Mathematics
Surds
Surds are irrational numbers that cannot be expressed as a ratio of two integers. They are typically represented as square roots of non-perfect squares or higher roots of numbers. Surds are commonly used in geometry, trigonometry, and other branches of mathematics.
Written by Perlego with AI-assistance
Related key terms
1 of 5
3 Key excerpts on "Surds"
eBook - PDFMathematics NQF2 SB
TVET FIRST
- M Van Rensburg, I Mapaling A Thorne(Authors)
- 2017(Publication Date)
- Macmillan(Publisher)
l Irrational numbers such as √ __ 3 and 3 √ ___ 10 are called Surds. Surd laws 1. √ __ a × √ __ a = a 2. √ _____ a × b = √ __ a × √ __ b 3. √ __ a __ b = √ __ a ____ √ __ b 4. m √ __ a + m √ __ b = m ( √ __ a + √ __ b ) Warning: Be careful not to make the following mistake: √ _____ a + b ≠ √ __ a + √ __ b For example, √ ___ 16 + √ __ 9 ≠ √ ___ 25 31 Module 2 Example 2.21 Simplify by applying the surd laws: √ ______ 5 __ 2 + 9 ___ 16 Solution: √ ______ 5 __ 2 + 9 ___ 16 = √ _________ 5 __ 2 × 8 __ 8 + 9 ___ 16 = √ _____ 40 + 9 ______ 16 [Use lowest common denominator and add the fractions] = √ ___ 49 ___ 16 = √ ___ 49 ____ √ ___ 16 [Apply the surd law √ __ a __ b = √ __ a ___ √ __ b ] = 7 __ 4 = 1 3 __ 4 Example 2.22 Simplify by applying the surd laws: 3 √ __ 7 × 2 √ __ 7 Solution: 3 √ __ 7 × 2 √ __ 7 = 3 × 2 × √ __ 7 × √ __ 7 [Multiply the coefficients and the Surds] = 3 × 2 × 7 [Simplify by applying the law √ __ a × √ __ a = a ] = 42 Example 2.23 Simplify by applying the surd laws: 4 √ __ 3 + 6 √ __ 3 − 2 √ __ 3 Solution: 4 √ __ 3 + 6 √ __ 3 − 2 √ __ 3 = 8 √ __ 3 [Add and subtract like terms (4 x + 6 x − 2 x )] Example 2.24 Simplify by using the surd laws: √ ___ 45 _____ √ __ 5 Solution: √ ___ 45 _____ √ __ 5 = √ _____ 9 × 5 _______ √ __ 5 [Write the number as a product of a perfect square and a prime number] = √ __ 9 √ __ 5 ______ √ __ 5 [Apply the surd law √ __ ab = √ __ a × √ __ b ] = 3 √ __ 5 ____ √ __ 5 [Simplify] = 3
eBook - PDF- Charles S. Peirce, Carolyn Eisele(Authors)
- 2016(Publication Date)
- De Gruyter(Publisher)
Illustration 36. Namely, except where the sequence passes the third value is greater than the second, and the second greater than the first. But where £ is between two of the three, that value [that] precedes is the greatest of the whole sequence and that which succeeds is the least of the whole sequence. The sequence formed by the rule that the object in the Nth place, 4 where Ν is any ordinal number is -——, progresses increasingly, its values being 1, 1.3333.., 2, 4, oo, -4 , -2 , -1.333..., -1 , -0.8, etc. Definition 87. The complete system of real quantity, rational and surd, or irrational, consists of all rational quantities in their order, together Surds 147 with the limits of all series of such quantities which progress increasingly, without completing the circuit, and whose limits are not rational. Gloss 19. The word surd is from the latin surdus, used in the same sense, but literally meaning deaf or idiotic. It is, as applied to quantity, a mistranslation of the Greek word άλογος, meaning without λόγος, where λόγος might signify either reason, or word, or ratio. A rational quantity is called in Greek βητή, that is, expressible. The idea probably intended to be embodied in the two Greek words probably was that rational numbers are capable of exact statement, irrational not. Yet it is possible that what may have been intended was the clearer idea that rational numbers are ratios of whole numbers, the irrational not. Though Euclid uses the two words in somewhat modified senses, it is probable that with early Greek mathematicians φητή and άλογος were precisely our rational and surd.
eBook - PDFTales of Impossibility
The 2000-Year Quest to Solve the Mathematical Problems of Antiquity
- David S. Richeson(Author)
- 2019(Publication Date)
- Princeton University Press(Publisher)
He wrote, “There are no absurd, irrational, irregular, inexplicable, or surd numbers.” 5 Nevertheless, even in the seventeenth century, mathematicians like Pascal and Barrow asserted that numbers such as √ 2 can be under-stood only geometrically— √ 2 is merely a collection of symbols to represent a magnitude, such as the diagonal of a square with unit-length sides. This was also the view Newton espoused in Universal Arithmetick . He presented numbers as abstract quantities, yet tied them to magnitudes: 6 By a “number” we understand not so much a multitude of units as the abstract ratio of any quantity to another quantity of the same kind which is considered to be unity. It is threefold: integral, fractional, and surd. An integer is measured by unity, a fraction by a multiple part of unity, while a surd is incommensurable with unity. IRRATIONAL AND TRANSCENDENTAL NUMBERS 351 A correct understanding of the real number line was a necessary step in making calculus fully rigorous. Newton, Leibniz, and their seventeenth-century contemporaries deduced the basic theorems of calculus, but the proofs were built upon a shaky foundation of unde-fined concepts such as “infinitesimals.” In his 1734 book The Analyst , Bishop George Berkeley (1685–1753) famously pointed out the lack of rigor in Newton’s calculus: 7 What are these . . . evanescent increments? They are neither finite quantities nor quantities infinitely small, nor yet nothing. May we not call them the ghosts of departed quantities? Mathematicians in the eighteenth century—most notably Euler— pushed calculus much further ahead. But there were still unresolved questions of the rigorous underpinnings. It was in the nineteenth cen-tury that mathematicians tied up the loose ends and came up with the definitions and theorems that can be found in today’s textbooks.
Index pages curate the most relevant extracts from our library of academic textbooks. They’ve been created using an in-house natural language model (NLM), each adding context and meaning to key research topics.
