1
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Known by 2 names, argued by ways
One of them sounds irregular, sometimes that pays
My legs are like a human's, usually alike
This is easy, known by a child's psyche

Name the mathematical concept

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1
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It's a

Trapezoid / Trapezium

Since

People debate over its name and it has two equal legs.

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  • $\begingroup$ Correct answer! $\endgroup$ – Xavier Stanton Aug 18 '18 at 15:19
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Integral?

Reasoning:

Integrals can be definite or they can be indefinite (2 names). 'Indefinite' also means uncertain, which suggests irregularity. I'm guessing that the 'pays' part of line 2 is there for rhyming purposes. Integrals can be approximated by the sum of the area of minute rectangles of the same width (the 'legs' are the same size). Not sure if children know how to do integrals, however, but knowledge of integrals is a prerequisite for a lot of math, physics...ect, so maybe the last line just means that integrals are elementary compared to a lot of stuff that requires knowledge about them?

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  • $\begingroup$ It isn't an integral $\endgroup$ – Xavier Stanton Aug 18 '18 at 4:34
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Is it

A star/asterisk *

Known by 2 names, argued by ways

Asterisk/star

One of them sounds irregular, sometimes that pays

Maybe asterisk sounds irregular?

My legs are like a human's, usually alike

The points of an asterisk are usually drawn the same way and same length

This is easy, known by a child's psyche

A child knows it as a star, but mathematicians use it for many purposes like multiplication, A* search, convolution etc

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It could be(partial one)

Pi

As,

Known by 2 names, argued by ways

It (the sounding word)is known by pi and it's homophone pie

One of them sounds irregular, sometimes that pays

Not sure on this part...

My legs are like a human's, usually alike

It is obvious, when represented with two vertical lines connected by a horizontal line at the top.

This is easy, known by a child's psyche

Child's psyche could be towards a sweet, usually( though an assumption!) - for which a possibility is pie

Name the mathematical concept

It is pi.

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