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Binary place value and algorithm traces

Start a set of 8
Step 6 of 6

Transfer the update convention

Euler defines the slope at the old pair. Sequential code is correct when it saves that slope first, then updates y and x. The generic assignment exercise checks state tracing; it does not by itself grade numerical-method implementation

Year 12, VCE Units 3 & 4, Chapter 6, 1 difficulty level, No calculatorBuild a worksheet

Binary and precise execution

State representation and execution conventions before tracing

  1. Place value Integer weights are powers of two from the right. Fractional weights are successive halves. Only 0 and 1 are binary digits
  2. Conversion Sum weighted bits to read a numeral. Repeated division by two yields bits in reverse order; handle zero explicitly
  3. Control flow Assignment is sequential. IF chooses a branch; FOR visits its inclusive range; WHILE tests before each body execution
  4. Correctness and termination Trace to find errors, explain what each update preserves, and justify that a finite loop ends. Passing sample tests alone is not a general proof

Worked example

Convert decimal 43 to binary, recording each place and checking the result

\[43=32+8+2+1\]
  1. \(\displaystyle 43-32=11\)The weight-32 digit is 1
  2. \(\displaystyle 11<16,\quad11-8=3\)The weight-16 digit is 0 and the weight-8 digit is 1
  3. \(\displaystyle 3<4,\quad3-2=1,\quad1-1=0\)The remaining digits at weights 4, 2 and 1 are 0, 1 and 1
  4. \(\displaystyle (101011)_2=32+8+2+1=43\)Keep the zero places and check the weighted sum

Common mistakes

Removing internal zeroes
✕\(\displaystyle 1001_2=11_2\)✓\(\displaystyle 1001_2=9,\quad11_2=3\)
Internal zeroes preserve place weights; leading zeroes alone may be omitted
Using old values
✕all assignments happen simultaneously✓execute one statement, then the next
Later right sides read the updated state
One extra loop
✕continue when total equals a strict upper threshold✓test the exact guard before the next iteration
For total < L, equality stops the loop

Difficulty levels

standardTrace each assignment using the current stored values

Questions about this topic

Does a WHILE loop always run once?+

No. Its condition may already be false at entry

Is independent coding automatically assessed?+

No. Generated activity checks supplied calculations and traces. Your own pseudocode, implementation and correctness arguments have ungraded tasks and models

Can I use a calculator?+

No. This topic is practised without a calculator

Which year level and chapter is this?+

Year 12, VCE Units 3 & 4, Specialist Mathematics, chapter 6: Differential equations and rectilinear motion

In this chapter

6A Models, verification and direction fields: Form differential models, verify solutions and read direction fieldsExponential growth and decay rate modelsSeparate variables, impose initial values and retain solution domainsAnti-differentiate a power of a linear functionDifferentiate parametric curves and integrate vector motionSolve and interpret continuous logistic models6C Logistic growth and applications: Form differential models, verify solutions and read direction fieldsApproximate solutions with explicit Euler stepsBinary place value and algorithm tracesUse acceleration forms and differential equations in rectilinear motionAnalyse straight-line motion through displacement, velocity and acceleration