Question

Let e = 2.71828183

a. Find the binary scientific notation of e with five bits after the binary point; i.e. find integer n and bits x1, x2, . . . , x5 such that e = 1.x1x2x3x4x5 × 2n.
b. Find the single-precision IEEE 754 representation of e. (Hint: Round the result of 223e to the closest integer like m. Convert m to its normalized binary scientific notation and round its fraction to 23 bits after the binary point. Finally, find the normalized binary scientific notation of m/223 by subtracting 23 from the power of two in the representation of m).

Answer 1

To express given numbers in scientific notation, we rewrite the numbers to have the decimal point after the first non-zero digit and multiply by 10 raised to the power of the positions moved. For division and multiplication, we directly add or subtract the exponents in scientific notation.

Step-by-step explanation:

To express 333,999,500,000 ÷ 0.00000000003396 in scientific notation, we need to rewrite both numbers in scientific notation first. For the numerator, 333,999,500,000 is written as 3.339995 × 1011. For the denominator, 0.00000000003396 is written as 3.396 × 10-11. When we divide these numbers, we get:

(3.339995 × 1011) ÷ (3.396 × 10-11) = (3.339995 ÷ 3.396) × 1011 - (-11) = 0.98341274 × 1022.
After rounding to three significant figures and expressing it properly in scientific notation, the result is:

9.83 × 1021

For expressing 6.022 × 1023 in standard notation, simply move the decimal point 23 places to the right, which gives:

602,200,000,000,000,000,000,000

To evaluate 0.00000000552 × 0.0000000006188 in scientific notation:

5.52 × 10-9 × 6.188 × 10-10 = (5.52 × 6.188) × 10-9 + (-10) = 34.15856 × 10-19.

After rounding to three significant figures and putting it in scientific notation,

we get:3.42 × 10-18