Number bases

Unit 3 Number bases

Key unit competence: Solve problems involving conversion, operations and equations of number bases.

Key ideas

• A number is an idea; a numeral is the symbol for it; a digit is one symbol. In base $b$, the digits are $0,1,\dots ,b-1$ and place values are powers of $b$.
• Base ten to base $b$: divide repeatedly by $b$ and read the remainders from bottom to top.
• Base $b$ to base ten: expand with place values, e.g. $231_{\text{five}}=2\times 25+3\times 5+1=66$.
• Between two non-decimal bases, go through base ten (or group binary digits: 3 binary digits = 1 octal digit).
• Add, subtract, multiply and divide in base $b$ as in base ten, but carry or borrow $b$, not 10.
• Equations in bases: convert each numeral to base ten in terms of the unknown, then solve. Example: $23_x=17$ means $2x+3=17$, so $x=7$.

Formulas to know

Common misconceptions

• "$5_{\text{five}}$ is a numeral." No: base five uses only the digits 0 to 4.
• "Read the remainders from top to bottom." No: the first remainder is the units digit, so read upwards.
• "In base two, $1+1=2$." No: $1+1=10_{\text{two}}$.

Worked examples

Practice questions

Easier and harder questions are mixed, just like in a real exam.

1. Convert 59 to base two and to base five.

   Show answer

   $59=111011_{\text{two}}$ (32+16+8+2+1) and $59=214_{\text{five}}$ (50+5+4).

   Common mistake: Reading remainders from top to bottom.

2. Work out $23_{\text{four}}\times 12_{\text{four}}$.

   Show answer

   $11\times 6=66=1\times 64+0\times 16+0\times 4+2=1002_{\text{four}}$.

   Common mistake: Multiplying as if in base ten.

3. Convert $3021_{\text{four}}$ to base ten.

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   $3\times 64+0+2\times 4+1=201$.

   Common mistake: Using powers of 10.

4. Convert $57_{\text{eight}}$ to base two.

   Show answer

   $5=101$, $7=111$: $101111_{\text{two}}$.

   Common mistake: Converting the whole number 57 instead of each octal digit.

5. Work out $1101_{\text{two}}+1011_{\text{two}}$ and check in base ten.

   Show answer

   $11000_{\text{two}}$; check $13+11=24=16+8$.

   Common mistake: Carrying 10 instead of 2.

6. Find the base $b$ in which $35_b+24_b=103_b$.

   Show answer

   $(3b+5)+(2b+4)=b^2+3$: $b^2-5b-6=0$, $(b-6)(b+1)=0$, $b=6$.

   Common mistake: Forgetting to reject the negative base.

7. Work out $403_{\text{five}}-124_{\text{five}}$.

   Show answer

   $403_{\text{five}}=103$, $124_{\text{five}}=39$, difference $64=224_{\text{five}}$.

   Common mistake: Borrowing 10 instead of 5.

8. Convert $2A_{\text{twelve}}$ to base ten, where A stands for ten.

   Show answer

   $2\times 12+10=34$.

   Common mistake: Treating A as 1.

Matching and drag-and-drop

1. Matching

   Match each item in column A with the correct answer in column B. Write the letter next to each number.

   Column A	Column B
   1. Work out $1000_{\text{two}}-11_{\text{two}}$.	a. $43_{\text{five}}$
   2. Work out $34_{\text{five}}+23_{\text{five}}$.	b. $1000_{\text{two}}$
   3. Convert 23 to base five.	c. $112_{\text{five}}$
   4. Work out $101_{\text{two}}+11_{\text{two}}$.	d. $101_{\text{two}}$

   Show answer

   1 → d ($101_{\text{two}}$) · 2 → c ($112_{\text{five}}$) · 3 → a ($43_{\text{five}}$) · 4 → b ($1000_{\text{two}}$)

   • Step 1. $1000_{\text{two}}=8$ and $11_{\text{two}}=3$; $8-3=5$.
     Step 2. $5=4+1=101_{\text{two}}$.
   • Step 1. $4+3=7=12_{\text{five}}$: write 2 carry 1.
     Step 2. $3+2+1=6=11_{\text{five}}$.
   • $23=4\times 5+3$.
   • $5+3=8=1000_{\text{two}}$.

2. Drag and drop

   Choose the correct word or phrase from the box for each gap. One word in the box is not needed.

   $101011_{\text{two}}$  ·  $101101_{\text{two}}$  ·  7 000  ·  $65_{\text{eight}}$
   1. In the numeral $7707$ (base ten), the value of the first 7 is .
   2. Convert $110101_{\text{two}}$ to base eight. →
   3. Convert 45 to base two. →
   Show answer

   1: 7 000 · 2: $65_{\text{eight}}$ · 3: $101101_{\text{two}}$

   • It is in the thousands place.
   • Group in threes from the right: $110|101$ gives 6 and 5.
   • Remainders read upwards: 1, 0, 1, 1, 0, 1.

3. Matching

   Match each item in column A with the correct answer in column B. Write the letter next to each number.

   Column A	Column B
   1. Convert $231_{\text{five}}$ to base ten.	a. 15
   2. Convert $17_{\text{eight}}$ to base ten.	b. 66
   3. Convert $1101_{\text{two}}$ to base ten.	c. 13
   4. Work out $12_{\text{four}}\times 3_{\text{four}}$.	d. $102_{\text{four}}$

   Show answer

   1 → b (66) · 2 → a (15) · 3 → c (13) · 4 → d ($102_{\text{four}}$)

   • $2\times 25+3\times 5+1$.
   • $1\times 8+7$.
   • $8+4+0+1$.
   • $6\times 3=18=1\times 16+0\times 4+2$.

4. Drag and drop

   Choose the correct word or phrase from the box for each gap. One word in the box is not needed.

   $31_{\text{five}}$  ·  $21_{\text{five}}$  ·  $10022_{\text{three}}$  ·  7
   1. Find $x$ if $23_x=17$. →
   2. Convert $324_{\text{five}}$ to base three. →
   3. Work out $143_{\text{five}}÷3_{\text{five}}$. →
   Show answer

   1: 7 · 2: $10022_{\text{three}}$ · 3: $31_{\text{five}}$

   • $2x+3=17$.
   • Step 1. To base ten: $3\times 25+2\times 5+4=89$.
     Step 2. $89=1\times 81+0\times 27+0\times 9+2\times 3+2$, so $10022_{\text{three}}$.
   • Step 1. $143_{\text{five}}=48$.
     Step 2. $48÷3=16=31_{\text{five}}$.

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