Grade 9 Number Base Two: Short Notes

9 ශ්‍රේණිය · 2 පරිච්ඡේදය · ද්විමය සංඛ්‍යා

80%

of Grade 9 term tests include it (32 of 40)

148

marks in past papers (46 questions)

9 min

full lesson with quick checks

🎯 What you'll learn

State the place values of a base-two number
Convert base-two numbers to base ten
Convert base-ten numbers to base two
Add and subtract base-two numbers

On or off?

Every photo, song and message on your phone is stored using only two signals: on and off, 1 and 0. With just these two digits a computer can write any number. That system is base two, or binary, and it is easier than it looks.

Place values in base two

In base ten each place is 10 times the one to its right: 1, 10, 100, 1000.In base two each place is 2 times the one to its right:… 32, 16, 8, 4, 2, 1that is 2⁵, 2⁴, 2³, 2², 2¹, 2⁰.Only the digits 0 and 1 are used. We write 1011₂ (read 'one zero one one, base two').

A base-two abacus

Each rod holds at most one bead. Reading the rods from left: 1, 0, 1, 1, 0, so the number is 10110₂. The blue numbers are the place values.

✏️ Base two to base ten

Write 10110₂ in base ten.

1Place values from the right: 1, 2, 4, 8, 16
210110₂ = 1×16 + 0×8 + 1×4 + 1×2 + 0×1
3= 16 + 4 + 2 = 22

✅ 22

⚠️ Start from the right

In the 2015 O/L paper many candidates gave the digits of a base-two number the wrong place values: using 1, 10, 100…, or starting the 1 at the left end. The rightmost digit is always worth 1. Write the place values 1, 2, 4, 8, 16… above the digits from right to left before adding.

Base ten to base two

Keep dividing by 2 and write down each remainder (0 or 1) until the quotient is 0.Then read the remainders from the bottom up.Another way: take away the largest power of 2 that fits, again and again. 37 = 32 + 4 + 1, so put 1s in the 32, 4 and 1 places.

✏️ Convert 37 to base two

Write 37 in base two.

137 ÷ 2 = 18 r 1; 18 ÷ 2 = 9 r 0
29 ÷ 2 = 4 r 1; 4 ÷ 2 = 2 r 0
32 ÷ 2 = 1 r 0; 1 ÷ 2 = 0 r 1
4Remainders bottom up: 100101₂
5Check: 32 + 4 + 1 = 37 ✓

✅ 100101₂

⚠️ Read the remainders upwards

Reading the remainders from the top gives the number backwards: 101001₂ = 41, not 37. The last remainder is the biggest place value. Always check by converting back.Also remember: a power of 2 is a 1 followed by zeros, e.g. 2⁵ = 32 = 100000₂.

Adding in base two

Only four facts:0 + 0 = 00 + 1 = 11 + 1 = 10₂ → write 0, carry 11 + 1 + 1 = 11₂ → write 1, carry 1Work column by column from the right, just like ordinary addition.

Subtracting in base two

1 − 0 = 1, 1 − 1 = 0, 0 − 0 = 0.For 0 − 1 you must borrow from the next place to the left. A borrowed 1 is worth 2 in this place (not 10!), so 0 − 1 becomes 10₂ − 1 = 1.Check by adding your answer back.

📌 Remember

• Base two uses only 0 and 1. Place values from the right: 1, 2, 4, 8, 16, 32…• To base ten: add the place values that have a 1.• To base two: divide by 2 repeatedly, read remainders bottom up.• 1 + 1 = 10₂: write 0, carry 1. A borrowed 1 is worth 2.• Always check by converting to base ten.

🧾 Quick revision sheet

Digits used in base two?	Only 0 and 1.
Place values in base two?	1, 2, 4, 8, 16, 32… (powers of 2) from the right.
1 + 1 in base two?	10₂: write 0, carry 1.
1111₂ in base ten?	8 + 4 + 2 + 1 = 15
How do you convert 37 to base two?	Divide by 2 repeatedly; read the remainders bottom up: 100101₂.
2⁴ in base two?	10000₂ (a 1 and four 0s).

Now test yourself: 5 quick checks and 2 more worked examples

The full lesson checks your answers as you go, and Gedara Sir tracks which parts you've got.

📖 Start the 9-minute lesson free ▶ Practise 5 past-paper questions