Irrational-Numbers Edexcel Higher

Irrational numbers

What is a rational number?

• A rational number is a number that can be written as a fraction in its simplest form

  • It must be possible to write in the form , where and are both integers

    • cannot be zero

  • This includes all terminating and recurring decimals

    • E.g. 0.15 (which is ) and 0.151515151515… (which is )

  • This also includes all integers

    • can be written as

    • can be written as or <img alt=”fraction numerator 3 over denominator negative 1 end fraction” data-mathml=”<math ><semantics><mfrac><mn>3</mn><mrow><mo>-</mo><mn>1</mn></mrow></mfrac><annotation encoding=”application/vnd.wiris.mtweb-params+json”>{“fontFamily”:”Times New Roman”,”fontSize”:”18″,”autoformat”:true,”toolbar”:”<toolbar ref=’general’><tab ref=’general’><removeItem ref=’setColor’/><removeItem ref=’bold’/><removeItem ref=’italic’/><removeItem ref=’autoItalic’/><removeItem ref=’setUnicode’/><removeItem ref=’mtext’ /><removeItem ref=’rtl’/><removeItem ref=’forceLigature’/><removeItem ref=’setFontFamily’ /><removeItem ref=’setFontSize’/></tab></toolbar>”}</annotation></semantics></math>” height=”47″ role=”math” src=”data:image/svg+xml;charset=utf8,%3Csvg%20xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F2000%2Fsvg%22%20xmlns%3Awrs%3D%22http%3A%2F%2Fwww.wiris.com%2Fxml%2Fmathml-extension%22%20height%3D%2247%22%20width%3D%2231%22%20wrs%3Abaseline%3D%2230%22%3E%3C!–MathML%3A%20%3Cmath%20xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmfrac%3E%3Cmn%3E3%3C%2Fmn%3E%3Cmrow%3E%3Cmo%3E-%3C%2Fmo%3E%3Cmn%3E1%3C%2Fmn%3E%3C%2Fmrow%3E%3C%2Fmfrac%3E%3C%2Fmath%3E–%3E%3Cdefs%3E%3Cstyle%20type%3D%22text%2Fcss%22%3E%40font-face%7Bfont-family%3A’math1da40657c9fece7e48d30af42d3’%3Bsrc%3Aurl(data%3Afont%2Ftruetype%3Bcharset%3Dutf-8%3Bbase64%2CAAEAAAAMAIAAAwBAT1MvMi7iBBMAAADMAAAATmNtYXDEvmKUAAABHAAAADRjdnQgDVUNBwAAAVAAAAA6Z2x5ZoPi2VsAAAGMAAAAcmhlYWQQC2qxAAACAAAAADZoaGVhCGsXSAAAAjgAAAAkaG10eE2rRkcAAAJcAAAACGxvY2EAHTwYAAACZAAAAAxtYXhwBT0FPgAAAnAAAAAgbmFtZaBxlY4AAAKQAAABn3Bvc3QB9wD6AAAEMAAAACBwcmVwa1uragAABFAAAAAUAAADSwGQAAUAAAQABAAAAAAABAAEAAAAAAAAAQEAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAACAgICAAAAAg1UADev96AAAD6ACWAAAAAAACAAEAAQAAABQAAwABAAAAFAAEACAAAAAEAAQAAQAAIhL%2F%2FwAAIhL%2F%2F93vAAEAAAAAAAABVAMsAIABAABWACoCWAIeAQ4BLAIsAFoBgAKAAKAA1ACAAAAAAAAAACsAVQCAAKsA1QEAASsABwAAAAIAVQAAAwADqwADAAcAADMRIRElIREhVQKr%2FasCAP4AA6v8VVUDAAABAIABVQLVAasAAwAwGAGwBBCxAAP2sAM8sQIH9bABPLEFA%2BYAsQAAExCxAAblsQABExCwATyxAwX1sAI8EyEVIYACVf2rAatWAAAAAQAAAAEAANV4zkFfDzz1AAMEAP%2F%2F%2F%2F%2FWOhNz%2F%2F%2F%2F%2F9Y6E3MAAP8gBIADqwAAAAoAAgABAAAAAAABAAAD6P9qAAAXcAAA%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%2BgAAAAAAAAAAAAAAAAAAAAAAAAAAuQcRAACNhRgAsgAAABUUE7EAAT8%3D)format(‘truetype’)%3Bfont-weight%3Anormal%3Bfont-style%3Anormal%3B%7D%3C%2Fstyle%3E%3C%2Fdefs%3E%3Cline%20stroke%3D%22%23000000%22%20stroke-linecap%3D%22square%22%20stroke-width%3D%221%22%20×1%3D%222.5%22%20×2%3D%2227.5%22%20y1%3D%2223.5%22%20y2%3D%2223.5%22%2F%3E%3Ctext%20font-family%3D%22Times%20New%20Roman%22%20font-size%3D%2218%22%20text-anchor%3D%22middle%22%20x%3D%2215.5%22%20y%3D%2216%22%3E3%3C%2Ftext%3E%3Ctext%20font-family%3D%22math1da40657c9fece7e48d30af42d3%22%20font-size%3D%2216%22%20text-anchor%3D%22middle%22%20x%3D%2210.5%22%20y%3D%2241%22%3E%26%23×2212%3B%3C%2Ftext%3E%3Ctext%20font-family%3D%22Times%20New%20Roman%22%20font-size%3D%2218%22%20tex