What this chapter is about
This chapter is all about having fun with numbers while discovering interesting patterns and properties. You already know how to add, subtract, multiply and divide. Now you will look at numbers in new ways — finding patterns in digits, playing games with numbers, and noticing how numbers behave in surprising situations.
You will explore ideas like reversing digits, finding special number patterns, and understanding what makes some numbers interesting. This helps you think like a mathematician — someone who looks for patterns and asks "why does this happen?"
After this chapter, you should be able to spot patterns in numbers, play number puzzles confidently, and explain why certain number tricks work.
Key ideas
- Digit means each single symbol in a number. The number 435 has three digits: 4, 3 and 5.
- You can reverse a number by writing its digits in the opposite order. The reverse of 123 is 321.
- Adding a two-digit number to its reverse often gives interesting results. For example, 25 + 52 = 77.
- Palindrome means a number that reads the same forwards and backwards. Examples: 11, 121, 4554.
- Multiplying certain numbers by 9 creates patterns in the digits of the answer.
- The sum of digits of a number can tell you things about that number, like whether it is divisible by 3 or 9.
- Playing with place value (ones, tens, hundreds) helps explain why number tricks work.
- Number patterns often repeat, and spotting them helps you predict what comes next.
Formulas and facts to remember
- Reverse of a two-digit number: If a number is written as (10 × a) + b, its reverse is (10 × b) + a. Here a is the tens digit and b is the ones digit.
- Sum of a two-digit number and its reverse: The answer is always 11 × (a + b). That is why results like 77, 99, 121 appear often.
- Divisibility by 9: If the sum of all digits of a number is divisible by 9, then the number itself is divisible by 9.
- Divisibility by 3: If the sum of all digits is divisible by 3, the number is divisible by 3.
- Palindrome check: Read the number from left to right and right to left. If both readings are the same, it is a palindrome.
Worked examples
Example 1: Adding a number to its reverse
Take the number 36. Step 1: Find the reverse. The digits are 3 and 6. Reverse is 63. Step 2: Add the number and its reverse. 36 + 63 = 99. Notice that 99 = 11 × 9, and 9 is the sum of the digits (3 + 6).
Example 2: Checking for a palindrome
Is 1331 a palindrome? Step 1: Write the digits in order: 1, 3, 3, 1. Step 2: Write them in reverse order: 1, 3, 3, 1. Step 3: Both are the same, so 1331 is a palindrome.
Example 3: Using the sum of digits
Is 738 divisible by 9? Step 1: Add the digits. 7 + 3 + 8 = 18. Step 2: Check if 18 is divisible by 9. Yes, 18 ÷ 9 = 2. Step 3: So 738 is divisible by 9.
Common mistakes
- Confusing the reverse of a number with subtracting from it → Reversing means flipping the order of digits, not doing any arithmetic.
- Thinking all numbers become palindromes after one addition with their reverse → Some numbers need several steps of adding and reversing before they become palindromes.
- Forgetting that a single-digit number is always a palindrome → Numbers like 5 or 7 read the same both ways.
- Adding digits wrongly when checking divisibility → Take your time; add each digit carefully, one by one.
- Believing number patterns always continue forever → Some patterns break after a few steps; always check.
Quick revision
- The reverse of 47 is 74. Adding them gives 121, a palindrome.
- A palindrome reads the same forwards and backwards.
- Sum of digits trick: add all digits to check divisibility by 3 or 9.
- Two-digit number plus its reverse always equals 11 times the sum of its digits.
- Look for patterns — they make big calculations easier.