Sign, Exponent and Fraction

A single 64-bit float can hold the mass of an electron (about 9 × 10⁻³¹ kg) and the mass of the Sun (about 2 × 10³⁰ kg). No fixed position for the point could cover both. So the point is allowed to float: the number stores its digits and, separately, where the point goes. You already know this idea as scientific notation.

The rule

A float is sign × 1.fraction × 2^exponent — scientific notation in binary.

In decimal, 6,500 is written 6.5 × 10³. A float does the same in base 2. Take 6.5: in binary it is 110.1 (4 + 2 + ½). Slide the point left until one digit is in front of it:

The standard layout (IEEE 754) stores three fields:

Two details keep it compact. The leading 1 before the point is always 1 for normal numbers, so it is not stored; a double effectively keeps 53 significant bits. And the exponent is stored with a bias added (1023 for doubles) so negative exponents need no sign: exponent 2 is stored as 1025.

53 significant bits are about 15–17 decimal digits; a 32-bit float's 24 bits are about 7 digits. Because the number of significant bits is fixed, the gap between neighbouring floats grows as numbers get bigger. Above 2⁵³ (about 9 × 10¹⁵) a double cannot even store every whole number.

The shortcut

To place any number, write it in binary, then slide the point until a single 1 is in front. The number of places you slid is the exponent: slid left means positive, slid right means negative.

Drills

Answer fast — trust the rule.

Remember

• A float is sign × 1.fraction × 2^exponent.
• Doubles: 1 + 11 + 52 bits, about 15–17 digits. Floats: 1 + 8 + 23 bits, about 7 digits.
• The trap: precision is fixed in digits, not in size — big numbers have big gaps between them.