Counting in Base 2 (Binary) - mathematics -jss1

Counting in Base 2 (Binary)

Meaning of Base 2

Base 2, also called binary, is a number system that uses only two digits: 0 and 1.

It works just like our normal counting (base 10) but after 1 we cannot write 2 (because the only digits are 0 and 1). So, when we finish writing 1 we carry over to the next place.

Place Values in Base 2

Place from right	Power of 2	Value
First (rightmost)	2⁰	1
Second	2¹	2
Third	2²	4
Fourth	2³	8
Fifth	2⁴	16
Sixth	2⁵	32

When we write a binary number like 1011₂, it means:

1 × 2³ = 8

0 × 2² = 0

1 × 2¹ = 2

1 × 2⁰ = 1

Add them up: 8 + 0 + 2 + 1 = 11 (decimal)

How to Count in Binary

We count like this:

0, 1, 10, 11, 100, 101, 110, 111, 1000, …

Notice that after 1 we write 10 (which means 2 in decimal). Whenever you cannot write the next digit (because we only have 0 and 1), you reset that place to 0 and add 1 to the next left position.

Converting Decimal to Binary — STEP METHOD

Divide the decimal number by 2.

Write down the remainder (it will be 0 or 1).

Take the quotient (the result) and divide by 2 again.

Repeat until the quotient is 0.

Read the remainders from bottom to top. That gives the binary number.

Examples — Decimal to Binary

Example 1: 2 → binary

2 ÷ 2 = 1 remainder 0
1 ÷ 2 = 0 remainder 1
Read from bottom to top → 10₂

Example 2: 3 → binary

3 ÷ 2 = 1 remainder 1
1 ÷ 2 = 0 remainder 1
Read from bottom to top → 11₂

Example 3: 4 → binary

4 ÷ 2 = 2 remainder 0
2 ÷ 2 = 1 remainder 0
1 ÷ 2 = 0 remainder 1
Read from bottom to top → 100₂

Example 4: 5 → binary

5 ÷ 2 = 2 remainder 1
2 ÷ 2 = 1 remainder 0
1 ÷ 2 = 0 remainder 1
Read from bottom to top → 101₂

Example 5: 6 → binary

6 ÷ 2 = 3 remainder 0
3 ÷ 2 = 1 remainder 1
1 ÷ 2 = 0 remainder 1
Read from bottom to top → 110₂

Example 6: 7 → binary

7 ÷ 2 = 3 remainder 1
3 ÷ 2 = 1 remainder 1
1 ÷ 2 = 0 remainder 1
Read from bottom to top → 111₂

Example 7: 8 → binary

8 ÷ 2 = 4 remainder 0
4 ÷ 2 = 2 remainder 0
2 ÷ 2 = 1 remainder 0
1 ÷ 2 = 0 remainder 1
Read from bottom to top → 1000₂

Example 8: 9 → binary

9 ÷ 2 = 4 remainder 1
4 ÷ 2 = 2 remainder 0
2 ÷ 2 = 1 remainder 0
1 ÷ 2 = 0 remainder 1
Read from bottom to top → 1001₂

Example 9: 10 → binary

10 ÷ 2 = 5 remainder 0
5 ÷ 2 = 2 remainder 1
2 ÷ 2 = 1 remainder 0
1 ÷ 2 = 0 remainder 1
Read from bottom to top → 1010₂

Example 10: 11 → binary

11 ÷ 2 = 5 remainder 1
5 ÷ 2 = 2 remainder 1
2 ÷ 2 = 1 remainder 0
1 ÷ 2 = 0 remainder 1
Read from bottom to top → 1011₂

Converting Binary to Decimal — STEP METHOD

Write the binary digits.

Under each digit write its place value (1, 2, 4, 8, …).

Multiply each digit by its place value.

Add all the results together.

Binary to Decimal

Example 1: 101₂ → decimal

Place values: 2² = 4, 2¹ = 2, 2⁰ = 1
Calculation: 1×4 + 0×2 + 1×1 = 4 + 0 + 1 = 5

Example 2: 110₂ → decimal

Calculation: 1×4 + 1×2 + 0×1 = 4 + 2 + 0 = 6

Example 3: 111₂ → decimal

Calculation: 1×4 + 1×2 + 1×1 = 4 + 2 + 1 = 7

Example 4: 1000₂ → decimal

Calculation: 1×8 + 0×4 + 0×2 + 0×1 = 8

Example 5: 1001₂ → decimal

Calculation: 1×8 + 0×4 + 0×2 + 1×1 = 8 + 0 + 0 + 1 = 9

Example 6: 1010₂ → decimal

Calculation: 1×8 + 0×4 + 1×2 + 0×1 = 8 + 0 + 2 + 0 = 10

Real-Life Uses of Base 2

Computers and phones: All information is stored in 0s and 1s.

Electronics: Circuits are either ON (1) or OFF (0).

Internet & Coding: Binary is the foundation of all computer programs.

Music & Digital Media: Sounds and pictures are stored in binary form.

CHECK OTHER RELATED TOPICS HERE

ADDITION AND SUBTRACTION

ADDITION AND SUBTRACTION OF POSITIVE AND NEGATIVE INTEGERS

ADDITION AND SUBTRACTION OF FRACTION

WORD PROBLEMS ON ADDITION AND SUBTRACTION OF FRACTIONS

MULTIPLICATIONS AND DIVISION OF FRACTIONS

SECONDARY