Since pi is transcendental, pi2 is also transcendental. So pi is the square root of the transcendental number pi2.
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A transcendental number is one which is not algebraic. An algebraic number is one which is a root of a non-zero polynomial with rational coefficients.
Pi is both an irrational number and a transcendental number.
1/pi = 0.3183098861837906715377675267450287240689192914809128974953346881177935952684530701802276055325061719121456854535159160737858236922291573057559348214633996784584799338748181551461554927938506153774347857924347953233867247804834472580236647602284453995114318809237801738053479122409788218738756881710574461998928868004973446954789192217966461935661498123339729256093988973043757631495731339284820779917482786972199677361983999248857511703423577168622350375343210930950739760194789207295186675361186049889932706106543135510064406495556327943320458934962391963316812120336060719962678239749976655733088705595101400324813551287776991426217602443987522953627555294757812661360929159569635226248546281399215500490005955197141781138055935702630504200326354920418496232124811229124062929681784969183828704231508151124017430532136044343182815149491654451954925707997503106587816279635448187165095941466574380813999518153154156986940787179656174346851280733790233250914118866552625373000522454359423064225199009...It is an irrational number (moreover, it is a transcendental number)
An irrational number is a number that cannot be represented as a fraction involving two integers. A transcendental number is a number that cannot be repesented as a polynomial with rational coefficients. Two notable transcendental numbers are pi and e.
Real, complex, quaternion, etc. But more specifically it's a transcendental real number, not being the root of an integer polynomial (and certainly not the ratio of two integers).