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.
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 |
|---|---|---|
| 2 | 0, 1 | \(101_2\) |
| 3 | 0, 1, 2 | \(210_3\) |
| 4 | 0, 1, 2, 3 | \(123_4\) |
| 5 | 0, 1, 2, 3, 4 | \(340_5\) |
| 6 | 0, 1, 2, 3, 4, 5 | \(451_6\) |
| 7 | 0, 1, 2, 3, 4, 5, 6 | \(341_7\) |
| 8 | 0, 1, 2, 3, 4, 5, 6, 7 | \(341_8\) |
| 9 | 0, 1, 2, 3, 4, 5, 6, 7, 8 | \(5037_9\) |
| 10 | 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 | \(95_{10}\) |
Try Answering First
Read the question, answer it in your mind, then press “Check Answer”.
State the digit value of the underlined digit in \(5\underline{0}37_9\).
Convert \(110111_2\) directly to a number in base eight.
Calculate \(110_2 + 111_2\) in vertical form.
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
A number written with larger digits always has a greater value than a number written with smaller digits.
The value depends entirely on the base. For example, \(100_2 = 4_{10}\), while \(21_3 = 7_{10}\).
Base ten numbers must always be written with a subscript 10.
Writing the base subscript for base ten is optional and can be left out. Subscripts are required for other bases to avoid ambiguity.
Direct grouping of three digits can be used between any two bases.
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.
