Two directions, one value
Binary → DecimalTurn each 1-bit into its power of two, then add.
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Decimal → BinaryFind the required powers of two, then write 1 or 0 in each place.
11010112 = 10710 = 11010112
Chapter 26 · Decimal and Binary Numeration
Convert Between Decimal and Binary
Keep the value fixed while changing its numeral. Read binary by adding powers of two, and write binary by finding which powers of two build the decimal value.
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Meaning before method
Keep the value fixed while the numeral changes
decimal numeral45₁₀
=
binary numeral101101₂
Conversion changes the writing, not the amount. Five objects remain five objects whether the label is 510 or 1012.
Direction checkpoint.
Need a hint?
Follow the arrow: identify the starting and ending base.
original foundation
Convert binary to decimal by place value
Count places from the right, beginning with exponent zero. A 1-bit contributes its power of two; a 0-bit contributes zero.
Legal binary numeral.
Rebuild the two original conversions.
Need a hint?
Write a power of two under every digit, starting with 1 on the right.
Worked example 5 and Practice 5
Switch on the powers inside 1101011₂
Click a bit to explore nearby numerals. The checkpoint below always asks about the printed numerals 1101011₂ and 1100100₂, even if you change the switches. Restore the first numeral when you want its diagram.
Complete expansion
1×26 + 1×25 + 0×24 + 1×23 + 0×22 + 1×21 + 1×20
Shortened expansion
26 + 25 + 23 + 21 + 1
Worked example 5 and Guided Practice 5.
Need a hint?
The leftmost bit is in the 64 place.
General binary reader
Normalize, expand, and check a binary numeral
normalized1010101₂
bits7
active places4
decimal value85
entered1 010 101₂
→
decimal85₁₀
→
converted back1010101₂
conversion exercises.
Need a hint?
Spaces group digits; they do not change the place count.
Added visual scaffold
Build a decimal value from the greatest powers of two
Show a hint
The greatest power that fits is 32. Subtract it and continue with what remains.
Choose the greatest power of two that fits, subtract it, and continue. The selected powers become 1-bits.
Greedy subtraction ledger
Bit result
binary numeral100101₂
Round-trip: 100101₂ = 37₁₀
Use selected powers to write binary.
Need a hint?
Choose the greatest power of two that fits, subtract, and repeat.
original repeated-division method
Divide by two and read the remainders upward
Show a hint
13=2×6+1, then 6=2×3+0. Continue until the quotient is zero.
Read this column from bottom to top
Why the first remainder is the ones bit: N = 2q + r separates pairs from the leftover unit. The first remainder gives the ones bit; dividing q next finds the twos bit, then the fours bit. Common error: the first remainder is the ones bit, so reading the list downward reverses the numeral.
Worked example 6.
Need a hint?
Each remainder is the next lowest bit because N=2q+r.
Two methods, one answer
Compare powers-of-two selection with repeated division
Method A · selected powers
Method B · division remainders
Round-trip proof: both routes must produce the same binary numeral, and converting that numeral back must recover the starting decimal value.
Guided practices and original exercises.
Need a hint?
Convert your answer back to decimal to check it.
Test 26 conversion transfer
Audit both directions without mixing them up
Questions 6 and 7 of the chapter test isolate the two conversion directions. Enter only the numeral requested; base labels are already shown.
Grouping spaces
101 010₂ and 101010₂ are the same numeral. Spaces are only reading separators.
Leading zeros
0011100₂ and 11100₂ have the same value. Normalizing removes grouping spaces and unnecessary leading zeros. A numeral consisting only of zeros becomes the single digit 0. Significant bits are the digits left after removing leading zeros; zero uses one bit.
Need a hint?
The first two answers are decimal; the last two are binary.
Independent practice
Conversion workshop
Correct all eight questions to complete the workshop.
Need a hint?
Answers: 53; 1001001₂; 65; 11111111₂; 10000000000₂; 73; 10010001₂; 145. Check 53=32+16+4+1, 73=64+8+1, and 145=128+16+1. The all-ones eight-bit numeral is 256−1=255.
Mastery check
Exit ticket and conversion certificate
Need a hint?
Answers: 100; 11011001₂; bottom to top; 1111101001₂; Yes. Check 217=128+64+16+8+1 and 1001=512+256+128+64+32+8+1.
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Decimal–Binary Conversion Navigator
This certificate recognizes the learner for reading binary place values, using powers of two, applying repeated division, and verifying conversions by round trip.
Completed