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Yes, I do believe that those are the same things. I have a long-lasting history in math and when my students ask me this question, I am always positive that, in math, these two terms can mean the same thing in different instances. For example, if a number is not a real number, it can be undefined as well (and vice versa). If someone came up to me and asked me this exact question, I would say that the two terms mean the same thing.

Q: Is undefined the same thing as not a real number?

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The binary number 10101010 is equivalent, in decimal representation, to 128 + 32 + 8 + 2 = 170. But this answer assumes that the given number is binary - an assumption for which there is no real justification. Besides, the relationship is an equivalence, which is not quite "the same thing".

It can be 82/1 or 164/2 or 246/3 or 328/5 and so on............Just multiply the numerator and denominator by the same number [any real number except zero(because it will become undefined{infinity} when a number other than zero is divided by zero).]

In the real numbers, the logarithm is only defined for positive numbers. The logarithm of zero or a negative number is undefined. (For calculators who work with complex number, only the logarithm of zero is undefined.) This follows from the definition of the logarithm, as the solution of: 10x = whatever "Whatever" is the number of which you want to calculate the logarithm. Since 10x is always positive, that means you can't find an "x" such that the power results in a negative number, or in zero. The same applies if you use a base other than 10, for example the number e = 2.718...

No. Natural numbers are a very small subset of real numbers.

Mixed numbers and mixed fractions are the same thing.

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Any two points on a vertical line. For example, the line x=7 is a vertical line with undefined slope. Every point (7,y) where y is any real number, lies on the same line.

The binary number 10101010 is equivalent, in decimal representation, to 128 + 32 + 8 + 2 = 170. But this answer assumes that the given number is binary - an assumption for which there is no real justification. Besides, the relationship is an equivalence, which is not quite "the same thing".

A line is undefined because all points on the line have the same x-coordinate.

As long as the number is not zero, the quotient remains unchanged. If the multiplier is zero then the quotient is undefined.

To divide a number by zero means the number will be unchanged, same as dividing by 1. Try dividing by zero on a calculator and you will get an undefined error message.

It can be 82/1 or 164/2 or 246/3 or 328/5 and so on............Just multiply the numerator and denominator by the same number [any real number except zero(because it will become undefined{infinity} when a number other than zero is divided by zero).]

No. It is the same thing as real paint.

Since the imaginary portion of a real number is zero, the complex conjugate of a real number is the same number.

Yes they mean the same thing.

In the real numbers, the logarithm is only defined for positive numbers. The logarithm of zero or a negative number is undefined. (For calculators who work with complex number, only the logarithm of zero is undefined.) This follows from the definition of the logarithm, as the solution of: 10x = whatever "Whatever" is the number of which you want to calculate the logarithm. Since 10x is always positive, that means you can't find an "x" such that the power results in a negative number, or in zero. The same applies if you use a base other than 10, for example the number e = 2.718...

yes.

An equation with an undefined slope is typically in the form x = a, where 'a' is a constant number. This indicates a vertical line on the coordinate plane, where every point on the line has the same x-coordinate and no defined slope because the line is perfectly vertical.