Study Guide: Decimal System
Tips
- When adding or subtracting decimal numbers, always line up the decimal points.
- When multiplying decimal numbers, count all the digits to the right of both decimal points, and move the decimal point in the answer that many places to the left. There is no need to line up the decimal points when multiplying.
- When dividing decimal numbers, always move the decimal point in the divisor to the right as many places as needed to make it a whole number. Then move the decimal point in the dividend the same number of places, adding zeroes if needed.
The Decimal Number System, also called Hindu-Arabic number system, is a positional numeral system that uses 10 as the base and requires 10 different symbols which are 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. It also uses a dot (the decimal point) to represent fractions. In this system, the numerals used to show a value take different place values depending upon position. In a base-10 system the number 543.21 represents the sum:
$$(5 \times 10^2) + (4 \times 10^1) + (3 \times 10^0) + (2 \times 10^{−1}) + (1 \times 10^{−2})$$
History
- Invented in India.
- Brought to Persia and Arabia around the 8th century CE.
- Brought to Europe in the 12th century CE.
- Brought to the Americas in the 15th century CE.
- Spread worldwide in the modern era.
The advantages the positional system are so numerous that the Hindu-Arabic numerals, and the base-10 system have been adopted almost everywhere. This might be the closest humanity has to a universal language.
Place Value
The place (or position) of a symbol determines its value (the actual numeric quantity).
In the below examples, the only thing that changes is the position of the symbol $1$, but the value of that symbol changes by multiples of $10$ from very large to very small.
\begin{array}{lrll}
\textbf{Name} & \textbf{Value} & \textbf{Prefix} & \textbf{Symbol} \\
trillion &1,000,000,000,000 & tera & T \\
billion &1,000,000,000 & giga & G \\
million &1,000,000 & mega & M \\
thousand &1,000 & kilo & k \\
hundred &100 & hecto & h \\
ten &10 & deka & da \\
unit &1 & & \\
tenth &0.1 & deci & d \\
hundredth &0.01 & centi & c \\
thousandth &0.001 & milli & m \\
millionth &0.000,001 & micro & µ \\
billionth &0.000,000,001 & nano & n \\
trillionth &0.000,000,000,001 & pico & p
\end{array}
Vocabulary |
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Decimal Number a fraction in the standard (base 10) numbering system, for example the fraction $\dfrac{37}{100}$ can be written $0.37$ in decimal notation. |
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Decimal Point A point (.) used to separate the whole and fractional parts of a decimal number, for example between the 3 (whole part) and 4 (fractional part) in $3.4$. |
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Dividend the number being divided, for example 12 is the dividend in $12 \div 3 = 4$. |
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Divisor the quantity by which another quantity, the dividend, is divided, for example 3 is the divisor in $12 \div 3 = 4$. |
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Quotient the solution to a division problem, for example 4 is the quotient in $12 \div 3 = 4$. |
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Decimal Number System the Decimal Number System, also called Hindu-Arabic number system, is a positional numeral system that uses 10 as the base and requires 10 different symbols which are 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. It also uses a dot (the decimal point) to represent fractions. In this system, the numerals used to show a value take different place values depending upon position. In a base-10 system the number 543.21 represents the sum:
\begin{array}{}
(5 \times 10^2) &+ &(4 \times 10^1) &+ &(3 \times 10^0) &+ &(2 \times 10^{−1}) &+ &(1 \times 10^{−2}) \\
500 &+ &40 &+ &3 &+ &0.2 &+ &0.001
\end{array}
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Place Value positional notation for numbers, allowing the use of the same symbols for different orders of magnitude, for example the "one's place" ($1$), "ten's place" ($10$), "hundred's place" ($100$), etc. |