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Chapter 26 · Decimal and Binary Numeration

Lesson 26.4 — Multiply, Divide, and Evaluate Binary Expressions

Build multiplication from shifted 0-or-1 partial products, choose quotient bits by comparing and subtracting, and keep the ordinary order of operations when several binary operations appear together.

10 interactive missionsBinary multiplication and divisionQuotient and remainderOffline core lesson
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Two ideas power the whole lesson

0 × N = 0,   1 × N = N,   N × 102 = N shifted left one place
D = d × q + r,   0 ≤ r < d

A multiplier bit decides whether a shifted copy is included. A quotient bit decides whether the divisor fits in the current partial dividend.

1
Multiplication building blocks

Let one bit decide whether a shifted copy appears

Not complete
N × 0₂ = 0₂the row contributes nothing
N × 1₂ = Ncopy the multiplicand
N × 10₂ = N0₂shift left one binary place

Shifted partial-product machine

What the shift means

Moving a binary numeral left by one place multiplies its value by:

102=210

Moving it left by k places multiplies by 2k. A multiplier bit of 0 turns the whole shifted row into zero.

Partial-product checkpoint.

Write the three products in binary. The printed questions stay fixed when you change the explorer.

Need a hint?

Each left shift multiplies the value by 2.

2
original foundation multiplication

Build 1101₂ × 101₂ from shifted rows

Not complete

Show a hint

That 1 is in the fours place, so shift 1101 two places left.


          
multiplicand13
multiplier5
decimal product65
binary product1000001₂

        

        

          
Inspect one multiplier bit

          

          

          

          

        

      

      

      

        Rebuild the worked example foundation multiplication.
        

        
Need a hint?
The 1-bits of 101 select the unshifted and twice-shifted copies.

        
      

    

  


  

    
3
Worked example 3 and Guided Practice 3
Audit any binary multiplication with partial products
Not complete

    

      

      

      
A in decimal38
B in decimal5
product190
nonzero rows2

      

      

        original multiplication results.
        

        
Need a hint?
Write a partial row for each multiplier place, then add.

        
      

    

  


  

    
4
original foundation exact division
Choose quotient bits in 110110₂ ÷ 110₂
Not complete

    
Show a hint
Compare their values: five is smaller than six, so the divisor does not fit.

      
At each step: bring down one dividend bit. If the divisor fits, write quotient bit 1 and subtract it. If it does not fit, write quotient bit 0 and keep the partial remainder.

      

        

          
dividend110110₂
÷
divisor110₂
=
quotient1001₂

          
dividend54
divisor6
quotient9
remainder0

        

        

          
Inspect one brought-down bit

          

          

          

          

        

      

      

      
1101102 = 1102 × 10012 + 0

      

        Exact-division checkpoint.
        

        
Need a hint?
Bring down a bit, compare, and subtract only when the divisor fits.

        
      

    

  


  

    
5
Worked example 4 and Guided Practice 4
Use the long-division algorithm on exact quotients
Not complete

    

      

      

      

      
dividend value99
divisor value9
quotient value11
remainder value0

      

      

        original exact-division results.
        

        
Need a hint?
Multiply the quotient by the divisor to check an exact answer.

        
      

    

  


  

    
6
original foundation non-exact division
Keep both the quotient and the remainder
Not complete

    

      

        

          
dividend101101₂
÷
divisor111₂
=
quotient110₂

          
1011012 = 1112 × 1102 + 112

          
45dividend
7divisor
6quotient
3remainder

        

        

          
Remainder safety check

          
0 ≤ 112 < 1112

          
The remainder is smaller than the divisor. If it were not, one more copy of the divisor would still fit, so the quotient would be too small.

          

          

        

      

      

      

        Quotient-and-remainder checkpoint.
Write both quotient and remainder in binary, in separate fields.

        

        
Need a hint?
The remainder is nonnegative and smaller than the divisor.

        
      

    

  


  

    
7
General division laboratory
Audit exact and non-exact binary divisions
Not complete

    

      

      

      

      

      

      

        original exercise audit.
Write all three answers in binary. The third division is exact and asks for its quotient only.

        

        
Need a hint?
Check dividend = divisor × quotient + remainder.

        
      

    

  


  

    
8
Mixed binary expressions
Keep the ordinary order of operations
Not complete

    

      

        
1 · ParenthesesComplete the innermost grouped work first.

        
2 · × and ÷Before + and −; equal priority goes left to right.

        
3 · + and −Finish equal-priority work from left to right.

      

      

      
Expression
decimal value
binary result
number of stages
first action
One legal evaluation path

      

        Mixed-expression checkpoint.
Write the first three results in binary. These printed expressions are independent of the selected explorer example.

        

        
Need a hint?
Inside the parentheses, division still comes before addition.

        
      

    

  


  

    
9
Independent workshop
Choose partial products, quotient bits, and operation order
Not complete

    

      

        
Write every answer in this workshop in binary. Correct all eight to complete it.

        

        

        

        

        

        

        

      

      
Need a hint?
Answers: 10110₂; 1000001₂; 10111110₂; 101101₂; 110₂; 11₂; 1000100110₂; 111₂. Decimal checks: 11×2=22, 13×5=65, 38×5=190, 225÷5=45, 45=7×6+3, 50×11=550, and (7+105÷5)÷4=7.

      
    

  


  

    
10
Objective exit ticket
Show that you can multiply, divide, and verify
Not complete

    
Write every answer below in binary. A division with no remainder field asks for its exact quotient.

      

        

        

        

        

        

      

      
Need a hint?
Answers: 1110101₂; 1001₂; 0; 101101₂; 1011₂. Verify 13×9=117, 117÷13=9, 225÷5=45, and 550÷50=11. Every exact division has remainder zero.

      
      

        
× ÷ ₂

        
Chapter 26 skill certificate

        
Binary Product, Quotient & Expression Navigator

        
This certifies that Binary Learner can build shifted partial products, find quotient and remainder, verify D=dq+r, and evaluate mixed binary expressions in the correct order.

        
Completed