Even though quite a few people have already contributed towards answering your question, I would like to offer some thoughts of mine that may help you.

I am not going to give you ways of showing that complex numbers are necessary and meaningful, rather an idea about why I think people find them meaningless and how this can be resolved.

I am prompted by the part of your question that says: "...most non-mathematicians find it hard to accept that such a number is meaningful. In contrast, they feel that real numbers have an obvious and intuitive meaning." I have thought about this a lot in the past and have come to the conclusion that the problem of finding complex numbers meaningful lies in the meaning we have attached to the number systems "preceding" them in the hierachical chain: $\mathbb{N}, \mathbb{Z}, \mathbb{Q}, \mathbb{R}$. The meaning we have attached is that of quantity! Numbers all the way up to and including the reals are scalar quantities: We use them for distance, area, volume, weight, speed, intensity and so on. However, when you get to the complex numbers all that has to go away. There is no such thing as $2 + 3i$ kilogrammes or $-4i$ dollars or $1-5i$ centimetres... Yet we ask the learner to call them numbers!

So, the learner is faced with an apparent contradiction in terms: They are asked to think of these strange entities as numbers while at the same time these new additions to the realm don't behave as the numbers of old. And you should take into account that the older numbers have been around for a long time. That's where I think the problem lies.

To circumvent this, I think that what one needs to do is explain to the learner that - from now on - numbers are going to take on a much wider role than before: They are going to be used not only to denote quantities, but also to denote directions, which is precisely what complex numbers do on the two-dimensional plane. Every complex number can be represented by a vector and this automatically suggests a direction. The "older" meaning is not lost, since a complex number has a modulus in addition to its real and imaginary parts and these three are quantities.

Of course, one might say "But real numbers denote directions as well: +1 denotes a unit displacement to the right of 0 along the x-axis while -1 denotes a displacement to the left". This is indeed true, however, I sincerely doubt that the untrained mind of a learner, who encounters complex numbers for the first time, will delve in that direction. And, if it does, then that's fine, because it shows that the direction concept was already lurking in the number hierarchy ever since negative integers were introduced!!

I hope all this is of help to you.

Even though quite a few people have already contributed towards answering your question, I would like to offer some thoughts of mine that may help you.

I am not going to give you ways of showing that complex numbers are necessary and meaningful, rather an idea about why I think people find them meaningless and how this can be resolved.

I am prompted by the part of your question that says: "...most non-mathematicians find it hard to accept that such a number is meaningful. In contrast, they feel that real numbers have an obvious and intuitive meaning." I have thought about this a lot in the past and have come to the conclusion that the problem in finding complex numbers meaningful lies in the meaning we have attached to the number systems "preceding" them in the hierachical chain: $\mathbb{N}, \mathbb{Z}, \mathbb{Q}, \mathbb{R}$. The meaning we have attached is that of quantity! Numbers all the way up to and including the reals are scalar quantities: We use them for distance, area, volume, weight, speed, intensity and so on. However, when you get to the complex numbers all that has to go away. There is no such thing as $2 + 3i$ kilogrammes or $-4i$ dollars or $1-5i$ centimetres... Yet we ask the learner to call them numbers!

So, the learner is faced with an apparent contradiction in terms: They are asked to think of these strange entities as numbers while at the same time these new additions to the realm don't behave as the numbers of old. And you should take into account that the older numbers have been around for a long time. That's where I think the problem lies.

To circumvent this, I think that what one needs to do is explain to the learner that - from now on - numbers are going to take on a much wider role than before: They are going to be used not only to denote quantities, but also to denote directions, which is precisely what complex numbers do on the two-dimensional plane. Every complex number can be represented by a vector and this automatically suggests a direction. The "older" meaning is not lost, since a complex number has a modulus in addition to its real and imaginary parts and these three are quantities.

Of course, one might say "But real numbers denote directions as well: +1 denotes a unit displacement to the right of 0 along the x-axis while -1 denotes a displacement to the left". This is indeed true, however, I sincerely doubt that the untrained mind of a learner, who encounters complex numbers for the first time, will delve in that direction. And, if it does, then that's fine, because it shows that the direction concept was already lurking in the number hierarchy ever since negative integers were introduced!!

I hope all this is of help to you.

