Read Place Value and Count in Binary
Discover why 102 means two, build values from powers of two, and watch a binary odometer carry when a place reaches two.
Big idea: a numeral is a place-value code
For example, 23 means two tens and three ones: 2 × 10 + 3. In binary, 101 means one four, no twos, and one one. The same value can be written with different numeral systems. Decimal uses powers of ten; binary uses powers of two.
Every digit must be smaller than the base: in base b, 0 ≤ digit < b. The rightmost place has exponent zero and value 1; each step left multiplies the place value by the base. The letters in this optional general formula name the digits and their positions. The worked example develops the same idea through decimal and binary examples.
Two systems, one value
Read the subscript. It tells you which place values the digits use.
Write one value in two systems
A value is an amount. A numeral is a written code for that amount. Move the slider and compare the decimal and binary numerals.
5×100
1×22+0×21+1×20
Need a hint?
Read the base first: the rightmost place is worth 1 in both systems.
Rebuild ordinary place value with powers of ten
Need a hint?
Count decimal places from the right: 1, 10, 100, 1000, … .
Use only 0 and 1 to select powers of two
Binary has only two digits. A bit of 1 switches its power of two on; a bit of 0 switches it off.
Need a hint?
Match the bits to 32, 16, 8, 4, 2, 1; add only the places with a 1.
Count by carrying whenever a place reaches two
| Decimal value | Binary numeral | What changed? |
|---|
Need a hint?
Add from the right. A trailing 1 becomes 0 and sends a carry left.
Expand every binary place, including zeros
Need a hint?
Keep every position, even those with a zero.
Remove leading zeros, but protect internal zeros
Both equal decimal 5.
They equal decimal 5 and 3.
The digit 2 cannot appear in a binary numeral.
Need a hint?
Only zeros before the first 1 can be removed without moving another bit.
Match decimal values to their binary numerals
The worked example lists the opening binary count. Read across each row: the value is unchanged, but the written numeral changes.
| Decimal | Binary | Powers of two used |
|---|
8 + 4 + 1
Need a hint?
Build each value with the largest power of two that fits.
Audit the first five original-test skills
These structured checks mirror the chapter test’s opening foundations without requiring you to type long algebraic expressions.
Need a hint?
The exponent starts at zero on the right. Each step left raises it by one.
Place-value and binary-counting workshop
Correct all eight questions to complete the workshop.
Need a hint?
Answers: 50,000; 2; 1100₂; 6; 16; 512; 10102₂; 1000₂. Moving left doubles a binary place value. Add one to 1011₂ by carrying through its two trailing 1s.
Exit ticket and certificate
All five answers must be correct. The certificate appears after every mission is complete.
Need a hint?
Answers: 0 and 1; 45; 1000₂; Yes; 2. For 101101₂, add 32+8+4+1. Leading zeros contribute nothing; a place containing two units sends one carry to the left.
Binary Place-Value Navigator
This certificate recognizes the learner for reading decimal and binary place value, selecting powers of two, and counting with binary carries.
Completed