Binary Arithmetic

Addition, Subtraction, and Multiplication in Base 2

Primary Seven Mathematics — Term I, Topic 2

What You Will Learn Today

By the end of this lesson you will be able to:

1. Add binary numbers up to 5 digits with carrying
2. Subtract binary numbers up to 5 digits with borrowing
3. Multiply binary numbers using long multiplication
4. Verify answers by converting to base 10

What You Already Know From Before

From whole-numbers-03 (base conversion):

• The only digits in base 2 are 0 and 1
• Binary column values: 1, 2, 4, 8, 16, 32, …

From P5–P6 Operations (base-10 arithmetic):

• Column addition with carry
• Column subtraction with borrow

Today: same procedures — overflow at 2, not 10.

The Key Insight: Overflow at 2

In base 10, a column overflows at 10:

In base 2, a column overflows at 2:

"2" is not a valid binary digit.

The Four Binary Addition Facts

Extra case: ← write 1, carry 1

Worked Addition 1:

  ¹ ← carry
  1 1
+ 1 0
─────
1 0 1

Verify: ; ✓

Worked Addition 2:

  1 1   ← carries
  1 1 0 1 1
+ 1 0 1 1 0
───────────
1 1 0 0 0 1

Verify: ; ✓

Verification Summary For Binary Addition

✓

The Four Binary Subtraction Facts

Borrow case: → (leaves 0 in lent column)

Worked Subtraction 1:

No borrowing needed in this example:

  1 1 0 1
- 1 0 0 1
─────────
  0 1 0 0

Verify: ; ✓

Worked Subtraction 2:

Cascading borrows — trace each borrow with a slash:

  0̶ 1̶ 1̶ 0  ← after borrowing
  1  0  0  0
-  0  0  1  1
──────────────
   0  1  0  1

Verify: ; ✓

The Borrow Cascade Explained Step By Step

• Ones: → must borrow
• Twos, Fours: hold 0 → cascade further
• Eights: holds 1 → lends down the chain
• Eights → 0; Fours, Twos, Ones each briefly hold

Result: ✓

The Four Binary Multiplication Facts

Multiply by 1: result is the number itself.
Multiply by 0: result is all zeros.

Binary Long Multiplication: The Procedure

Same as base-10 long multiplication:

1. For each multiplier digit (right to left): form a partial product
2. Shift the partial product left by its column position
3. Add all partial products (binary addition)

Multiply by 1 → copy the number. Multiply by 0 → write zeros.

Worked Multiplication 1:

      1 0 1   (= 5)
  ×     1 1   (= 3)
  ───────────
      1 0 1   ← 101 × 1 (ones)
  + 1 0 1 0   ← 101 × 1 (twos, shift left 1)
  ───────────
    1 1 1 1

Verify Multiplication 1 In Base 10

The Beauty of Binary Multiplication

The partial products in binary are always either the number itself (×1) or zero (×0) — never anything else.

Worked Multiplication 2:

      1 0 1 1 0   (22)
    ×     1 0 1   (5)
    ───────────────
      1 0 1 1 0   ×1
    0 0 0 0 0 0   ×0
+ 1 0 1 1 0 0 0   ×1
  ───────────────
  1 1 0 1 1 1 0

Verify Multiplication 2 In Base 10

7-digit binary column values: 64, 32, 16, 8, 4, 2, 1

Misconception 1: Writing "2" and Carry Errors

Error 1 — writing digit "2":

Error 2 — forgetting the carry:

Misconception 2: Shift Errors and Borrow Tracking

Error 3 — not shifting partial products:
Each partial-product row must start under its multiplier digit's column.

Error 4 — lost borrow in a cascade:
Slash the column lent from; write its new value above.

Summary: Addition And Subtraction Facts

Summary: Multiplication And The Big Idea

Carry and borrow at 2, not 10. Always verify in base 10.