Math Is Fun Forum

mathxyz

Member

From: Brooklyn, NY

Registered: 2024-02-24

Posts: 1,053

Number "Closest" to Zero

Is there a positive real number "closest" to zero?

Bob

Administrator

Registered: 2010-06-20

Posts: 10,619

Re: Number "Closest" to Zero

No.

Suppose n was the number. Let's say its decimal form is 0.00000...x...... Let x be the first non zero digit. There must be such a digit or else the number is zero.  And say the next digits are y and z.

ie. n = 0.00000...xyz...... where every digit before x is zero.  Change x to zero too. New number is 0.00000.....0yz....... so it's smaller. So the number we thought we'd found isn't the smallest.

Bob

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Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

mathxyz

Member

From: Brooklyn, NY

Registered: 2024-02-24

Posts: 1,053

Re: Number "Closest" to Zero

No.

Suppose n was the number. Let's say its decimal form is 0.00000...x...... Let x be the first non zero digit. There must be such a digit or else the number is zero.  And say the next digits are y and z.

ie. n = 0.00000...xyz...... where every digit before x is zero.  Change x to zero too. New number is 0.00000.....0yz....... so it's smaller. So the number we thought we'd found isn't the smallest.

Bob

Another great reply for my online math files.

Europe2048

Member

Registered: 2024-01-03

Posts: 38

Re: Number "Closest" to Zero

No.

Suppose n was the number. Let's say its decimal form is 0.00000...x...... Let x be the first non zero digit. There must be such a digit or else the number is zero.  And say the next digits are y and z.

ie. n = 0.00000...xyz...... where every digit before x is zero.  Change x to zero too. New number is 0.00000.....0yz....... so it's smaller. So the number we thought we'd found isn't the smallest.

Bob

Sure, this does work if the number is non-terminating, but what about a situation like this:
Let's suppose n is in the form x × 10^-p, where x and p are counting numbers. For example, let's suppose n = 125 × 10^-6 = 0.000125.
Let's make a sequence S where S₁ = n, and S_{z+1} is S_z but with the first non-zero digit replaced with 0.
Then, the sequence will go like this:

S_1 = 0.000125
S_2 = 0.000025
S_3 = 0.000005
S_4 = 0.000000 (not positive)

So we can see that this doesn't work all the time.

Let's make another sequence T, where T₁ = n, and T_{z+1} is T_z / 2. Here's how it goes now:

T_1 = 0.000 125
T_2 = 0.000 0625
T_3 = 0.000 03125
T_4 = 0.000 015625
...

Let's try it with a non-terminating number:

T_1 = 0.000 333333...
T_2 = 0.000 166666...
T_3 = 0.000 083333...
T_4 = 0.000 041666...
...

So halving does work all the time.

mathxyz

Member

From: Brooklyn, NY

Registered: 2024-02-24

Posts: 1,053

Re: Number "Closest" to Zero

No.

Suppose n was the number. Let's say its decimal form is 0.00000...x...... Let x be the first non zero digit. There must be such a digit or else the number is zero.  And say the next digits are y and z.

ie. n = 0.00000...xyz...... where every digit before x is zero.  Change x to zero too. New number is 0.00000.....0yz....... so it's smaller. So the number we thought we'd found isn't the smallest.

Bob

Sure, this does work if the number is non-terminating, but what about a situation like this:
Let's suppose n is in the form x × 10^-p, where x and p are counting numbers. For example, let's suppose n = 125 × 10^-6 = 0.000125.
Let's make a sequence S where S₁ = n, and S_{z+1} is S_z but with the first non-zero digit replaced with 0.
Then, the sequence will go like this:

S_1 = 0.000125
S_2 = 0.000025
S_3 = 0.000005
S_4 = 0.000000 (not positive)

So we can see that this doesn't work all the time.

Let's make another sequence T, where T₁ = n, and T_{z+1} is T_z / 2. Here's how it goes now:

T_1 = 0.000 125
T_2 = 0.000 0625
T_3 = 0.000 03125
T_4 = 0.000 015625
...

Let's try it with a non-terminating number:

T_1 = 0.000 333333...
T_2 = 0.000 166666...
T_3 = 0.000 083333...
T_4 = 0.000 041666...
...

So halving does work all the time.

Interesting reply.

Keep_Relentless

Member

From: Queensland, Australia

Registered: 2024-05-05

Posts: 61

Re: Number "Closest" to Zero

Suppose we guess 0.001 (with three zeros) is the smallest number. We can find a smaller one: 0.0001 (with four zeros). And yet a smaller one: 0.00001 (with five zeros). This process can continue forever; there is no limit to the number of zeros we can add. So there is no smallest positive number.

My explanation is not as fancy as the others, but it works for me.

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"The most incomprehensible thing about the world is that it is comprehensible." -Albert Einstein.

mathxyz

Member

From: Brooklyn, NY

Registered: 2024-02-24

Posts: 1,053

Re: Number "Closest" to Zero

Suppose we guess 0.001 (with three zeros) is the smallest number. We can find a smaller one: 0.0001 (with four zeros). And yet a smaller one: 0.00001 (with five zeros). This process can continue forever; there is no limit to the number of zeros we can add. So there is no smallest positive number.

My explanation is not as fancy as the others, but it works for me.

Your explanation works for me as well. Thanks.