Using a number lineWhen we talk about bigger and also smaller numbers which can be an unfavorable or positive, the is helpful to describe a number heat to assist us protect against confusion about exactly what we are talking about.

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As optimistic numbers obtain bigger, the number gets bigger. The smallest optimistic number is the one the is closest to zero however still positive.As an unfavorable numbers get bigger, the number actually gets smaller (further far from zero). The smallest an unfavorable number is the number that is closest to zero yet still negative.

The largest and also smallest valuesImagine you have a byte provided to organize a floating-point number. In ours byte, we will enable 5 bits for the mantissa and 3 bits because that the exponent. Remember, both the mantissa and also the exponent are always held together 2’s complement numbers.

The largest positive number

The biggest number that can be held in our byte is as soon as both the mantissa and the exponent room the largest positive number that deserve to be held.The biggest mantissa that have the right to be held is 01111. (Note the you should assume the mantissa is normalised, and also so positive numbers constantly begin through 01).The biggest exponent that can be hosted is 011. (Remember, this is a 2s enhance number so the left most little bit is negative, and also in this situation represents -4).The largest number that have the right to be held is 0.1111 x 2011This is the same as 111.1, or 7.5 in decimal.

The smallest confident number

The smallest positive number that have the right to be organized is when the mantissa is the smallest hopeful number and the exponent is the largest an unfavorable number.The smallest positive (normalised) mantissa is 01000.The largest an unfavorable exponent is 100The smallest hopeful number that can be hosted is therefore 0.1000 x 2100This is the very same as 0.00001, or 1/32, or 0.03125

The largest negative number (or the the smallest number, depending upon how you view it!!)

The largest an unfavorable number that can be held (the an unfavorable number the is furthest indigenous zero) is once the mantissa is the biggest an adverse number that can be held and also the exponent is the biggest positive number possible.The biggest an adverse number you have the right to hold in the mantissa is 10000 (Remember that all negative numbers must begin with 10).The best positive number the exponent can hold is 011.The largest an adverse number is because of this 1.0000 x 2011This is the very same as -(1.0000) x 23 or -1000, or -8.

The smallest an unfavorable number (or the biggest an adverse number, depending on how you watch it!!)

The smallest an adverse number that can be hosted (the an unfavorable number that is closest to zero) is as soon as the mantissa holds the smallest an adverse number and also the exponent stop the largest an adverse number.The smallest normalised negative number the mantissa can hold is 10111 (Remember that the mantissa must begin with 10 for normalised an adverse numbers).The largest an unfavorable number the exponent can hold is 100.The smallest an unfavorable number that this byte have the right to hold as such is 1.0111 x 2100This is the exact same as -0.1001 x 2-4 or -0.00001001, or -0.03515625

To summariseYou have actually a fixed variety of bits. You have to decide how many bits to use for the mantissa and also the exponent.

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The an ext bits you enable for the exponent, the better the selection of number you deserve to represent - the down side being that the accuracy that the mantissa is reduced.The fewer bits you allow for the exponent, the smaller the variety of number you deserve to represent, yet the great news is you have an ext bits obtainable for the mantissa - so the number you put in it deserve to be more accurate.

How execute you decide? You need to look at every situation and decide what range of number you have to keep and to what degree of accuracy!