98 lines
2.1 KiB
Markdown
98 lines
2.1 KiB
Markdown
---
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tags:
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- Mathematics
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- Prealgebra
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---
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# The set of whole numbers
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We recall the set of whole numbers:
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$$ \mathbb{W} = {0, 1, 2, 3, ...} $$
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# The properties of $\mathbb{W}$
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>
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> In mathematics, a **property** is any characteristic that applies to a given set.
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## The commutative property
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### Addition
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When **adding** whole numbers, the placement of the addends does not affect the sum.
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Let **a**, **b** represent whole numbers, then:
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$$ a + b = b + a $$
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### Multiplication
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When **multiplying** whole numbers the placement of the [multiplicands](https://www.notion.so/Symbols-and-formal-conventions-80aeaf1872f94a0d97a2e8d07e3855bd) does not affect the [product](https://www.notion.so/Symbols-and-formal-conventions-80aeaf1872f94a0d97a2e8d07e3855bd).
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Let **a, b** represent whole numbers, then:
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$$ a \cdot b = b \cdot a $$
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### Subtraction
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**Subtraction** is not commutative, viz:
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$$ a - b \neq b - a $$
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### Division
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Division is not commutative, viz:
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$$ a \div b \neq b \div a $$
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## The associative property
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### Addition
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When grouping symbols (parentheses, brackets, braces) are used with addition, the particular placement of the grouping symbols relative to each of the addends does not change the sum.
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Let **a**, **b, c** represent whole numbers, then:
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$$ (a + b) + c = a + (b + c) $$
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### Multiplication
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Let **a, b, c** represent whole numbers, then:
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$$ a \cdot (b \cdot c) = (a \cdot b) \cdot c $$
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### Subtraction
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Subtraction is not associative, viz:
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$$ (a - b) - c \neq a - (b - c) $$
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### Division
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Division is not associative
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$$ (a \div b) \div c \neq a \div (b \div c) $$
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## The property of additive identity
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If **a** is any whole number, then:
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$$ a + 0 = a $$
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We therefore call zero the additive identity: whenever we add zero to a whole number, the sum is equal to the whole number itself.
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## The property of multiplicative identity
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If **a** is any whole number, then:
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$$ (a \cdot 1 = a) = (1 \cdot a = a) $$
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## Multiplication by zero
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If **a** is any whole number, then:
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$$ (a \cdot 0 = 0) = (0 \cdot a = 0) $$
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## Division by zero
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Division by zero is **undefined** but zero divided is zero.
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