IGCSE Mathematics 0580 - Number

This document provides detailed notes for the Number topic in the Cambridge IGCSE Mathematics 0580 syllabus. It covers ordering, standard form, estimation, bounds, indices, and surds.

1. Ordering

1.1 Comparing Numbers

To compare numbers, consider the following:

• Magnitude: The size of the number.
• Place Value: The position of each digit determines its value (e.g., in 456, the 4 is in the hundreds place, worth 400).
• Comparing Decimals: Compare digits from left to right. If they are the same, move to the next digit.

1.2 Ordering Sets of Numbers

To order a set of numbers:

1. Identify the largest and smallest numbers.
2. Arrange the remaining numbers in ascending or descending order.
3. Place the largest and smallest numbers at the appropriate ends of the sequence.

Example

Order the following numbers from smallest to largest: 0.75, 0.5, 0.9, 0.25

Solution: 0.25, 0.5, 0.75, 0.9

2. Standard Form

2.1 Expressing Numbers in Standard Form

Standard form is a way of expressing very large or very small numbers in a concise way. It is written in the form $a \times 10^n$, where 1 ≤ |a| < 10 and n is an integer.

To convert a number to standard form:

1. Move the decimal point until there is only one non-zero digit to the left of the decimal point.
2. Count the number of places the decimal point was moved.
3. If the original number was greater than 1, the exponent of 10 will be positive.
4. If the original number was less than 1, the exponent of 10 will be negative.

Example

Convert 67890 to standard form.

Solution: $6.789 \times 10^4$

2.2 Calculations in Standard Form

When adding or subtracting numbers in standard form:

1. The exponents of 10 must be the same.
2. Add or subtract the coefficients.
3. Keep the same power of 10.

When multiplying numbers in standard form:

1. Multiply the coefficients.
2. Add the exponents of 10.

When dividing numbers in standard form:

1. Divide the coefficients.
2. Subtract the exponents of 10.

3. Estimation

3.1 Rounding Numbers

Rounding numbers involves finding a simpler number that is close to the original number.

Rules for rounding:

• If the digit in the next place is 0, 1, 2, 3, or 4, round down.
• If the digit in the next place is 5, 6, 7, 8, or 9, round up.

3.2 Using Estimation

Estimation is a way of finding an approximate answer to a calculation.

To estimate, round the numbers in the calculation to a convenient value and then perform the calculation.

Example

Estimate the value of $3.7 \times 4.2$

Solution: Round 3.7 to 4 and 4.2 to 4. Estimate $4 \times 4 = 16$

4. Bounds

4.1 Finding Bounds

Bounds are the minimum and maximum possible values for a given number.

To find bounds:

1. Identify the relevant information (e.g., minimum and maximum values, rounding).
2. Determine the smallest and largest possible values.

Example

A rectangular garden has a length of 8.3 m and a width of 5.7 m. Find the minimum and maximum possible areas of the garden.

Solution: Minimum area: $8.3 \times 5.7 = 47.31 \, m^2$. Maximum area: $8.4 \times 5.8 = 48.72 \, m^2$

5. Indices

5.1 Laws of Indices

The laws of indices are rules that govern how exponents work.

• $a^m \times a^n = a^{m+n}$
• $a^m \div a^n = a^{m-n}$
• $(a^m)^n = a^{m \times n}$
• $(a \times b)^n = a^n \times b^n$
• $(a \div b)^n = a^n \div b^n$
• $a^0 = 1$ (where $a \neq 0$)
• $a^{-n} = \frac{1}{a^n}$

Example

Simplify: $2^3 \times 2^4 \div 2^2$

Solution: $2^{3+4} \div 2^2 = 2^7 \div 2^2 = 2^{7-2} = 2^5 = 32$

6. Surds

6.1 Simplifying Surds

A surd is the square root of a number that is not a perfect square.

To simplify a surd:

1. Find the highest perfect square factor of the number under the radical.
2. Separate the surd into the square root of the perfect square factor and the square root of the remaining number.

Example

Simplify $\sqrt{36}$

Solution: $\sqrt{36} = \sqrt{4 \times 9} = \sqrt{4} \times \sqrt{9} = 2 \times 3 = 6$

Example

Simplify $\sqrt{12}$

Solution: $\sqrt{12} = \sqrt{4 \times 3} = \sqrt{4} \times \sqrt{3} = 2\sqrt{3}$

6.2 Simplifying Expressions with Surds

You can combine surds if the numbers under the radicals are the same.

Example

Simplify $3\sqrt{2} + 5\sqrt{2}$

Solution: $(3+5)\sqrt{2} = 8\sqrt{2}$