meant that the symbols could have a different
value depending on where they were positioned in the number – which freed people of the difficulty the Mesopotamians had faced. Today, the concept of place-value is taken for granted. But the
idea that the 8 in 80 is worth eight tens, and yet could be used, with the help of those friendly zeros, to also mean 800 or 8 million was revolutionary at the time. In fact, some European scholars
were deeply suspicious of this heathen method of calculating, despite its advantages.
In the
Book of Addition and Subtraction According to the Hindu Calculation
Al-Khwarizmi describes how to do arithmetic using these new numbers. His translators referred to him by the
Latinized name Algorism. Over time Al-Khwarizmi’s methods of calculation became known as algorithms
,
a word still in use today and which refers to a set of instructions to
perform a calculation – which is exactly what Al-Khwarizmi provided.
Transforming Mathematics
Al-Khwarizmi also wrote
The Compendious Book on Calculation by Transformations and Dividing
, which set out to show how to solve different types of quadratic equations (equations in which the unknown numbers are squared). ‘Transformations’ in Arabic is
Al-Jabr
, from which we derive (via Latin) the English term algebra . While Al-Khwarizmi himself did not replace unknown numbers with letters, he did pave the way for this to happen.
O MAR K HAYYÁM (1048–1131)
Persian scholar Omar Khayyám is best known for his
The Rubaiyat of Omar Khayy
á
m,
a selection of poems that were later translated into English in the
nineteenth century by the poet Edward Fitzgerald. Multi-talented, Khayyám spent a great proportion of his life as a court astronomer to a sultan, while also working as a scientist and
mathematician.
Khayyám’s mathematical works were far-reaching. He expanded on Al-Khwarizmi’s earlier work in algebra, and he was one of the first mathematicians to use the replacement of
unknown numbers with letters to make solving equations easier. He also devised techniques for solving cubic equations , where the unknown term has been cubed. Khayyám’s insight
enabled him to be one of the first people to connect geometry and algebra, which had until that point been separate disciplines.
Endless possibilities
Khayyám also investigated something now called the binomial theorem . This has many applications in mathematics, many of which involve rather tricky algebra. One
side product of binomial theorem is something called Pascal’s triangle , named after the seventeenth-century French mathematician Blaise Pascal, who borrowed the triangle from
Khayyám, who in turn borrowed it from the Chinese (see here ). Unlike the binomial theorem, Pascal’s triangle is simple to understand: the number in each cell of a triangle is made
by adding together the two numbers above it.
Pascal’s triangle is useful because each horizontal row shows us the binomial coefficients that the binomial theorem spits out. These can tell us how many
combinations of two different things it is possible to have.
For example, imagine you have planted a row of four flowerbulbs. It says on the packet that the flowers can be blue or pink, with an equal chance of having either.
There is one way for you to grow four blues:
BBBB
There are four ways for you to grow three blues and one pink:
BBBP
BBPB
BPBB
PBBB
There are six ways for you to end up with two of each:
BBPP
BPPB
PBBP
PPBB
BPBP
PBPB
There are four ways for you to have three pinks and one blue:
PPPB
PPBP
PBPP
BPPP
And one way for you to have four pinks:
PPPP
If you look across the fourth row of the triangle, it says 1, 4, 6, 4, 1, which corresponds to the number of ways worked out in the example above. Because there is an equal
chance of a flower being either pink or blue, you can also see that you’re most likely to end up with two of each colour because there are 6 out of 16 total ways this could
value depending on where they were positioned in the number – which freed people of the difficulty the Mesopotamians had faced. Today, the concept of place-value is taken for granted. But the
idea that the 8 in 80 is worth eight tens, and yet could be used, with the help of those friendly zeros, to also mean 800 or 8 million was revolutionary at the time. In fact, some European scholars
were deeply suspicious of this heathen method of calculating, despite its advantages.
In the
Book of Addition and Subtraction According to the Hindu Calculation
Al-Khwarizmi describes how to do arithmetic using these new numbers. His translators referred to him by the
Latinized name Algorism. Over time Al-Khwarizmi’s methods of calculation became known as algorithms
,
a word still in use today and which refers to a set of instructions to
perform a calculation – which is exactly what Al-Khwarizmi provided.
Transforming Mathematics
Al-Khwarizmi also wrote
The Compendious Book on Calculation by Transformations and Dividing
, which set out to show how to solve different types of quadratic equations (equations in which the unknown numbers are squared). ‘Transformations’ in Arabic is
Al-Jabr
, from which we derive (via Latin) the English term algebra . While Al-Khwarizmi himself did not replace unknown numbers with letters, he did pave the way for this to happen.
O MAR K HAYYÁM (1048–1131)
Persian scholar Omar Khayyám is best known for his
The Rubaiyat of Omar Khayy
á
m,
a selection of poems that were later translated into English in the
nineteenth century by the poet Edward Fitzgerald. Multi-talented, Khayyám spent a great proportion of his life as a court astronomer to a sultan, while also working as a scientist and
mathematician.
Khayyám’s mathematical works were far-reaching. He expanded on Al-Khwarizmi’s earlier work in algebra, and he was one of the first mathematicians to use the replacement of
unknown numbers with letters to make solving equations easier. He also devised techniques for solving cubic equations , where the unknown term has been cubed. Khayyám’s insight
enabled him to be one of the first people to connect geometry and algebra, which had until that point been separate disciplines.
Endless possibilities
Khayyám also investigated something now called the binomial theorem . This has many applications in mathematics, many of which involve rather tricky algebra. One
side product of binomial theorem is something called Pascal’s triangle , named after the seventeenth-century French mathematician Blaise Pascal, who borrowed the triangle from
Khayyám, who in turn borrowed it from the Chinese (see here ). Unlike the binomial theorem, Pascal’s triangle is simple to understand: the number in each cell of a triangle is made
by adding together the two numbers above it.
Pascal’s triangle is useful because each horizontal row shows us the binomial coefficients that the binomial theorem spits out. These can tell us how many
combinations of two different things it is possible to have.
For example, imagine you have planted a row of four flowerbulbs. It says on the packet that the flowers can be blue or pink, with an equal chance of having either.
There is one way for you to grow four blues:
BBBB
There are four ways for you to grow three blues and one pink:
BBBP
BBPB
BPBB
PBBB
There are six ways for you to end up with two of each:
BBPP
BPPB
PBBP
PPBB
BPBP
PBPB
There are four ways for you to have three pinks and one blue:
PPPB
PPBP
PBPP
BPPP
And one way for you to have four pinks:
PPPP
If you look across the fourth row of the triangle, it says 1, 4, 6, 4, 1, which corresponds to the number of ways worked out in the example above. Because there is an equal
chance of a flower being either pink or blue, you can also see that you’re most likely to end up with two of each colour because there are 6 out of 16 total ways this could
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