Even though quite a few people have already contributed towards answering your question, I would like to offer some thoughts of mine that may help you.

I am not going to give you ways of showing that complex numbers are necessary and meaningful, rather an idea about why I think people find them meaningless and how this can be resolved.

I am prompted by the part of your question that says: "...most non-mathematicians find it hard to accept that such a number is meaningful. In contrast, they feel that real numbers have an obvious and intuitive meaning." I have thought about this a lot in the past and have come to the conclusion that the problem of finding complex numbers meaningful lies in the meaning we have attached to the number systems "preceding" them in the hierachical chain: $\mathbb{N}, \mathbb{Z}, \mathbb{Q}, \mathbb{R}$. The meaning we have attached is that of quantity! Numbers all the way up to and including the reals are scalar quantities: We use them for distance, area, volume, weight, speed, intensity and so on. However, when you get to the complex numbers all that has to go away. There is no such thing as $2 + 3i$ kilogrammes or $-4i$ dollars or $1-5i$ centimetres... Yet we ask the learner to call them numbers!

So, the learner is faced with an apparent contradiction in terms: They are asked to think of these strange entities as numbers while at the same time these new additions to the realm don't behave as the numbers of old. And you should take into account that the older numbers have been around for a long time. That's where I think the problem lies.

To circumvent this, I think that what one needs to do is explain to the learner that - from now on - numbers are going to take on a much wider role than before: They are going to be used not only to denote quantities, but also to denote directions, which is precisely what complex numbers do on the two-dimensional plane. Every complex number can be represented by a vector and this automatically suggests a direction. The "older" meaning is not lost, since a complex number has a modulus in addition to its real and imaginary parts and these three are quantities.

Of course, one might say "But real numbers denote directions as well: +1 denotes a unit displacement to the right of 0 along the x-axis while -1 denotes a displacement to the left". This is indeed true, however, I sincerely doubt that the untrained mind of a learner, who encounters complex numbers for the firs time, will delve in that direction. And, if it does, then that's fine, because it shows that the direction concept was already lurking in the number hierarchy ever since negative integers were introduced!!

I hope all this is of help to you.

Even though quite a few people have already contributed towards answering your question, I would like to offer some thoughts of mine that may help you.

I am not going to give you ways of showing that complex numbers are necessary and meaningful, rather an idea about why I think people find them meaningless and how this can be resolved.

I am prompted by the part of your question that says: "...most non-mathematicians find it hard to accept that such a number is meaningful. In contrast, they feel that real numbers have an obvious and intuitive meaning." I have thought about this a lot in the past and have come to the conclusion that the problem of finding complex numbers meaningful lies in the meaning we have attached to the number systems "preceding" them in the hierachical chain: $\mathbb{N}, \mathbb{Z}, \mathbb{Q}, \mathbb{R}$. The meaning we have attached is that of quantity! Numbers all the way up to and including the reals are scalar quantities: We use them for distance, area, volume, weight, speed, intensity and so on. However, when you get to the complex numbers all that has to go away. There is no such thing as $2 + 3i$ kilogrammes or $-4i$ dollars or $1-5i$ centimetres... Yet we ask the learner to call them numbers!

So, the learner is faced with an apparent contradiction in terms: They are asked to think of these strange entities as numbers while at the same time these new additions to the realm don't behave as the numbers of old. And you should take into account that the older numbers have been around for a long time. That's where I think the problem lies.

To circumvent this, I think that what one needs to do is explain to the learner that - from now on - numbers are going to take on a much wider role than before: They are going to be used not only to denote quantities, but also to denote directions, which is precisely what complex numbers do on the two-dimensional plane. Every complex number can be represented by a vector and this automatically suggests a direction. The "older" meaning is not lost, since a complex number has a modulus in addition to its real and imaginary parts and these three are quantities.

Of course, one might say "But real numbers denote directions as well: +1 denotes a unit displacement to the right of 0 along the x-axis while -1 denotes a displacement to the left". This is indeed true, however, I sincerely doubt that the untrained mind of a learner, who encounters complex numbers for the first time, will delve in that direction. And, if it does, then that's fine, because it shows that the direction concept was already lurking in the number hierarchy ever since negative integers were introduced!!

I hope all this is of help to you.