\)
70, 1, 2, 3, 4, 5, 6\(341_7\)
80, 1, 2, 3, 4, 5, 6, 7\(341_8\)
90, 1, 2, 3, 4, 5, 6, 7, 8\(5037_9\)
100, 1, 2, 3, 4, 5, 6, 7, 8, 9\(95_{10}\)
1
Identify the base
Check the subscript next to the number. For example, \(341_8\) is in base eight.
2
Find each digit value
Multiply each digit by its place value. In \(341_8\), the digit value of \(3\) is \(3 \times 8^2 = 192\).
3
Add all digit values
Sum all digit values to get the number value in base ten.
4
Convert to another base if needed
Use repeated division by the new base, reading remainders from bottom upwards.

Try Answering First

Read the question, answer it in your mind, then press “Check Answer”.

1

State the digit value of the underlined digit in \(5\underline{0}37_9\).

Answer: The place value of the underlined digit is \(9^2 = 81\). The digit value is \(0 \times 9^2 = 0\).
2

Convert \(110111_2\) directly to a number in base eight.

Answer: Group into sets of three digits from right to left: \((110)(111)_2\). For the left group: \(1(2^2) + 1(2^1) + 0(2^0) = 4 + 2 + 0 = 6\). For the right group: \(1(2^2) + 1(2^1) + 1(2^0) = 4 + 2 + 1 = 7\). Combining gives \(67_8\).
3

Calculate \(110_2 + 111_2\) in vertical form.

Answer: From right to left: \(0 + 1 = 1_2\); \(1 + 1 = 2_{10} = 10_2\) (write 0, carry 1); \(1 + 1 + 1 = 3_{10} = 11_2\). Result is \(1101_2\).

Common Mistakes

  • ! Using a digit that is equal to or greater than the base, for example writing the digit 5 in a base five number or digit 2 in binary.
  • ! Forgetting that the rightmost place value is always \(a^0 = 1\), not \(a^1\).
  • ! Borrowing 10 instead of the base value during vertical subtraction, for example borrowing 6 in base six, or 8 in base eight, not 10.
  • ! Carrying groups of 10 instead of groups equal to the base during vertical addition.
  • ! Reading the remainders from top to bottom instead of bottom upwards during repeated division.
  • ! Grouping binary digits from left to right instead of right to left when converting to base eight.

Misconceptions

Misconception

A number written with larger digits always has a greater value than a number written with smaller digits.

Correct Fact

The value depends entirely on the base. For example, \(100_2 = 4_{10}\), while \(21_3 = 7_{10}\).

Misconception

Base ten numbers must always be written with a subscript 10.

Correct Fact

Writing the base subscript for base ten is optional and can be left out. Subscripts are required for other bases to avoid ambiguity.

Misconception

Direct grouping of three digits can be used between any two bases.

Correct Fact

Direct grouping of three digits is specifically applicable between base two and base eight because \(2^3 = 8\). It cannot be used directly between arbitrary bases such as base five and base seven.

Summary

  • A number in base \(a\) only uses digits from \(0\) to \(a - 1\).
  • Place values are powers of the base: \(a^n, \dots, a^2, a^1, a^0\).
  • Digit value = \(\text{digit} \times \text{place value}\).
  • Number value in base ten = sum of all digit values.
  • Conversion from base ten to another base uses repeated division by the base value, with remainders read bottom-up.
  • Conversion between non-decimal bases goes through base ten: \(\text{Base } p \rightarrow \text{Base } 10 \rightarrow \text{Base } q\).
  • Base 2 to base 8 conversion groups binary digits into 3-digit clusters from right to left; base 8 to base 2 expands each octal digit into 3 binary digits.
  • Addition and subtraction in base \(a\) follow standard arithmetic, where carries and borrows operate in groups of \(a\).

Short Activity

Complete all parts first, then press the check answer button.

A. Objective Quiz

1 Which of the following numbers is an invalid representation for a number in base six?

2 What is the value of the number \(341_7\) when converted to base ten?

B. Fill in the Blanks

3 In base eight, the place value of the third digit from the right is 8 to the power of .

4 In vertical subtraction of numbers in base seven, when 1 is borrowed from the next higher place value, the value transferred to the current position is equal to .

C. Matching: Octal Digit to Binary Equivalent

Match each octal digit (base 8) to its correct 3-digit binary equivalent (base 2). On a phone, tap the answer choice first, then tap the matching box.

Octal Digit (Base 8)
\(2_8\)
\(5_8\)
\(6_8\)
\(7_8\)
Binary Representation (Base 2)
\(110_2\)CorrectTry againEmpty
\(010_2\)CorrectTry againEmpty
\(111_2\)CorrectTry againEmpty
\(101_2\)CorrectTry againEmpty
 

Key Terms

Number base Place value Digit value Number value Decimal system Binary system Octal system Repeated division
The views expressed in the notes here are those of the contributor and don’t necessarily represent the views of Pandai.
Chapter : Number Bases
Topic : Number Bases
Form 4 Mathematics
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