<aside> 💡 Will take precedence over longs, and ints.

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$$ (-1)^sM 2^e $$

Guide: Converting Denary to Float

<aside> 💡 Guide: Converting to Floating Point Numbers

Denary to Float: 4.75 (8bit representation)

1. Find the 2s complement representation of 4 Keep dividing 4 by 2 and keep track of the remainder

0100 - 4

1. Find the 2s complement representation of 75 (keep multiplying by 2 and turn into 1 when it is 1 or greater

0011

1. Add the exponent Part

   $100.0011*2^0$

2. Normalize the Mantissa $1.000011*2^2$

3. Calculate the Exponent with $bias = 2^{k-1}-1$$E = exponent+bias$ $E = 2+3 = 5$ in binary that is 101

4. Represent Exponent 101

5. Represent Mantissa we take the 0011 (rest of the 1 area)

6. Represent Sign Bit negative so -1

Reversing the Floating Point Encoding - doing the backwards operation

Exponent Bits = 3, thus the bias is 3

E = exp + bias 5 = exp + 3 = EXP = 2

1.0011 * 2^2 100.11 = 2^0 4.75

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Normalized Numbers

• When the exponent values ≠ 00.00 and ≠ 11..111
• $bias =2^{k-1}-1$
• Smallest Normalized Number in 8 bit is 1 0001 000
• Biggest Normalized Number in 8 bit is 0 1110 111

Denormalized Values

• when the exponent = 0000 and the frac = 0.00
• when the exponent = 0.000 and the frac ≠ 0.00
• $E = 1-2^{k-1}-1$ (bias for denormalized)

Rounding

• Round to nearest WHOLE even number, regardless of whether it follows regular convention or not

Addition/Multiplication